Transient molecular diffusion in microfluidic channels: Modeling and experimental verification of the results

Accepted Manuscript Title: Transient Molecular Diffusion in Microfluidic Channels: Modeling and Experimental Verification of the Results Author: Faramarz Hossein-Babaei Ali Hooshyar Zare Vahid Ghafarinia PII: DOI: Reference:

S0925-4005(16)30585-8 http://dx.doi.org/doi:10.1016/j.snb.2016.04.103 SNB 20077

To appear in:

Sensors and Actuators B

Received date: Revised date: Accepted date:

24-12-2015 31-3-2016 17-4-2016

Please cite this article as: Faramarz Hossein-Babaei, Ali Hooshyar Zare, Vahid Ghafarinia, Transient Molecular Diffusion in Microfluidic Channels: Modeling and Experimental Verification of the Results, Sensors and Actuators B: Chemical http://dx.doi.org/10.1016/j.snb.2016.04.103 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Transient

Molecular

Diffusion

in

Microfluidic

Channels:

Modeling and Experimental Verification of the Results

Faramarz Hossein-Babaei1*, Ali Hooshyar Zare1, Vahid Ghafarinia2

1

Electronic Materials Laboratory, Industrial Control Center of Excellence, Electrical

Engineering Department, K. N. Toosi University of Technology, Tehran 16315-1355, Iran 2

Department of Electrical and Computer Engineering, Isfahan University of Technology,

Isfahan, 84156-83111, Iran *

Corresponding author: Tel.: +98 (0) 21 88734172; Fax: +98 (0) 21 88768289;

E-mail

addresses:

[email protected],

[email protected]

(F.

Hossein-Babaei),

[email protected] (A. Hooshyar Zare), [email protected] (V. Ghafarinia)

1

Abstract Almost all microfluidic devices operate at non-equilibrium transient conditions. Quantitative predictions regarding fluid flow within the components of such devices at the assumed conditions are a prerequisite for their systematic design. Here, we present a mathematical model for the transient dynamics of a target molecule (TM) diffusing along a microfluidic channel driven by a time-varying concentration gradient, and experimentally verify its predictions on a number of case studies. The model is the outcome of coupling the free molecular diffusion equation and Langmuir surface adsorption isotherm, both of which hold the specific geometrical and operational features of the microfluidic system at isothermal and isobar conditions. The TM flux fluctuations, caused by a sudden change in the imposed boundary conditions, are predicted to be highly uneven along the microchannel for a long time after the event. In complicated cases, such as a pulse train TM concentration modulation at the inlet of a background gas-filled microfluidic channel, the model correctly predicts the experimental results in both “diffuse-in” and “diffuse-out” conditions.

Keywords: Microfluidic channel; Molecular diffusion; Flux fluctuations; Modeling; Langmuir isotherm; Concentration modulation;

1. Introduction The involvement of microfluidic devices in diagnostic applications and biological analyses is short-term and, almost in all cases, such devices operate at non-equilibrium transient conditions [1-3]. These devices are mostly low-cost and disposable, but their design and prototype optimization tasks are time consuming and costly [4-6]. This is mainly due to the unavailability of mathematical models able to predict the transient behavior of the system components, which would allow rapid performance simulation at different configurations [7,8]. Such models would be invaluable for the determination of the optimum structure as, due to the awkwardness, inaccuracy, and high cost of the available microflow measurement techniques, the experimental approach based on trial-and-error is immensely time consuming, if not impractical. The presence of different driving and deterring forces in a microfluidic channel makes flow predictions in the transition mode difficult [9,10]. A mathematical description of the 2

