Theoretical Modeling and Experimental Evaluation of a Microscale

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Anal. Chem. 2004, 76, 2725-2733

Theoretical Modeling and Experimental Evaluation of a Microscale Molecular Mass Sensor Colin D. Costin,† Adam D. McBrady,† Milton E. McDonnell,‡ and Robert E. Synovec*,†

Center for Process Analytical Chemistry (CPAC), Department of Chemistry, Box 351700, University of Washington, Seattle, Washington 98195-1700, and Specialty Materials, Analytical Sciences Laboratory, Honeywell International, 101 Columbia Road, Morristown, New Jersey 07962

A theoretical model for a recently developed microscale molecular mass sensor (µ-MMS) is presented. The µ-MMS employs a widely applicable technique of measuring the refractive index gradient (RIG) in a microchannel created after two adjacent streams merge: a “sample stream” containing analyte(s) of interest in a host solvent and a “mobile-phase” stream containing only the host solvent. Because the flow in the microchannel is laminar, the analytes in the sample stream mix with the mobile-phase stream primarily by diffusion. The diffusion-induced RIG in the microchannel is measured by monitoring the deflection angle of a diode laser probe beam, which is orthogonal to both the direction of flow and the direction of analyte diffusion. The µ-MMS samples the RIG with probe beams at two positions along the direction of flow, and the ratio of the downstream to the upstream signal monitors the diffusion coefficient. Following calibration for a given class of compounds, the molecular mass of an analyte of interest can be determined. Along with the analyte diffusion coefficient, the theoretical model indicated three other specific parameters are important to interpret the µ-MMS output: the radius of the interrogating light probe beams, the time intervals between each of the detection positions, and the merge point relative to the detection positions. A series of experiments were conducted at different beam radii and flow rates to investigate these parameters, and the results are consistent with the model. The model shows that by using smaller beam radii and altering flow rates the molecular mass range of the µ-MMS can be, in principle, tuned from less than 102 g/mol to greater than 108 g/mol. The ratio data from the µ-MMS is also demonstrated to readily provide a “universal calibration”, from which the determination of unknown diffusion coefficients can be readily obtained. The development of micro total analytical systems (µ-TAS) stands to revolutionize chemical analysis across a wide range of applications such as national defense, health care, and drug development.1,2 These devices can provide a number of benefits such as high-throughput analysis, less reagent usage and waste * Corresponding author: (e-mail) [email protected]. † University of Washington. ‡ Honeywell International. 10.1021/ac030405c CCC: $27.50 Published on Web 04/10/2004

© 2004 American Chemical Society

generation, higher sensitivity, and increased portability due to their small size. While there have been only a few accounts of completely integrated µ-TAS,3,4 microfluidic devices have seen recent success in sample preparation for HPLC,5 2D on-chip separations,6 biomolecule analysis,7,8 and high-throughput screening.9 Yet, there continue to be technological hurdles for the µ-TAS field to overcome before more completely integrated systems garner widespread implementation. One barrier is the development of a sensitive, universal, and informative detection technique. A recently developed system, the micro refractive index gradient detector (µ-RIG detector), has been shown to be a sensitive universal detector for microfluidic devices.10 This system measures the refractive index gradient (RIG) created between two merging streams in a microchannel: the “sample stream” containing analyte(s) of interest in a host solvent and the “mobile-phase” stream containing only the host solvent. Due to the low Reynolds number inherent to microfluidic channels, when these two streams merge into a single analysis channel, the mixing between them is due primarily to diffusion.11 This behavior creates a concentration gradient of analyte(s) across the microchannel, orthogonal to the direction of flow, which is manifested as a RIG. The RIG is measured by monitoring the angle of deflection of a diode laser probe beam interrogating the microchannel orthogonal to both the direction of flow and the concentration gradient. This deflection is measured on a position-sensitive detector (PSD). The µ-RIG detector was further developed into a system that can also determine the molecular mass and diffusion coefficient of an injected analyte, known as a micro molecular mass sensor (1) Reyes, D. R.; Iossifidis, D.; Auroux, P. A.; Manz, A. Anal. Chem. 2002, 74, 2623-2636. (2) Auroux, P. A.; Iossifidis, D.; Reyes, D. R.; Manz, A. Anal. Chem. 2002, 74, 2637-2652. (3) Grover, W. H.; Skelley, A. M.; Liu, C. N.; Lagally, E. T.; Mathies, R. A. Sens. Actuators, B 2003, 89, 315-323. (4) Lagally, E. T.; Emrich, C. A.; Mathies, R. A. Lab Chip 2001, 2001, 102107. (5) Jandik, P.; Weigl, B. H.; Kessler, N.; Cheng, J.; Morris, C. J.; Schulte, T.; Avdalovic, N. J. Chromatogr., A 2002, 954, 33-40. (6) Ramsey, J. D.; Jacobson, S. C.; Culbertson, C. T.; Ramsey, J. M. Anal. Chem. 2003, 75, 3758-3764. (7) Peterson, D. S.; Rohr, T.; Svec, F.; Frechet, J. M. J. Anal. Chem. 2002, 74, 4081-4088. (8) Mao, H.; Yang, T.; Cremer, P. S. Anal. Chem. 2002, 74, 379-385. (9) Sinclair, J.; Pihl, J.; Olofsson, J.; Karlsson, M.; Jardemark, K.; Chiu, D. T.; Orwar, O. Anal. Chem. 2002, 74, 6133-6138. (10) Costin, C. D.; Synovec, R. E. Talanta 2002, 58, 551-560. (11) Kamholz, A. E.; Weigl, B. H.; Finlayson, B. A.; Yager, P. Anal. Chem. 1999, 71, 5340-5347.

