multicomponent mass transfer in polymer-coated chemical sensors

Sensors and Actuators B 99 (2004) 273–280

Multicomponent mass transfer in polymer-coated chemical sensorsCynthia Phillips, Andrei G. Fedorov∗

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

Received 6 August 2003; accepted 20 November 2003

Abstract

A methodology for treating mass transfer in chemical sensors involving multiple interacting chemical species is presented based onthe Maxwell–Stefan theory for multicomponent diffusion. The importance of the so-called cross-diffusion terms, which are neglectedin commonly used pseudo-binary Fickian diffusion approximation for modeling transport of multiple interacting chemical species isinvestigated. Quantitative limits are also developed in terms of the sensor characteristics to define when non-binary diffusion effectsbecome significant and must be accounted for during data interpretation.© 2003 Elsevier B.V. All rights reserved.

Keywords:Chemical sensor; Infrared sensor; Multicomponent diffusion; Maxwell–Stefan

1. Introduction and current state-of-the-art

Independent of the specific transduction mechanism, allchemical sensors generate signals upon molecular interac-tion of their selective chemical recognition interface withthe desired target analyte. To increase the detection thresh-old, the analyte(s) of interest is usually preconcentrated byvarious techniques, utilizing hydrophobic polymer layers aspreferred implementation in many sensing applications op-erated in aqueous environments. In depth understanding ofanalyte enrichment in the polymers due to bulk solvationeffects has been a major issue of physical and analyticalchemistry over several decades prompted. In the chemicalsensing community, major efforts have been invested intodiffusion modeling and optimization of polymer-based en-richment layers[1,2], whereas optimization of the sensorflow cell geometry and mass transport in the flow cell wasalmost completely untouched[3]. This is a surprising fact, inparticular since data acquisition for chemical sensors can bea complex process; thus, appropriate simulation models ofchemical sensors can serve as highly valuable tool for sensordesign and data interpretation. During the past decade, onlya few relatively simplistic models have been developed fordifferent types of chemical sensors, ranging from fiber-opticchemical sensors[4–6] to dopamine biosensors[7] to ther-moelectric gas sensors[8]. In addition, the pseudo-binary

∗ Corresponding author. Tel.:+1-404-385-1356; fax:+1-404-894-8496.E-mail address:[email protected] (A.G. Fedorov).

diffusion approximation[5] is commonly used for treatingmixtures containing multiple analytes of interest, and thismay be questionable under certain conditions encountered inpractice. In particular, rigorous treatment of multiple analytetransport is important to a wide range of sensing applica-tions from pollution monitoring in subsea environment[9] tosimultaneous determination of phenolic compounds by mul-ticomponent biosensors[10] to multicomponent analysis ofhuman blood by fiber-optic evanescent-wave spectroscopy[11].

A schematic of the baseline configuration for the planarinfrared evanescent wave (IR-EW) chemical sensor flow cell[12,13] is shown inFig. 1. Aqueous solutions of analyte(s)of interest flow over a polymer coated attenuated total reflec-tion (ATR) crystal acting as a mid-infrared waveguide[14].Induced by bulk solvation effects, the analyte molecules par-tition from the aqueous phase into the hydrophobic poly-mer, and the polymer layer becomes enriched with analytemolecules as the flow continues to pass over the membrane.IR radiation coupled into the ATR waveguide incident at thecrystal/polymer interface at angles larger than the criticalangle creates an evanescent field guided along that interface.The polymer thickness is designed such that the evanescentfield, which decays exponentially into the adjacent medium,attenuates completely before reaching the aqueous solution,thus effectively eliminating interference from strong IR ab-sorption bands of water in the mid-infrared (MIR) part ofthe spectrum (3–20�m). Analyte molecules enriched in thepolymer absorb energy of the evanescently guided field atwavelengths in resonance with molecule specific vibrational

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274 C. Phillips, A.G. Fedorov / Sensors and Actuators B 99 (2004) 273–280

Fig. 1. Schematic of (a) the baseline configuration for planar IR-EW chemical sensor, (b) IR radiation wavefront propagation within the ATR and theevanescent wave formation.

transitions. In particular, for organic molecules the MIRspectrum serves as a chemical fingerprint allowing identifi-cation and quantification of individual components and mul-tiple analytes[15–17]. Signal detection and processing areachieved by Fourier transform infrared (FT-IR) analysis.