flow rates has to accommodate and balance verves of different natures including pressure gradient [11,12], surface tension [13], concentration gradient [14-16], osmotic drag [17], friction force [18], adsorption to and desorption from the channel walls [19-23], which are effective at different magnitudes at different stages of the flow process depending on the materials and conditions assumed; for example, the ionic or molecular adsorption to the channel wall is of much higher impact at the transient mode of operation [24,25]. In the steady-state condition, however, the state of the system is described with a smaller number of parameters; the phenomena related to the molecular interactions with the channel wall, for instance, balance each other out at the steady-state. Moreover, the mathematics is simplified by the fact that the time derivatives of the state variables approach zero as the steady-state prevails. The steady-state fluid flow in microfluidic channels has been the subject of numerous research projects [26-31], but the transient flux variations along a microchannel due to the externally imposed concentration fluctuations has yet remained unattended. It has been demonstrated that monitoring the molecular diffusion rate of a target molecule (TM) in a microfluidic channel provides valuable information on its nature [15,21,32]. The device utilized for the experimental rate recordings is made by coupling a microchannel with a microcavity enclosing a chemoresistive chip on a PMMA substrate [14]. It is conceived that the results of such recordings can provide the verification data for the mathematical models developed to describe the transient flow rates in the microfluidic devices. Here, a mathematical model is proposed, which predicts the transient microflow modes caused by the externally imposed time varying concentration gradient in a microchannel in isostatic and isothermal conditions. The model is applicable for the practical cases such as an ionic or molecular solute diffusion in a solvent-filled microchannel or a gaseous analyte diffusion progress in a channel filled with a background atmosphere. The model predictions on the transient flow rates of a number of target molecules through a microchannel are experimentally verified.

2. Mathematical Model A cylindrical microchannel is positioned so that its open end, located at x = 0, acts as the inlet during the diffuse-in process and as the outlet in the diffuse-out (Fig. 1a). The internal diameter and length of the channel are d and L, respectively. The channel is filled and surrounded by a background atmosphere, such as clean air. The system is at isobar and 3

isothermal conditions. The channel inlet is exposed to a constant target molecule concentration, Co, in the 0 to te time interval as shown in Fig. 1b. Beyond this time interval, the inlet is open to the clean background atmosphere with zero TM concentration. The established concentration gradient drives the TM along the channel during both the diffuse-in and diffuse-out modes which result in a temporal variation of the analyte concentration at any point along the channel, C(x, t). As a typical example, Fig. 1b shows the variations of the C(L, t) with time in the 0 to tf time interval. The aim is to calculate C(x, t) based on the geometrical and material parameters of the system. In a microfluidic channel, by definition in the 1 mm to 1 µm diameter range, the only processes of significance at isothermal and isobar conditions are the concentration gradientcaused molecular diffusion and the molecular interactions with the channel walls. Then, C(x, t) is determined by the following “diffusion-physisorption equation” [21]. C ( x, t )  2C ( x, t ) C S ( x, t ) D  t t x 2

(1)

wherein, D is the diffusivity of the TM in the background gas, and CS(x, t) is the amount of TMs physisorbed to the channel wall per unit volume of the channel. D is assumed to be independent from the TM concentration, which is valid if C(x, t) remains below 1% of the background atmosphere for all of time and space. At these conditions, diffusivity is determined by the collisions between the TMs and the background gas molecules, rather than those between TMs themselves. In all the cases considered theoretically and/or experimentally below, the partial pressure of the target gas is less than 0.1%, i.e. 1000 ppm. Eq. (1) is valid for the larger diameter capillaries, as well, wherein the second term in the right hand side vanishes and (1) reduces to the free molecular diffusion equation (see below). However, (1) is unable to describe diffusion process in extra smaller cross-section channels, d < 1 µm, wherein the Knudsen [33] and surface diffusion mechanisms [34] would also become important, none of which have been accounted for in (1). According to Fig. 1a, in a volume of dV, considered along the channel, CS(x, t) can be written in terms of the fractional wall surface coverage,  ( x, t ) : CS ( x, t ) dV   ( x, t ) Ca dA

(2)

4

in which, Ca is the surface density of the adsorption sites and dA is the wall surface enclosing dV. Considering a circular cross-section for the channel, the geometrical relationship between dA and dV, (2) reduces to: C S ( x, t ) 

4Ca  ( x, t ) d

(3)

wherein, d is the channel diameter. At low TM concentrations,  ( x, t ) is determined by the Langmuir isotherm:

 ( x, t ) 

bC( x, t ) 1  bC( x, t )

(4)

in which, b is the adsorption (physisorption) constant defined as b

kRgT 2MRgT

e

Q / Rg T

(5)

where k is a constant of proportionality, Rg is the universal gas constant, T is the absolute temperature of the system, M is the molar mass of the TM, and Q is the heat of physisorption of the TM molecules to the channel wall. Replacing (4) for b in (3) results in C S ( x, t ) 

4Ca bC( x, t ) d 1  bC( x, t )

(6)

The time derivative of (6) is CS ( x, t ) 4Ca b C ( x, t )  2 t d (1  bC( x, t )) t

(7)

Inserting (7) in (1) results in  4Ca  C ( x, t ) b  2C ( x, t ) 1    D d (1  bC( x, t ))2  t x 2 