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the system without presenting any rigorous mathematical derivation.13 The true potential of the µ-MMS is further explored in the current report by presenting a theoretical model to determine what parameters are most important in the design and operation of the instrument. The model was also used to demonstrate how these parameters can be altered to easily tune the detector to the requirements of a given set of analytes. This work provides a detailed mathematical derivation of the theory inherent to this device and demonstrates the tremendous potential the µ-MMS has for both microfluidic and traditional-sized analytical platforms. The theoretical findings are then experimentally evaluated to demonstrate concurrence.

Figure 1. (A) Schematic of µ-MMS microchannel and optical configuration. A sample stream (dark for illustration purposes only) enters the T-junction from the top while a mobile-phase stream (light) enters from the bottom. The two streams merge and flow down a single channel in the z-direction producing a time-dependent concentration gradient of the analyte in the sample stream. Diode laser beams transmitted by fiber-optic cables and focused by GRIN lens assemblies produce beams with a radius ro. The probe beams are deflected by the concentration gradient, and the deflected beams are detected by PSDs. The measured deflection angles θA and θB are from rays at two different locations along the axis of flow. The angular deflection averaged over the whole beam provides diffusion coefficient information that is readily correlated to molecular mass. (B) A planar wave front PQ is incident to a RIG. Segment P′Q′ is the planar wave after the spherical wave fronts propagate through the RIG. During the advance of the wave front by ds, the RIG produces a displacement dx. Since the incident and refracted fronts are not parallel, the beam is deflected by an angle ∆θ.

(µ-MMS).12,13 This system works under the same principles as the µ-RIG detector. The µ-MMS uses a fiber-optic splitter coupled to a laser diode to give two interrogating probe beams. The current version of the µ-MMS measures the deflection of both probe beams simultaneously by two PSDs (Figure 1A). The addition of a second detection position (downstream signal) to the first detection position (upstream signal) yields the critical data to obtain diffusion coefficient information. The ratio, R, of the downstream signal divided by the upstream signal monitors the diffusion coefficient, which for dilute solutions is essentially independent of analyte concentration. Furthermore, for a given class of compounds and a fixed set of experimental conditions such as constant flow rate and constant optical configuration, the ratio R has been shown to readily correlate to the molecular mass. Hence, the µ-MMS has been experimentally demonstrated as a sensitive, universal, and informative device for microscale applications and has also been demonstrated as a multidimensional detector for HPLC.13 Previous reports have focused on instrument design, initial characterization, and an empirical theory to qualitatively describe (12) Costin, C. D.; Synovec, R. E. Anal. Chem. 2002, 74, 4558-4565. (13) Costin, C. D.; Olund, R. K.; Staggemeier, B. A.; Torgerson, A. K.; Synovec, R. E. J. Chromatogr., A 2003, 1013, 77-91.