The “minor” defects of the design can have a profoundinfluence on the accuracy, precision, response time, and se-lectivity of the polymer coated, infrared evanescent wave(IR-EW) chemical sensor. Indeed, the choice of the optimalgeometry of the sensor flow cell, the thickness and transportproperties of the polymer layer, the geometry and opticalproperties of the ATR crystal are often not obvious but re-quire a use of the experimentally verified theoretical modelsof the analyte mixture transport in the flow cell, the analytepartitioning into and diffusion within the polymer coupledwith the analysis of the IR light propagation and attenua-tion (by both internally reflected and evanescent waves) inthe ATR crystal. No adequate integrated model for all theseprocesses exists at this moment[3], thereby making designand application of chemical sensors still very much an art.

Our recent work on the model-based optimal design ofthe sensor flow cell[3] is the first step in development suchan integrated sensor model, and it serves as a foundation forthis work. We developed the state-of-the-art model of theIR-EW chemical sensor and performed the detailed simu-lations to obtain fundamental understanding of the variousmass and IR radiation transport mechanisms governing sen-sor dynamics and to identify the major parameters affectingthe sensor performance (i.e., the sensor sensitivity, selectiv-ity, and response time). Using this fundamental insight intosensor behavior, we then developed a simple design equationthat can be used by the sensor developers to find the opti-mal dimensions and properties of the flow cell and the poly-mer layer for the baseline planar configuration of IR-EWchemical sensors. Finally, armed with this knowledge, weproposed and demonstrated the alternative sensor configura-tions which result in superior sensor performance (sensitiv-ity, selectivity, and response time) relative to the commonlyused (baseline) design of the IR-EW sensor shown inFig. 1.

2. Motivation for the present study

Ideal chemical sensor systems should simultaneously pro-vide high sensitivity and selectivity and should be able todetect and quantify individual analytes of interest amongstmany analytes in a complex sample matrix with similarchemical fingerprints. For example, IR-EW optical sensorsmust be able to differentiate between analytes with similarinfrared absorption bands. The presence of more than oneanalyte in the solution may significantly affect mass trans-fer and, in turn, selectivity and response time of the sensor.Multicomponent diffusion may result in behavior that can-not be explained by Fick’s law for binary (non-interacting)diffusion: for example, diffusion in the absence of a driv-ing force due to concentration gradients or absence of dif-fusion in the presence of a driving force or even diffusioncounter to the driving force induced by the concentrationgradient.

Sound analysis of multicomponent mass transfer inchemical sensor systems is important, as it would allowcalibration of the sensor such that analytes with similarchemical fingerprints could be uniquely identified and mea-sured. In the case of infrared spectroscopy, for example,the amplitude (or peak area) of the light energy absorp-tion peaks obtained by FT-IR analysis is correlated to theconcentration via Beer–Lambert’s law. At trace concentra-tions, small peaks that are distinct to various compounds ofinterest, have small signal to noise ratios, forcing the useof stronger peaks (absorption bands) for analyte identifica-tion. Unfortunately, many of the substances being detectedhave similar or overlapping strong absorption peaks, thusmaking differentiation critical. Accurate identification andconcentration measurement of analytes with similar ab-sorption bands require additional information that can beused to differentiate molecule-specific signatures from acomplex FT-IR measurement. A mathematical model ofmulticomponent mass transfer coupled with FT-IR data canprovide facile calibration for sensor equipment, allowingmapping of possible analyte pairs’ concentrations and the

C. Phillips, A.G. Fedorov / Sensors and Actuators B 99 (2004) 273–280 275

corresponding FT-IR signal, so that all analytes present canbe properly identified and measured.

3. Model formulation for multicomponent diffusion

In the case of binary diffusion, Fick’s law of diffusionadequately explains mass transport. However, Fick’s lawis only valid when applied to the following three cases:(i) binary mixtures, (ii) diffusion of dilute species in amulticomponent mixture, or (iii) in the absence of elec-trostatic or centrifugal forces[18,19]. The most widelyused approach to describe multicomponent diffusion isthe Maxwell–Stefan (M–S) approach, pioneered in thelate 19th century[19]. This approach was developed inorder to describe multicomponent diffusion in gases byusing molecular dynamics and was later extended to mul-ticomponent diffusion in non-ideal fluids (including liq-uids). M–S diffusivities are physically similar to a dragcoefficient.