(8)

Equation (8) describes the diffusion process inside the microfluidic channel at the presence of physisorption/desorption interactions with the channel walls, and is valid in the transient as well as the steady-state regime according to the initial and boundary conditions imposed. For large bore size channels, e.g. d >300 µm in the case of 1-propanol vapor (see below), TM

5

physisorption/desorption rates are insignificant, and, hence, (8) reduces to the free molecular diffusion equation, C ( x, t )  2 C ( x, t ) D t x 2

(9)

Eq. (8) is numerically solved for diffuse-in and diffuse-out processes utilizing MATLAB software “parabolic-elliptic PDE in 1-D” resulting in C1(x, t) and C2(x, t), respectively. The boundary and initial conditions imposed by the system geometry and the utilized methodology on the diffuse-in process are C ( x, t ) 0 xL x

(10)

C1 (0, t )  C0

(11)

0  t  te

C1 ( x,0)  0

(12)

while in the diffuse-out process, the conditions are C ( x, t ) 0 x x  L

(13)

C2 ( x, te )  C1 ( x, te )

(14)

C2 (0, t )  0 te  t  t f

(15)

Combining C1 and C2 affords calculating the analyte concentration at any point and time within the channel, C(x, t). Example solutions are given below. Based on the solutions of (8) and using the parameters related to 1000 ppm (0.044 mol/m3) of 1-propanol vapor in air, TM diffusion in a borosilicate glass channel of d = 50 µm is analyzed. The open end of the channel at x = 0 is exposed to the contaminated atmosphere during the t = 0-35 s period. Fig. 2a demonstrates 1-propanol diffusion progress along the channel by presenting concentration profile snapshots at different timeframes indicating the gradual buildup of the TM concentration within the channel. At t = 35 s the channel is reconnected to the clean air and the channel is depleted out from 1-propanol during the diffuse-out mode. The predicted concentration profiles for t > 35 s are given in Fig. 2b. The

6

temporal TM concentration variations at a point along the channel, say at x = L, for channels of different internal diameters are presented in Fig. 3a, showing that in large diameter channels (d > ~300 µm, for 1-propanol) the predictions of (8) approach those of the free diffusion equation. Fig. 3a also shows that the channel-related diffusion retardation becomes substantial as the channel diameter goes below 50 µm. Significant departures from the free diffusion depends on the physisorption tendency of the TMs to the channel wall, as well. Fig. 3b presents the results of similar calculations for the cases of hydrogen contamination depicting that the channel retardation is insignificant for the channels with d >30 µm in the case of hydrogen. The transient flux of the TM from the channel cross-section at point x is calculated based on the time derivative of the TM concentration spatially integrated in the channel volume from x to L. That is 𝐿 𝐿 𝑑 𝑑 4𝐶𝑎 𝑏𝐶(𝑥, 𝑡) 𝑑2 ∅(𝑥, 𝑡) = {∫ [𝐶(𝑥, 𝑡) + 𝐶𝑆 (𝑥, 𝑡)]𝐴𝑑𝑥 } = {∫ [𝐶(𝑥, 𝑡) + ] (𝜋 )𝑑𝑥} 𝑑𝑡 𝑥 𝑑𝑡 𝑥 𝑑 1 + 𝑏𝐶(𝑥, 𝑡) 4

= 𝐿 𝜋𝑑 2 (𝐶(𝑥, 𝑡) + 𝑏𝐶 2 (𝑥, 𝑡)) + 4𝜋𝑑𝐶𝑎 𝑏𝐶(𝑥, 𝑡) 𝑑 {∫ 𝑑𝑥} 𝑑𝑡 𝑥 4(1 + 𝑏𝐶(𝑥, 𝑡))

(16)

∅(x, t) is calculated by determining C(x, t) and Cs(x, t) functions from (8) and (6) and inserting the results in (16). The predicted transient flow rates at four different points, along the typical channel shown in Fig. 1, are given in Fig. 4a-b in the 0 to tf time interval for two different TMs. Fig. 4, the first mathematical visualization of the transient molecular flux caused by an ephemeral concentration gradient rise in a microchannel, shows a surprisingly uneven transient molecular flux along the channel at the early stages of both diffuse-in and diffuseout regimes. According to Fig. 4a, the flux of 1-propanol at x = 3 mm rises in 0.3 s to its maximum at 2.0 x 10-5 µmol/s, while at 20 mm the flux maximum occurs at t = 30 s and the maximum rate is 2.2 x 10-6 µmol/s. Nevertheless, the rates decay almost evenly along the channel beyond t = 20 s. A similar situation prevails during the diffuse-out regime; quite uneven rates along the channel at the early seconds while the rates decay evenly later on. The time integrals of the diffuse-in and diffuse-out temporal fluxes are equal in absolute values. 7