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THEORY To model the µ-MMS, we seek an expression for the ratio of the average deflection angle of two beams of finite width as they probe the RIG at upstream and downstream positions. The merged sample and mobile-phase streams flow through a “detection cell” in the z-direction (Figure 1A) at a linear speed vf. At z ) 0, the two streams begin parallel flow with a sharp concentration difference co. As the z-coordinate increases the streams diffuse into one another, reducing the magnitude of the RIG and increasing its width in the x-direction. A laser probe beam some distance A down the z-axis of the microchannel from the merge point of the two streams, and centered in the channel at x ) 0, travels through the detection cell of thickness w in the y-direction. A second, parallel laser probe beam passes through the center of the microchannel at some distance B, which is greater than A. The RIG deflects each beam and their deflections are measured on two PSDs. The normalized Gaussian beam irradiance profile in the x,z plane for the TEM00 mode centered at position A with radius, ro, is given by

p dx dz )

(

)

x2 + (z - A)2 2 exp -2 dx dz πro2 ro2

(1)

The probe radius ro is defined in eq 1 as two standard deviations of the Gaussian beam profile. For each beam in the µ-MMS, each ray of light is deflected by a different part of the RIG. The average beam deflection corresponds to the deflection of all the rays integrated over the appropriate part of the RIG. Consider the deflection of a single ray, such as the center ray shown in Figure 1A. For each incremental advancement of the light ray ds, there is a small displacement dx perpendicular to the incident direction, which defines the angle of deflection ∆θ

∆θ ) sin

(dxds) ) dxds

(2)

The angle can be described as a function of the time ∆t since the light entered the detection cell, the light velocity vl, and the distance traveled in the cell, ∆s ) vl‚∆t, by

∆θ )

n dθ dθ ∆t ) ∆s dt vo dt

(3)

where the light wave velocity vl ) vo/n depends on the speed of light in a vacuum vo and the refractive index of the medium n.

The propagation of a planar wave specified by line segment PQ (Figure 1B) can be described by the interference of spherical wave fronts originating from all points along the segment. In an isotropic medium, the constructive interference of the emanating spheres of constant radius indicates the position of the propagated wave as a new front parallel to PQ. When the wave travels parallel to a RIG, the radii of the emanating spheres decrease with increasing refractive index. The propagating front determined by the constructive interference of the spheres is no longer parallel to segment PQ as is shown by segment P′Q′ in Figure 1B. If the segments PQ and P′Q′ are extended to their intersection point, the enclosed angle has the same value as the angle describing the deviation of the beam. The velocity of the point P which makes a circular arc of radius x about the intersection point is given by

v)

vo dθ )x n dt

(4)

Since the wave velocity v is always perpendicular to this radius x, geometry demands that angular relations between changes in the direction of v and x, that is, ∆x and ∆s, must be the same as those between v and x, themselves. It thus follows from eq 4 that

dθ v dv vo dn ) ) ) dt x dx n2 dx

(5)

which indicates that eq 3 can be written as

∆θ )

1 dn 1 dn dc ∆s ) ∆y n dx n dc dx

(6)

It has been explicitly acknowledged in the final expression of eq 6 that the RIG is produced by a concentration gradient dc/dx and that the net displacement of the beam ∆s is essentially equivalent to its displacement in the y-direction, ∆y. The result given in eq 6 is in agreement with related work in this field.14-16 From the solution to the diffusion equation given by Crank,17 the concentration profile in the µ-MMS configuration at arbitrary time and position can be described by

co c ) [1 + erf(ξ)] 2

(7)

where co is the initial concentration and erf(ξ) is the error function of argument ξ ) x/(4Dt)1/2 for analytes that have been diffusing since time t ) 0 from a step profile at x ) 0 with diffusion coefficient D. The concentration gradient orthogonal to the direction of flow, dc/dx, can then be written as (14) Pawliszyn, J. Anal. Chem. 1986, 58, 3207-3215. (15) Pawliszyn, J.; Weber, M. F.; Dignam, M. J.; Mandelis, A.; Venter, R. D.; Park, S. Anal. Chem. 1986, 58, 236-238. (16) Pawliszyn, J. Anal. Chem. 1988, 60, 2796-2801. (17) Crank, J. The mathematics of diffusion, 2nd ed.; Clarendon Press: Oxford, U.K., 1975.