Although the dilute approximation may be applied to ana-lyte concentrations in both the aqueous flow and the polymermembrane of the sensor, and Fick’s law can be used accord-ing to guideline (ii), the diffusion coefficient for the Fick’slaw must be obtained experimentally due to lack of existingmodels for its prediction, making this approach a difficulttool for modeling. Unlike Fickian diffusivities for dilutediffusion in a multicomponent system, the M–S diffusivitiescan be predicted from more readily available data. Currently,very few multicomponent diffusivities are reported for liq-uid solutions[18], which are necessary for application ofFick’s law to a multicomponent system. Fickian diffusioncoefficients in liquid solutions are concentration-dependent;thus, reporting of liquid diffusion coefficients is evenmore complicated. Hence, the Maxwell–Stefan approachoffers a number of potential benefits over the Fickiandescription of diffusion, because the diffusion coeffi-cients can be described in terms of known quantities,the Maxwell–Stefan binary diffusion coefficients, and themixture composition.

3.1. Maxwell–Stefan approach to multicomponentdiffusion in gases and liquids

The Maxwell–Stefan approach was developed by consid-ering perfectly elastic molecular collisions and making amomentum balance on a molecule of speciesi. Then by ap-plying the definition of molar flux, an expression for thedriving force can be obtained[19]:

dCi ≡ − xi

RT∇Tµi (1)

where µi is the chemical potential of speciesi, R theuniversal gas constant,T the temperature,xi the molefraction of speciesi, and ∇T is the gradient operator atconstant temperature. The driving force given inEq. (1)

is the driving force due to concentration gradients; diffu-sion among the species can also result due to temperatureand pressure gradients, and external forces. The drivingforce for thermal diffusion resembles the driving forcefor ordinary multicomponent diffusion, but it is causedby the temperature gradient and is defined as follows[19]:

dTi ≡ − xi

RT∇µT

i (2)

The driving force due to thermal diffusion can be expressedin terms of local species “thermal” velocitiesuT

i and thethermal diffusion coefficientDT

i as follows:

dTi ≡ − xi

RT∇µT

i = −n∑

j=1

j �=i

xixj(uTi − uT

j )

–Dij,

uTi = ui +

(DT

i

ρi

)∇T

T, i = 1,2, . . . , n (3)

Thermal diffusion, the Soret effect, becomes very impor-tant in chemical vapor deposition processes where signifi-cant temperature gradients exist[20] and is not expected tobe significant under essentially isothermal conditions char-acteristic to most chemical sensors.

The terms representing the pressure diffusion and theforced (due to external forces) diffusion can be developedsimultaneously. A slightly different form of the drivingforce equation is more convenient for this development[19]:

dP–Fi = − xi

RT∇Tp + xi

RTF̃i (4)

wherep is the pressure and̃Fi is the external body forceacting on speciesi. By combiningEqs. (1)–(4), the totaldriving force due to all effects is given by:

di ≡ dCi + dT

i + dP–Fi

= − xi

RT∇Tµi − xi

RT∇µT

i − xi

RT∇Tp + xi

RTF̃i (5)

In most diffusion processes, mechanical equilibrium isreached well before thermal equilibrium, and this assump-tion can be used to modifyEq. (5) [19]:

di = − xi

RT∇Tµi − xi

RT∇µT

i − xi

RT∇p

+ ωi

RT

(F̃i −

n∑k=1

ωkF̃k

)(6)

hereωi = ρxi is the mass fraction of speciesi and the bodyforcesF̃k are given per unit mass of the speciesi.

In addition to mass conservation, the momentum con-servation equation also needs to be modified slightly fora multicomponent mixture by expressing the externalforce term as a sum of the forces acting on each species


present:

∂v∂t

+ v∇v = −1

ρ∇P + υ∇2v +

n∑i=1

ωiF̃i (7)

3.2. Diffusion in solids

Multicomponent diffusion of analytes in solids must beaddressed differently than multicomponent diffusion in liq-uids and gases. Diffusion in solids is much slower than thatin liquids due to denser packing of the matrix molecules.For this physical reason, diffusion in solids of severalspecies is usually treated as binary diffusion between thespecies and matrix only. Extensive experiments proved thevalidity of this approximate approach for most polymers[21].