In comparison, the initial transient flow rates calculated for hydrogen in similar conditions (Fig. 4b) are more uneven along the channel, but, owing to the 6 times higher hydrogen diffusivity, evenness approaches in less than 10 s. The direct applications of the presented theory are related to the gas recognition and analysis as the temporal diffusion rates recorded for an unknown TM analyte can be used as its finger print. Comparing such recordings with those theoretically generated for known target gases can result in analyte classification. By quantitatively relating the analyte flux through a microchannel to its nature, Eq. (16) broadens the experimental pattern generation possibilities. Using room temperature-operating microbalances equipped with surface capture agents [35], the analyte flux via a microchannel upon a predetermined transient exposure can experimentally be evaluated and used as the discriminating information. The presented mathematical descriptions can also be utilized for the design of such microfluidic-based miniaturized gas analysis systems. Design applications outside the field of gas analysis are also numerous covering all cases involving transient behaviors of gases and gas mixtures in microchannels; determining the optimum design for a microfluidic vapor-diffusion barrier is an example [36]. Both (8) and (16) can be utilized for the simulation and design work on the microfluidic channels and devices whose operation is driven by the concentration gradients imposed. A major limitation arises from the isothermal conditions assumed, as many of such devices either contain or gradually acquire a temperature gradient established along the microchannel, which would diminish the accuracy of the model. A useful extension of the presented model would be the introduction of a temperature gradient parallel to the channel. Another immediate extension would be the coverage of the multi-TM cases in which the concentrations of two or more molecular gases are simultaneously or sequentially varying. Our preliminary experiments on the case of two component gas mixtures support applicability of (8) for each component as long as the total concentration of the TMs remains below a few tenth of a percent of the background gas. The latter extension would pave the route for investigations on the selective diffusion and partial filtering of the components of gas mixtures in air-filled microchannels Using (16) for estimating the flux of a molecular solute, such as glucose, driven by a transient concentration gradient in a liquid-filled microchannel, would lead to similar results though in a different time scale. The diffusivities involved are three orders of magnitude 8

lower in liquids and, hence, the uneven flux along the channel is expected to remain significant for tens of minutes rather than seconds, after the removal of the diffusion source. Similar to the gas-filled microchannels, the boundaries are determined by the channel geometry, while the boundary fuzziness caused by the double layer formation at solid-liquid interfaces is compensated for by defining effective surface adsorption parameters. Similarly, Eq. (8) can be utilized for determining the solutes concentration at any time and space within the microchannel if the effective adsorption parameters are determined as the parameters of the best fitting to the experimental data. The diffusion-adsorption of ionic solutes is further complicated by the simultaneous diffusion of cations and anions with different diffusivities and adsorption parameters, while being coupled by the coulombic forces acting in between. While both (8) and (16) are mathematically sound for predicting the diffusion rates and fluxes caused by the transient concentration gradients in liquid-filled microchannels, the availability of experimental results is the prerequisite for further development in this respect.

3. Experimental Verification The parameters of the gases used are given in Table-I [37]. The experimental layout is schematically depicted in Fig. 5a and b. The system consists of two closed chambers filled with two predetermined different composition atmospheres. A pneumatic arm exposes the open end of a bundle of 50 µm bore borosilicate glass channels to these atmospheres according to the programmed experimental procedure. The diffusion progress rate in the channels is continuously monitored by the sensory system installed at the other end of the 50 mm long channels. Each experiment includes diffuse-in and diffuse-out modes of operation. In the former, the open end of the channel is exposed to the TM-contaminated atmosphere, when the TMs are allowed to diffuse into the channel, and, in the latter, the channel is reconnected to the background gas chamber to allow TMs to diffuse out. Further details on the experimental system are provided in Reference [15]. During the whole process, the TM concentration at the end of the channel is continuously monitored. The monitored signal is the response of the generic gas sensor positioned at x=L, as shown in Fig.5b. The relationship between the output of the sensor and the prevailing TM concentration, according to the experimental relationship given in Fig. 5c, is nonlinear. In our experiments, the TM concentration is limited to 44 mmol/m3 and, as shown in Fig. 5c, this relationship is approximately linear. In these cases, the slope of the red 9