( )

-x2 dc dc dξ ) ) co(4πDt)-1/2 exp dx dξ dx 4Dt

(8)

Equation 8 reveals that the concentration gradient probed by the RIG sensing mechanism is a Gaussian function. Thus, the general expression for the angular deflection is

∆θ )

( )

co dn -x2 (4Dt)-1/2 exp ∆s 4Dt xπn dc

(9)

Equation 9 appears deceptively simple since x is actually a function of ∆θ,

x)

∫

x

0

dx )

∫ ∆θ ds y

(10)

0

so one needs all preceding values of ∆θ at the corresponding ds values in the detection cell to obtain the current x to evaluate the current incremental ∆θ. The complexity is, however, inconsequential if ∆θ is so small that x essentially retains its initial value as a ray travels across the detection cell. If x is constant, the exponential term causes no difficulty in the integration of eq 9 to give

θ)

( )

( )

cow dn -x2 -x2 (4Dt)-1/2 exp ) Kwco(4Dt)-1/2 exp 4Dt 4Dt xπn dc (11)

where w is channel depth or path length and the optical constants of the solution are combined as into one constant, K ) (dn/dc)/ (π1/2n). The value for the total deflections, θA and θB, of the center ray at the upstream and downstream positions, respectively, is shown in Figure 1A. To find the average of the angular deviations over the beam irradiance profile, one must integrate eq 11 over the beam profile of eq 1. Although there is an insignificant change in x upon going through the detection cell, the variation of the laser spot intensity across the face of the cell is extremely important. The output of the detector depends on the relative size of the two lengths: ro, which characterizes the beam size, and (4Dt)1/2, which describes the extent of the diffusion gradient. The appropriate integrals for the average deflection of all the rays in the x,z plane is

θ h)

2Kwco πro2

∫∫

x,z plane

( ) 4Dz vf

-1/2

(

( )

vfx2 exp 4Dz

)

x2 + (z - A)2

exp -2

ro2

dx dz (12)

The limits of integration can cover the entire x,z plane since the second exponential term ensures negligible contribution to the integral from the first exponential term far from the beam center. Equation 12 is simply the three-dimensional overlap integral of two Gaussian functions: the concentration gradient being probed by a probe beam (eqs 8 and 11) and the probe beam itself (eq 1). Placing a restriction on the basic length parameters that ro is much less than A provides a closed-form solution to eq 12 at z ) A: Analytical Chemistry, Vol. 76, No. 10, May 15, 2004

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θ h)

(

)

x2Kwco 8DA 1+ ro vr 2 f o

-1/2

) x2Kwco(ro2 + 8Dt)-1/2 (13)

Equation 13 expresses the average deflection of a laser beam of a given radius ro in terms of the diffusion coefficient of the analyte creating the RIG and the amount of time the analyte sample has had to diffuse from the instant the sample and mobile-phase streams have merged. The signal from a single laser beam provides a concentration-dependent signal similar to a traditional refractive index detector but with added diffusion information. In the µ-MMS, an analyte diffusion coefficient is determined by measuring the deflection angle at a position just past the merging point (upstream position) and a second position some distance further downstream (downstream position) as shown in Figure 1A. Diffusion coefficient information that is essentially independent of analyte concentration is obtained by taking the ratio, R, of the deflection signal at the downstream position divided by the upstream position, following from eq 13,

R)

θB θA

)

(

vfro2 + 8DA

vfro2 + 8DB

) ( 1/2

)

ro2 + 8DtA

)

1/2

ro2 + 8DtB

(14)

where tA and tB are the times since the streams merged for the upstream and downstream position, respectively, ro is the probe beam radius defined in eq 1, and D is the analyte diffusion coefficient. One can see when D becomes sufficiently small for a given flow rate and probe beam radius, the ratio R approaches 1 and size differentiation of diffusing analytes is lost. Alternately, when the probe beam radius becomes sufficiently small, i.e., a single ray, the ratio R is a function of only the measurement times and the analyte size and information is again lost. Between these two extremes the diffusion coefficient is probed. The expression in eq 14 can be readily rearranged to provide a means to explicitly determine the diffusion coefficient of an “unknown” analyte with a beam of known beam size and transit times:

D)

ro2 (1 - R2) 8 (t R2 - t ) B

(15)

A

Thus, the output of the µ-MMS detector provides not only the analyte concentration information via the signal data at either detection position, as is typical for a refractive index detector, but also the diffusion coefficient of the analyte. By determining ro, tB, and tA for a given set of detection conditions, one can simply measure the ratio R for an unknown analyte and obtain the diffusion coefficient, D, by application of eq 15. Furthermore, the ratio R data can be related to analyte molecular mass, M, with calibration required for a given class of compounds in a particular solvent system. This relation is provided by the following general equation, which allows one to convert between diffusion coefficient and molecular mass,

D ) a(M)-b 2728

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(16)

where D is diffusion coefficient, M is molecular mass, and a and b are constants that are often readily available in the literature, such as for many polymer systems. Alternatively, if the constants a and b are not readily available, one can often empirically relate the ratio, R, data obtained experimentally to the molecular mass of analytes of interest. This is analogous to the practice of calibration in size exclusion chromatography, in which analyte molecular mass is correlated to the retention time, if a correlation is known to exist for a class of compounds of interest.13 EXPERIMENTAL SECTION Instrument Construction and Microfabrication. The µ-MMS was constructed as previously reported13 with the exception that the microfluidic channels were made in borosilicate glass instead of poly(dimethylsiloxane). The process to make the glass microchips involved isotropic etching with HF and closely followed a previously published procedure.18 Borosilicate wafers, 4 in. diameter × 1.1 mm thick (Silicon Valley Microelectronics Inc., San Jose, CA) were first cleaned by immersion in Nanostrip 2X (Cyantec Corp., Freemont, CA) for 10 min, rinsed with DI water, and then annealed in an oven at 540 °C for 30 min. The wafers were then cleaned again in Nanostrip 2X for 10 min followed by a rinse in DI water. A 200-Å layer of Cr followed by a 3000-Å layer of Au were then deposited by electron beam evaporation on the glass wafer to act as a protective mask during HF etching. The microchannel pattern was transferred to the gold and mask layer using photolithography techniques. In these experiments, the channels were T-shaped (Figure 1A) where the sample inlet and mobile-phase inlet channels had dimensions of 1 cm × 100 µm wide × 200 µm deep and the main flow channel had dimensions of 5 cm × 500 µm wide × 200 µm deep. The Au layer was then etched for 107 s in TFA Au etchant (Transene Corp., Danvers, MA) and the Cr layer was subsequently etched for 25 s in TFD Cr etchant (Transene Corp.). This left a portion of the glass surface exposed in the pattern of the microfluidic network being made. The wafer was then immersed in 49% HF for 15 min, and the etch depth was checked using a profilometer (Alpha Step 200, Tencor, San Jose, CA); additional time in the HF was often required to obtain a final depth of 200 µm. Fluidic access holes were then drilled into the etched wafer using 0.75-mm diamondtipped drill bits (C.R. Laurence Co., Los Angeles, CA). The etched wafer and an unetched bonding wafer were then put into a final cleaning solution of H2O/NH4OH/H2O2 at 70 °C for 30 min and rinsed with DI water. The microfluidic channel was finished by thermally bonding the etched and unetched wafers together. The two wafers were first brought into contact and pressed together and Newton ring continuity was used to ensure the surfaces were clean enough for thermal bonding. Additional time in the cleaning solution was required if particulate disruption was observed in the Newton ring pattern. The pressed wafers were sandwiched between quartz plates, put in a programmable oven, and slowly brought to 680 °C, where they were held for 3 h. If portions of the wafers did not bond, additional oven cycles were used until the microchannels were completely sealed. Microfluidic chips were inserted into the µ-MMS instrument and connected to PEEK tubing (Upchurch Scientific, Oak Harbor, WA) using Nanoport assemblies (N-126S, Upchurch Scientific). (18) Fan, Z. H.; Harrison, J. D. Anal. Chem. 1994, 66, 177-184.