4. Maxwell–Stefan formulation for the polymer-coatedchemical sensor

The Maxwell–Stefan formulation can be used to expressthe governing equations for multicomponent mass transferin the flow cell and polymer layer of the chemical sensor.First, the relevant contributions to diffusion must be de-termined. In the chemical sensor, the external body forcesare small and can be neglected. No significant temperaturegradients are expected to occur, therefore, the thermal dif-fusion Soret effect can also be neglected. Thus, the diffu-sion driving force in the chemical sensor is comprised ofcontributions from the composition and pressure gradientsonly:

di = − xi

RT∇Tµi − xi

RT∇p = −

n−1∑j=1

Γij∇xj − xi

RT∇p (8)

The analytes in the sensor are present in trace amounts inboth the aqueous phase and polymer layer. Therefore, thedilute approximation or assumption of ideality can be in-voked that results in the activity coefficient (γ) for chemicalpotential equal to unity and the expression for the coeffi-cient of the composition gradient,Γij , in Eq. (8) becomessimply:

Γij = δij + xi∂ ln γ

∂xj

= δij + xi∂ ln 1

∂xj= [I], i, j = 1,2, . . . , n − 1 (9)

As a result, the new simpler expression for the driving forcecan be obtained[19]:

di = −∇xi − xi

RT∇p =

n∑j=1

j �=i

xjJi − xiJj

CtDij= 1

Ct

BijJj (10)

Bii = xi

–Din+

n∑k=1k �=1

xk

–Dikand

Bij(i�=j) = −xi

(1

–Dij− 1

–Din

), i, j = 1,2, . . . , n − 1

(11)

HereBa = {Bij } is called a matrix of Maxwell–Stefan (M–S)diffusion coefficients,Ji is the local molar flux of the speciesi, and–Dij are the so-called Maxwell–Stefan (M–S) binarydiffusivities of the species “i” in respect to the species “j” .Eq. (10)can be conveniently recast into a form that resem-bles the Fick’s law for binary diffusion:

Ji = −CtBij−1∇xj − CtBij

−1 xj

RT∇P (12)

In turn, the species mass conservation equation becomes:

∂Ci

∂t+ v · ∇Ci = ∇

(Bij

−1∇Cj + Bij−1Cj

RT∇P

)(13)

As an example of the application of the above theoreti-cal developments, below we present the governing speciesmass conservation equations for a ternary mixture in the sen-sor (i.e. one solvent and two solutes). Since no interactionof species in the polymer solid was assumed, the transportequations in the polymer layer are those of binary diffusion,where the transport of each species is only dependent on itsown concentration gradient.

4.1. Species mass conservation

Aqueous phase

∂Ca1

∂t+ u

∂Ca1

∂x+ v

∂Ca1

∂y

= ∇[B−1

11 ∇Ca1 + B−112 ∇Ca2 + B−1

11∇P

RTCa1

+B−112

∇P

RTCa2

](14)

∂Ca2

∂t+ u

∂Ca2

∂x+ v

∂Ca2

∂y

= ∇[B−1

21 ∇Ca1 + B−122 ∇Ca2 + B−1

21∇P

RTCa1

+B−122

∇P

RTCa2

](15)

Polymer

∂Cp1

∂t= Dp1∇2Cp1 (16)

∂Cp2

∂t= Dp2∇2Cp2 (17)

whereCai is the concentration of speciesi in the aqueousphase,Cpi is the concentration of speciesi in the polymer


phase, andB−1ij are elements of the inverted matrix of M–S

coefficient (Eq. (11)) given by:

B−1a =

Ca2/–D23 + Ca1/–D21 + Ca3/–D23

Ra

Ca1(1/–D12 + 1/–D13)

Ra

Ca2(1/–D21 + 1/–D23)

Ra

Ca1/–D13 + Ca2/–D12 + Ca3/–D13

Ra

, Ra = det(Ba) (18)

The boundary conditions are similar to those in the singlecomponent case for each component; except the interfaceboundary condition becomes slightly more complicated.

4.2. Boundary conditions

Ja1|int = Jp1|int and Ja2|int = Jp2|int (interface)

(19)

or[−B−1

11 ∇Ca1 − B−112 ∇Ca2 − B−1

11∇P

RTCa1 − B−1

12∇P

RTCa2

]= −Dp1∇Cp1 (20)

[−B−1

21 ∇Ca1 − B−122 ∇Ca2 − B−1

21∇P

RTCa1 − B−1

22∇P

RTCa2

]= −Dp2∇Cp2 (21)

K1 = Cp1

Ca1and K2 = Cp2

Ca2(interface) (22)

whereKi is the partition coefficient for speciesi.