line segment in Fig. 5c determines the constant relating the sensor responses to the TM concentration (αi, where i defines the TM). This constant is different for different TMs (see the different slopes of the lines related to 1-propanol and methanol in Fig. 5c), causing problems when the system is used for analytical purposes and the TM is unknown prior to the experiment. The problem is solved by normalization of all the recorded responses to fit the 01 range. The generation of the absolute values of the recordings for a known TM is possible as the related αi would be available from the calibration data (see Fig. 1b and Fig. 3a-c), but in analytical applications (the main concern in our laboratory [14, 15, 38]), and for an unknown TM, the comparisons and identifications is carried out on the normalized recordings. A typical recording result, obtained for 1-propanol in an air-filled borosilicate channel of 50 µm diameter at T = 25 °C is given in Fig. 6a, where it is compared with the best fitting solution of (8). The list of fitting parameters is provided as legend in Fig. 6a. Similar experiments are carried out on methanol, iso-propanol, tert-butanol, methyl isobutyl ketone, 2-pentanone, methane and carbon tetrachloride; in all cases the model-predicted rates fit the corresponding experimental results. The fitting results, along with the fitting parameters are given in Fig. 6b-h. The close agreement of the experimental results with the predictions of the model verifies the model utilized for the above described transient flow analyses. In a different set of experiments, 1-propanol vapor concentration is pulse modulated at the open end of the microchannel while monitoring its concentration variation at x = 25 mm. (Shorter channels are utilized for this set of experiment than the previous.) In these complicated experiments, each test includes different consecutive diffuse-in and diffuse-out modes, while the terminal conditions of each step is the initial condition for the next. As shown in Fig. 7, our model, using only two fitting parameters has successfully described the temporal variation of the TM concentration at the closed end of the channel.

4. Conclusions A mathematical model is presented for the quantitative description of the flow rate fluctuations caused by the transient concentration variations in microfluidic channels operating at isostatic and isothermal conditions. This model accounts for the interactions between the diffusing species and the channel walls during both diffuse-in and diffuse-out 10

modes, and is shown to describe the experimental transient diffusion rates recorded in gasfilled microchannels with acceptable accuracy at various conditions. For a sudden variation in concentration at an open end, the predicted flow rate fluctuations are distinctly non-uniform along the microchannel for considerably long monitoring periods. The quantitative relationship between this non-uniform flux and the structural parameters of the microchannel are determined for eight different TMs, demonstrating that the temporal profile of the concentration fluctuations depend on the nature of the TM as well as the channel specifications. The model-predicted transient rates are verified by monitoring the diffusion rates of a TM in air-filled microchannels. We anticipate that, after appropriate parameter modifications, the presented theory would be applicable to the cases of molecular solute diffusion in the liquid-filled microchannels. The availability of experimental data is a prerequisite for further investigations.

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Author Biographies: Faramarz Hossein-Babaei received the B.Sc. degree in electronic engineering from Amir-Kabir University of Technology (Tehran Polytechnic), Iran, in 1971, and the M.Sc. degree in materials science and the Ph.D. in electrical engineering from the Imperial College, London, UK, in 1975 and 1978, respectively. He is Professor of Electronic Materials at the Electrical Engineering Department of K.N. Toosi University of Technology, Tehran, Iran. He has founded a number of hi-tech spin-off companies mostly active in the field of high temperature materials and technology. His present research interests include electric heating, microfluidics, metal oxide gas sensors and artificial olfaction. Prof. Hossein-Babaei received the Khwarizmi International Award for his outstanding R and D work on high temperature systems in 2006.

Ali Hooshyar Zare received his B.Sc. and M.Sc. degree in electrical engineering from K.N. Toosi University of Technology, Tehran, Iran in 2007 and 2009, respectively. He is currently a Ph.D. candidate at the Electrical Engineering Department of KNTU. His main areas of interest are microfluidics, electronic nose, gas and humidity sensors.

Vahid Ghafarinia received the B.S. degree from Razi University, Kermanshah, Iran in 2003 and the M.S. degree from K.N. Toosi University of Technology, Tehran, Iran in 2005, and Ph.D. in 2010 in electronic engineering. He is currently assisted professor at the Department of Electrical and Computer Engineering, Isfahan University of Technology. His research interests include machine olfaction, chemoresistive gas sensors, pattern recognition and soft computing.