Laser Beam Radius Measurements. The laser probe beam radius was measured using the same equipment configuration as the µ-MMS instrument; however, the microfluidic chip was removed and replaced with a razor blade and the PSDs were replaced by a simple photodiode. The razor blade was then passed through the laser beam in 10-µm increments with a translational stage. The resulting beam intensity was monitored with the photodiode. A plot of intensity as a function of razor blade position was made, and the derivative of these data was taken to obtain a beam profile. Laser beam radii were measured at two different distances between the GRIN lens and the razor blade (see Figure 1A). “Position 1” was 1 mm from the GRIN lens, which is the actual working distance of the µ-MMS for most of the studies where the probe beams pass through 1 mm of glass wafer before encountering the microchannel. “Position 2” was 10 cm away from the GRIN lens. The 2σ beam radii (ro), as defined earlier, were determined to be an average of 79 µm for both beams at position 1 and an average of 293 µm at position 2. For a given beam position, both beams were identical within ∼5%. Experimental Evaluation of the Theory. Data sets were obtained to compare with the theory starting with 1 part-perthousand (ppth) (w/v) solutions of poly(ethylene glycol) (PEG) calibration standards of different molecular mass (ALO-2774, Phenomenex, Torrance, CA) in deionized water (Nanopure, Barnstead, Dubuque, IA). Experiments varied the two important parameters of the µ-MMS: laser beam radii and flow rate. Two laser beam radii, and three flow rates (i.e., changing tA and tB) were experimentally investigated. Throughout these experiments. the upstream detection position was located 1 mm past the T-junction where the mobile phase and sample streams merge and the downstream detection position was 4.9 cm past the upstream position. Data were collected from the microchip with the laser probe beam radii of 79 and 293 µm for the two different probe beam detection “positions”. Note that since the two probe beams are coherent light, interference patterns and speckling of each beam spot is observed on the face of each PSD. However, since the PSDs measure the average deflection of each beam, these properties of coherent light average out, and did not appear to adversely influence the data measured. A 5-µL volume of each PEG solution was injected on to the microchip via off-chip flow injection sample introduction, where the flow rate of both the sample and the mobile-phase streams were 20 µL/min. For the flow rate experiments, all data were collected with the microchip at the 1-mm position, where ro ) 79 µm. Data were also collected with sample and mobile-phase flow rates at 50 µL/min, and a subsequent experiment was conducted with sample and mobile-phase flow rates at 8 µL/min. For the flow rate experiments, the solutions were run at 500 ppm and 1 ppth to investigate the significance of viscosity effects. RESULTS AND DISCUSSION Before going into the theoretical evaluation, it is useful to provide examples of the data provided by the sensor. In its simplest form, the µ-MMS can be used as a sensitive, reproducible, nondestructive, widely applicable detection technique for microscale and benchtop instrumentation. Figure 2A shows signals collected at the upstream detection position for three 5-µL injections of aqueous solutions of 22 500 molar mass PEG at 50 ppm concentration and representative baselines. Using these data,

Figure 2. (A) Plots of three injections of PEG 22500 with representative baseline segments. Each peak is a 5-µL injection of 50 ppm (w/v) aqueous PEG solution at a flow rate of 20 µL/min. The 3σ limit of detection was calculated to be 0.9 ppm. PEG peaks are offset 0.50 µrad from the baseline sections for clarity. (B) Signal versus time plots of upstream and downstream signals for three different molecular mass PEG solutions. The number after each PEG is the molecular mass (in g/mol). Each PEG solution has an upstream and downstream signal overlaid and aligned in time, where the taller peak is the upstream signal and the shorter peak is the downstream signal for each PEG. Data were collected at a 20 µL/min flow rate with a 5-µL injection volume using PEG solutions at a concentration of 1 ppth. The downstream/upstream ratio R, which provides diffusion coefficient and molecular mass information, is also indicated.