4.3. Maxwell–Stefan diffusion coefficients

One of the benefits of the Maxwell–Stefan approach is theability to estimate the multicomponent M–S diffusivities orB−1-matrix elements. These diffusion coefficients can be de-termined from the Maxwell–Stefan binary diffusivities–Dij

of the species “i” in respect to the species “j”. Most esti-mation equations are based on the Vignes equation and thedefinition of the infinite dilution diffusivities–D0

ij , which areavailable for most liquids from experiments[22]:

–Dij ,xi→1 = –D0ji , –Dij ,xj→1 = –D0

ij (23)

An improvement on the Vignes equation prediction definesthe M–S diffusivities as a function of the binary infinitedilution diffusivities and the mole fractions[23]:

–Dij = (–D0ij )

(1+xj−xi)/2(–D0ji )

(1+xi−xj)/2 (24)

In order to use this result, the infinite dilution bi-nary diffusivities are necessary. For liquid mixtures, theStokes–Einstein equation yields a theoretical prediction forthe infinite dilution diffusion coefficients[24]. The use ofthe Stokes–Einstein equation for prediction of infinite di-lution coefficients is limited to solutions where the solvent

molecules are much larger than those of the solute; how-ever, many semi-empirical formulas have been developed

based on this equation. Wilke and Chang have developedthe most commonly used correlation[25]. For aqueous so-lutions, there are correlations reported to yield reasonablepredictions for the infinite dilution diffusivity such as theHayduk–Laudie correlation[26,27] given below:

–D0ij = 13.26× 10−5µ−1.14

j V−0.589i (25)

HereVi is the molar volume of speciesi, µ is the viscos-ity of the solvent (speciesj), and T is the temperature inKelvin. The values for viscosity and molar volume are ob-tained from published experimental values. When experi-mental data is unavailable, the molar volumes can be com-puted using the additive method based on chemical compo-sition [18]. It should be noted that the infinite dilution dif-fusivities are not symmetric, whereas the M–S diffusivitiesare. For a ternary system, for example, there would be sixinfinite dilution diffusion coefficients and three M–S diffu-sion coefficients.

5. Results and discussion

The goal of the multicomponent diffusion model is todetermine the threshold for importance of cross-term diffu-sion effects when the binary diffusion approximation failsto correctly predict the multiple analyte transport. The mul-ticomponent effects will become significant when the diffu-sive flux term of speciesj (the cross-term) in the transportequation for speciesi becomes significant in comparison tothe diffusive flux term of speciesi. To this end, the relativestrength of the analyte concentration gradients must be ad-dressed followed by the analysis of the relative strength ofthe pressure diffusion effects. Without the loss of general-ity, the ternary mixture (i.e., two analytes and the solvent)is considered.

5.1. Relative strength of the cross-diffusion terms

To provide a suitable reference for evaluation of mul-ticomponent effects due to concentration induced massdiffusion, the concentration gradients for both analytes isassumed to be of similar magnitude. Therefore, the relativestrength of the primary diffusion of main species “i” tosecondary diffusion of the auxiliary species “j” will dependon the M–S coefficients rather than the concentration gra-dients, essentially, on the relative strength ofB−1

11 versusB−1

12 , or equallyB−121 versusB−1

22 . From the inverted matrixof coefficients (Eq. (18)), the cross-termsB−1

ij depend on


Table 1Chemical properties of TeCE and DCB

Viscosity (mPa s) Molar volume Molecular weight

TeCE 0.844 70 165DCB 1.324 112 147

the analyte concentrations as well as the M–S analyte dif-fusivities in the aqueous phase. For most chemical species,the aqueous binary diffusion coefficients have been exper-imentally determined to be of the order of 10−5 cm2/s [18].As the concentrations of the analytes approach zero, thecross-terms will as well, and the diffusion of the analyteswill become that of binary diffusion. Therefore, at traceconcentration levels of interest to sensor applications, thecross-terms are expected to be very small.