15

(a) dx

C0

d = 50 µm x=L

x=x

10 (b)

L = 50 [mm] d = 50 [µm]

8

44 40

6

30

4

20

2

10

0 0

20

40

60 Time (s)

80

100

Co (mmol/m3)

TM Concentration at x = L (mmol/m3)

x=0

0 120

Fig. 1. (a) The schematic presentation of the microchannel and the one-dimensional spatial framework assumed for the mathematical analysis. (b) The temporal profile of a TM concentration imposed to the channel inlet (green), and a typical TM concentration profile recorded at the channel end (blue).

16

50

(a)

1-Propanol C0 = 44 [mmol/m3] D = 0.0993x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 3.011 [m3/mol] Ca = 2.89x10-6 [mol/m2]

C (mmol/m3)

40 30 20

t = 35 s

10 0

1s t=0s

0

(b)

30

36 s 37 s 38 s 39 s 40 s 45 s

20 10

25 s 10 s 15 s 2 s5 s

55 s 75 s t = 115 s

0 10

20 30 x (mm)

40

50

1-Propanol C0 = 44 [mmol/m3] D = 0.0993x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 3.011 [m3/mol] Ca = 2.89x10-6 [mol/m2]

t = 35 s

40

C (mmol/m3)

50

0

10

20

30 x (mm)

40

50

Fig. 2. The predicted concentration profiles of 1-propanol in the channel atmosphere during the diffuse-in (a) and diffuse-out modes (b); during the diffuse-in mode the channel is exposed to the TM-contaminated air for 35 s.

17

12

d=

1000 µm

10

44 40

8

30

100 µm 50 µm

6

20 30 µm

4

10

15 µm

2

Co (mmol/m3)

300 µm

d = 10 µm

0

0

20

40

60 80 Time (s)

0 120

100

D = 0.61x10-4 [m2/s] b = 0.1 [m3/mol] Ca = 5x10-6 [mol/m2] L = 50 [mm]

(b) Hydrogen 40

25 d = 

30

1000 µm 300 µm 100 µm 50 µm 30 µm

24.5

15 µm 10 µm

17

18

19

20

21

10

0

40

30

24

20

44

20

Co (mmol/m3)

TM concentration at x = L (mmol/m3) TM concentration at x = L (mmol/m3)

D = 0.0993x10-4 [m2/s] b = 3.011 [m3/mol] Ca = 2.89x10-6 [mol/m2] L = 50 [mm]

(a) 1-Propanol

10

0

20

40

60 80 Time (s)

100

0 120

Fig. 3. The predicted temporal variations for 1-propanol (a), and hydrogen (b) concentration at x=L in channels with the stated diameters.

18

-5

x 10

44

Flux (µmol/s)

1.5

x = 3 mm

30

1

15

5 mm 20 mm

0.5 0

Co (mmol/m3)

(a)

2

0

x = 50 mm 1-Propanol

-0.5

D = 0.0993x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 3.011 [m3/mol] Ca = 2.89x10-6 [mol/m2]

-1 -1.5 0

20

40

60 Time (s)

80

100

120

-4

x 10

44 x = 3 mm

30

Flux (µmol/s)

0.5 5 mm

15

20 mm

0

Co (mmol/m3)

(b)

1

0 x = 50 mm

Hydrogen D = 0.61x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 0.1 [m3/mol] Ca = 5x10-6 [mol/m2]

-0.5

-1 0

20

40

60 Time (s)

80

100

120

Fig. 4. Temporal variations of the transient concentration gradient-driven flux at the stated points along the channel shown in Fig. 1 for 1-propanol (a) and hydrogen (b) calculated using (16). The line related to point x = 50 mm predicts zero flux as it coincides the closed end of the channel.

19

Arm Control

(a)

Interface Circuit

Computer

Reference clean air

Analyte-contaminated air

Electrical connection Leads

(b) Identical microfluidic channels of 50 micron bore size

Gas sensor

50 mm Impermeable package

(c)

Response (volt)

4

Methanol 1-Propanol

3 α2

2

α1

1 0

0

88

176

264

Concentration

352

440

(mmol/m3)

Fig. 5. Schematics of the experimental setup used for the continuous monitoring of the diffusion progress rates in microfluidic channels (a) and the microchannels integrated with a tin oxide-based chemoresistor (b); the sensing characteristics of the sensor relates its output to the concentrations of the stated contaminants in air (c).