the 3σ concentration limit of detection (LOD) was determined to be 0.9 ppm at the detector for the dn/dc ) 1.3 × 10-4 RI/ppth of this system. Furthermore, as shown in previous work,10 the detector has a large dynamic range. By using both detection positions, the µ-MMS also provides diffusion coefficient information as illustrated by the data in Figure 2B. For each sample injection, a signal is collected at the upstream position (larger peaks) just past the merging point of the two streams and another signal at the downstream position (smaller peaks) some point further down the microchannel along the axis of flow. The downstream/upstream signal ratio of a given analyte gives a diffusion coefficient dependent ratio R (eq 14), which can be readily correlated to molecular mass as demonstrated in Figure 2B. We now turn our attention to applying the theory, primarily through eq 14 to model the µ-MMS, which expresses the ratio R in terms of the probe beam radius, analyte diffusion coefficient, and times after the merging points for the upstream and downstream detection positions. Our purpose was to first explore the predicted dependence of the ratio on the relevant variables and then to compare the theoretically simulated results with experimental data. The first variable to consider is the probe beam radius, ro. Figure 3A shows a plot of ratio R versus the analyte Analytical Chemistry, Vol. 76, No. 10, May 15, 2004

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Figure 4. Ratio R versus molecular mass plots of data with theory plots overlaid. Data at ro ) 79 µm were collected with the probe beams and microchip 1 mm apart. Data at ro ) 293 µm were collected with the probe beams and microchip 10 cm apart. The flow rate was 20 µL/min with tA ) 0.14 s and tB ) 8 s. The theory curves were generated using the model (eq 14) with actual beam widths of 79 and 293 µm; the tA and tB values and the diffusion coefficients for the PEGs related to molecular mass using eq 17. Error bars for the data are too small to be seen in the plot.

Figure 3. (A) Theoretical plots of ratio (R) versus diffusion coefficient for a set of arbitrary analytes as a function of probe beam radius. The numbers next to each curve represent the value of ro (in µm) used to generate that particular curve using eq 14, where tA and tB were 0.14 and 8 s, respectively. (B) Theoretical plots of ratio versus molecular mass for a range of molecular mass in PEG, as a function of beam width. The numbers next to each curve represent the value of ro (in µm) used to generate that particular curve, where tA and tB were 0.14 and 8 s, respectively.

diffusion coefficient D where each curve represents a different beam radius, ro. For this simulation, the times at tA and tB were held constant at 0.14 and 8 s, respectively. The model clearly shows that as the beam radius changes the effective range of the sensor for the differentiation of diffusion coefficients also changes. Although some useful trends can be obtained from Figure 3A, it is also useful to relate the R values to the molecular mass of a particular class of compounds. For this study, we chose to relate the diffusion coefficients to the molecular masses of PEGs. For PEG standards, eq 1719 is applied to convert between PEG diffusion coefficient, DPEG, and molecular mass, M,

DPEG ) (1.25 × 10-4)(M)-0.55

(17)

From the simulated data in Figure 3A, in terms of the molecular mass of PEGs, the relation to the ratio R at the various probe beam radii is shown in Figure 3B. In this figure, the ability to tune the sensor to a particular molecular mass range most suitable for a given application is more evident. For smaller random-coiled analytes, one should use a beam radius of ∼100 µm, while for larger analytes one should use a smaller beam radius, closer to ∼10 µm. Alternatively, as we shall see, one can also tune the mass range by changing the flow rate. Experimental evaluation of the radius dependence was obtained with the microchip placed at two different distances from (19) Murugaiah, V.; Synovec, R. E. Anal. Chem. 1992, 64, 2130-2137.