To verify this simple scaling analysis, two analytes thatappear frequently as pollutants were chosen, specificallytetrachloroethylene (TeCE) and dichlorobenzene (DCB),whose chemical properties are shown inTable 1. The an-alyte concentrations in the aqueous stream are consideredto be on the order of 5–500 ppm (mg/l). We used theHayduk–Laudie correlation to determine the infinite dilu-tion diffusion coefficients, which were then used to computethe M–S diffusivities according toEq. (24). This choiceallowed the computation of the M–S diffusivities, by firstobtaining the values of viscosity from reported experimen-tal values[18], and by using the additive method to obtainthe molar volumes[18]. The M–S diffusivities,–D23 and–D13, were found to be on the order of 10−5 cm2/s over thegiven range of concentrations, which corresponds well tothe experimentally obtained values. Also, the molar fractionof the solvent (denoted as species 3 in the analysis) is somuch larger than the molar fraction of the analytes in theternary mixture that it can be assumed to be unity to sim-plify the analysis. From the coefficient matrix (B−1

a ), theterms to be compared can be simplified by canceling-liketerms, the resulting comparison matrix is as follows:

B−1a =

Ca2/–D23 + Ca1/–D21 + Ca3/23

Ra

Ca1(1/–D12 + 1/–D13)

Ra

Ca2(1/–D21 + 1/–D23)

Ra

Ca1/–D13 + Ca2/–D12 + Ca3/–D13

Ra

→[Ca2/–D23 + Ca3/–D23 Ca1/–D13

Ca2/–D23 Ca1/–D13 + Ca3/–D13

](26)

From the inverse coefficient matrix above andEq. (14),it follows that the strength of the cross-term (∼ B−1

12 /B−111 )

depends on the M–S diffusivities of both analytes in the sol-vent as well as analyte concentrations. If concentrations ofboth analytes in the solvent are approximately equal, the ra-tio –D23/–D13 would need to be at least 10 for the cross-termof the second (auxiliary) analyte to exhibit any influence onthe transport of the first (primary) analyte. When the ratioof M–S diffusivities reaches 100, the primary diffusion term

and the cross-term due to secondary species diffusion be-come equivalent in magnitude, depending on the concentra-tion of the analytes. This estimate was made assuming thehighest expected concentrations. Many analyte diffusivitieswere measured in the aqueous phase, and all reported to bein the range of 10−5 cm2/s [18]. In order to assess the ex-pected ratio of diffusivities–D23/–D13, Maxwell–Stefan dif-fusivities were computed for a number of analytes using theHayduk–Laudie correlation to compute the infinite diffusioncoefficients. All the M–S diffusivities were on the order of10−5 cm2/s. These results signify that the possibility for theanalytes of interest to have a ratio of M–S diffusivities be-yond the threshold of 100 is highly unlikely; thus, the influ-ence of (non-binary diffusion) cross-term is expected to beinsignificant.

On the other hand, if the M–S diffusivities for both an-alytes are approximately equal, significant influence of thediffusion cross term based on the secondary analyte diffu-sion is expected only when (1) the baseline concentrationsof both analytes to be at least 100 ppm and (2) the primaryanalyte concentration needs to be at least two orders of mag-nitude larger than the concentration of the secondary ana-lyte. Clearly, these requirements are seldom if ever met inpractical applications of chemical sensors.

In summary, the non-binary diffusion (cross-talk) effectsare not expected to be significant in analyzing multiple ana-lyte transport in chemical sensors. Since the cross-diffusionterms are negligible for the system considered, the masstransport for a multicomponent mixture can be treated asbinary diffusion of multiple non-interacting species andthe sensor design optimization performed in our earlierpaper[3] should be valid regardless of the number of an-alytes present in the flow. In support of this conclusion,Krupiczka and Rotkegel[28] found that the cross-termeffects for a ternary system were considerable under twoconditions (i) the ratio of the molar fluxes of interactingspecies,Nj/Ni, is large or (ii) when the binary diffusioncoefficients of the components show a considerable differ-

ence, which corresponds to the threshold criteria developedhere.