20

1

0.8 1-Propanol

30

D = 0.0993x10-4 [m2/s] L = 50 [mm] 20 d = 50 [µm] b = 3.011 [m3/mol] Ca = 2.89x10-6 [mol/m2] Experimental Response 10 Solution of Eq. (8)

0.6 0.4 0.2 0

C(L, t) (normalized)

44 40

(a)

Co (mmol/m3)

C(L, t) (normalized)

1

0 0

20

40

60 Time (s)

80

100

(b)

0.8 0.6 Methanol D = 0.1520x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 0.100 [m3/mol] Ca = 2.10x10-6 [mol/m2]

0.4 0.2 0 0

120

1

1

C(L, t) (normalized)

C(L, t) (normalized)

0.6 Iso-propanol D = 0.1013x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 1.483 [m3/mol] Ca = 2.06x10-6 [mol/m2]

0.4 0.2 0

1

20

40

60 Time (s)

80

100

100

120

0.8 0.6 Tert-butanol D = 0.0852x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 1.615 [m3/mol] Ca = 1.74x10-6 [mol/m2]

0.4 0.2

120

0

40

60 Time (s)

80

100

120

(f)

(e)

0.8 0.6 Methyl isobutyl ketone 0.4

D = 0.0709 x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 1.937 [m3/mol] Ca = 1.85x10-6 [mol/m2]

0.2 0 0

20

1

C(L, t) (normalized)

C(L, t) (normalized)

60 80 Time (s)

0 0

20

40

60 Time (s)

80

100

0.8 0.6 2-pentanone 0.4

D = 0.0793x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 0.338 [m3/mol] Ca = 1.65x10-6 [mol/m2]

0.2 0 0

120

1

20

40

60 Time (s)

80

100

120

1

(h)

(g) 0.8

C(L, t) (normalized)

C(L, t) (normalized)

40

(d)

(c) 0.8

20

0.6 Methane D = 0.2240x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 1.834 [m3/mol] Ca = 4.65x10-6 [mol/m2]

0.4 0.2 0

0.8 0.6 Carbon tetrachloride 0.4

D = 0.0828x10-4 [m2/s] L = 50 [mm] d = 50 [µm] b = 6.632 [m3/mol] Ca = 2.33x10-6 [mol/m2]

0.2 0

0

20

40

60 Time (s)

80

100

120

0

20

40

60 Time (s)

80

100

120

Fig. 6a-h. (a) The experimentally recorded variations of 1-propanol concentration at x = 50 mm from t = 0 to 115 s, when the open end of the air-filled channel is exposed to the contaminated air in the t = 0 to 35 s time interval, and the best fitting solution of Eq. (8); fitting parameters are given as legends. The green line shows the imposed TM exposure at the

21

open end of the channel. (b-h) The same as (a) plotted for the stated target molecules, all depicting fine fitting of the theoretical predictions to the experimental results.

Experimental Response Solution of Eq. (8)

C0 modulation

44

1 0.8

30 0.6 TM: 1-propanol D = 0.0993x10-4 [m2/s] L = 25 [mm] d = 50 [µm] b = 3.011 [m3/mol] Ca = 2.89x10-6 [mol/m2]

0.4 0.2

20

Co (mmol/m3)

C(L, t) (normalized)

40

10

0

0 60 s

0

50

20 s 40 s

100

150

200

250

300

Time (s)

Fig. 7. The experimental and the model predicted TM temporal concentration at x = 25 mm, when a pulse-modulated 1-propanol concentration (green) is applied to the open end of the microfluidic channel.

22

Table 1. Diffusivities [37] and the parameters utilized for fitting the model predictions to the experimental results.

Target

Molecule

Diffusivity

Fitting parameters

at 25 °C

b

Ca

[10-4 m2/s]

[m3/mol]

[10-6 mol/m2]

Methanol

CH4O

0.1520

0.100

2.10

1-propanol

C3H8O

0.0993

3.011

2.89

Iso-propanol

C3H8O

0.1013

1.483

2.06

Tert-butanol

C4H10O

0.0852

1.615

1.74

Methyl isobutyl ketone

C6H12O

0.0709

1.937

1.85

2-pentanone

C5H10O

0.0793

0.338

1.65

Methane

CH4

0.2240

1.834

4.65

Carbon tetracloride

CCl4

0.0828

6.632

2.33

23