5.2. Relative strength of pressure diffusion terms

In order to compare the pressure diffusion terms to theordinary (binary) diffusion terms, the maximum pressuregradient expected for each configuration of the sensor flow


Table 2Comparison of ordinary multicomponent and pressure diffusion terms

∇C (∇P/RT)C

Baseline,H = 50�m 120,000 2Top entry 90,000 30Tapered height 100,000 1

cell must be determined. In this analysis, we compare pres-sure and concentration gradients we computed in the pre-ceding companion paper for the baseline flow cell geom-etry, the top entry geometry, and the tapered channel ge-ometry with the exit channel height set to 50�m for eachgeometry. The maximum pressure diffusion terms and thenormal (due to concentration gradients) diffusion terms inEq. (14)are shown inTable 2for each analyte (i.e., TeCEand DCB) for three geometries, assuming inlet concentra-tions of both analytes to be 500 ppm. The concentrationgradients and the pressure gradient terms of both analytesare similar in magnitude; therefore, only one is shown forsimplicity.

Clearly, the concentration gradients are at least three tofive orders of magnitude greater than the pressure gradientterms, depending on the flow cell geometry. This scalinganalysis supports neglecting the pressure diffusion terms inthe sensor analysis; for the geometries and concentrationsconsidered, pressure diffusion is not important. Pressure dif-fusion would become significant relative to the normal dif-fusion induced by the concentration gradients if the pres-sure drop was increased by two orders of magnitude for thetop entrance flow cell geometry, or five orders of magnitudefor the baseline and tapered height geometries. Since thepressure drop is inversely proportional to the channel heightin the fourth power for the constant flow rate, the pressureinduced analyte diffusion will become comparable to nor-mal (concentration induced) diffusion when the height ofthe flow cell is reduced by at least a factor of 3 (∼1001/4)to ∼17�m for the top entry geometry and by a factor of15 (∼100,0001/4) to ∼3�m for the baseline and taperedheight geometries. These dimensions of the flow cell are notonly well below the optimal heights for the given flow rates[3], but also are hardly practical from the sensor fabricationprospective.

6. Conclusions

Transport of multiple analytes in a chemical sensor systemwas investigated aiming at establishing the threshold for im-portance of multicomponent diffusion effects. Detailed anal-ysis based on the Maxwell–Stefan formalism demonstratesthat:

• In the case of approximately equal analyte concentrations,the multicomponent effects due to secondary analyte dif-fusion become significant when the ratio of Maxwell–

Stefan diffusivity of the secondary analyte to the primaryanalyte (–D23/–D13) is equal to or greater than 10.

• In the case of approximately equal analyte M–S diffusivi-ties, the multicomponent effects due to secondary analytediffusion become significant when concentrations of bothanalytes are at least 100 ppm or greater, and the concen-tration of one analyte is two orders of magnitude greaterthan that of the other.

• Pressure diffusion exhibits effects so small that it can beneglected unless the flow cells with extremely small flowcell channel heights are considered.

In general, the multicomponent effects on the mass trans-fer are expected to be small for the operating conditions anddesign configurations found in the chemical sensors. Thus,the transport of multiple analytes can be treated as binarydiffusion of multiple non-interacting species.

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Biographies

Cynthia Phillips received her undergraduate degree in mechanical engi-neering in May 2000 from the University of Michigan at Ann Arbor.While attending a graduate school at Georgia Tech, she spent one year(2000–2001) at Georgia Tech Lorraine in France, taking graduate coursesas well as performing research in the laboratory of Professor Ambari atENSAM, Anger, France. She completed her graduate studies and researchunder the direction of Professor Fedorov and received her MS degree

in mechanical engineering in August 2002. She is an author of threearchival journal publications in the area of heterogeneous catalysis andmodel-based design of chemical sensors. In the Fall of 2002, she joinedthe technical staff of United Technologies Fuel Cell, Inc.

Andrei G. Fedorovis an Assistant Professor in the Woodruff School ofMechanical Engineering at the Georgia Institute of Technology. ProfessorFederov received his PhD (1997) in mechanical engineering from PurdueUniversity. From 1997 until 1999, he was a postdoctoral research asso-ciate at Purdue University. In January 2000, he joined Georgia Instituteof Technology as an assistant professor where his research efforts havefocused on transport phenomena (heat, mass, and radiation transfer) inmaterials processing, micro/nano scale catalysis and reaction engineer-ing, and chemical sensors and multifunctional scanning probes. Profes-sor Federov authored over 40 archival papers in major technical journalsand refreed conference/symposia proceedings as well as 17 patents andinvention disclosures. He was an invited speaker nationally and interna-tionally and won several professional awards. He is a member of ASMEand ASEE, and he has organized and chaired technical sessions on majornational and international conferences.

multicomponent mass transfer in polymer-coated chemical sensors

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