2730 Analytical Chemistry, Vol. 76, No. 10, May 15, 2004

the GRIN lens assembly. Since the light diverges slightly after leaving each GRIN lens, this procedure gave data at two different beam radii, 79 and 293 µm. Figure 4 is a plot of the ratio R of a series of aqueous PEG standard solutions of different molecular masses collected at these two different beam radii. For each position, the PEG series was run in triplicate, where each run was done with a 5-µL injection loop and the flow rate was 20 µL/ min at both beam radii. The plot also includes lines representing the theoretical simulation generated with eq 14 using experimentally determined values for beam radius and tA and tB. The experimental data show excellent correlation with the theory for the results at a beam radius of 79 µm. The results for the larger beam radius, 293 µm, qualitatively agree with the model, but there is deviation from the theory for the smaller analytes. We suspect this is a consequence of beam radius ro becoming comparable to the upstream detector offset A while (4DtB)1/2 approaches the cell depth w. Both these conditions are contrary to assumptions made in the theory. While the ratio of angular displacements appears to be insensitive to either condition as a consequence of the various averaging processes in the analysis, deviation from the prediction occurs when both happen simultaneously. Generally the agreement between the theory and experiment for the beam radius dependence is very supportive of the theory, and as we shall see, additional support was obtained by the flow rate study to follow. The other important parameters that dictate performance of the µ-MMS according to eq 14 are the times since the merging of the sample and mobile-phase streams, tA and tB. They are controlled by the location of the two probe beams and the flow rate. Recall, tA is the time that has passed since the merging of the two streams at the T-junction and the detection of the analyte at the upstream position, and tB is the time since the merging of the two streams and the detection of the analyte at the downstream position. Figure 5A shows the simulated theoretical plots of the ratio R while changing tA, at two different beam radii, while keeping tB constant at 8 s. This figure shows that, as tA is reduced, the molecular mass range of the detector increases. While this is true for both beam radii, either 10 or 100 µm, it is more important for the smaller beam radius.

Figure 5. (A) Ratio R versus molecular mass simulation plots for PEGs using eqs 14 and 17 at two different beam radii, ro, at three different upstream position times (tA) indicated holding downstream time (tB) constant at 8 s. There are three plots for each beam radius representing the three different tA values. (B) Ratio R versus molecular mass simulation plots for PEGs using eqs 14 and 17 at three different tB indicated, holding tA ) 0.14 s and ro ) 100 µm.

An even more effective way of tuning the molecular mass range for a given analysis is by changing the tB, which can be accomplished either by changing the flow rate (as presented here using only two fixed detection positions) or by implementing multiple sensing beams and, thus, multiple detection positions at a constant flow rate. Future development of the multiple sensing beam approach at constant flow rate has the advantage of not having to adjust the flow rate. Figure 5B shows the theoretical simulation of the ratio R as a function of molecular mass for three different tB values using a constant 100-µm beam radius and tA held constant at 0.1 s. This figure indicates that, by increasing the time between merging and detection at the downstream position, one can readily tune the µ-MMS to a molecular mass range most suitable for a given application. If tB is too short as shown for the 2-s simulation curve, then the molecular mass range is limited to only small, fast-diffusing analytes. As tB is increased to 8 s, it can be seen that the molecular mass range has expanded to include some higher molecular mass compounds without losing the resolution for the smaller analytes. When tB increases further, to the 32-s curve, it can be seen that now even larger analytes can be measured; however, the resolution for smaller analytes is beginning to be lost. For larger analytes with relatively smaller diffusion coefficients, a combination of a smaller beam radius and longer tB may be most appropriate for many applications. Herein, we changed the flow rate with fixed detection positions in order to change tB. Thus, changing the flow rate leads to changes of the same proportion to both tB and tA; however, eq 14 indicates the change in tB is far more important for any typical

operating conditions. On the basis of the simulations of Figure 5, we conclude that changing the flow rate is an effective means of tuning the detector to the class of analytes of interest. With this in mind, the flow rate dependence on the performance of the theoretical model for the µ-MMS was evaluated by performing an experiment using aqueous PEG solutions of varying molecular mass at three different flow rates: 50, 20, and 8 µL/min. The data are shown in Figure 6A, overlaid with simulated curves generated using the model and parameters based on the experimental conditions. All samples were run at both 500 ppm and 1 ppth concentrations. At the slower flow rates of 8 and 20 µL/min, the analytes were not significantly affected by sample stream viscosity, so the data from the two concentrations overlapped. Thus, only 1 ppth data are shown for the 8 and 20 µL/min data, and 500 ppm data have been omitted for clarity. The data for these two slower flow rates demonstrate an excellent correlation to the theoretical curves generated using eq 14 for the experimental conditions. Two other data sets in Figure 6A were taken at 50 µL/min, one at 1 ppth concentration and the other at 500 ppm. These two data sets demonstrate a well-documented viscosity effect of microfluidics:11,13 the volume occupied by each stream after a T-junction is proportional to its viscosity. If this viscosity effect is large enough, it causes a significant shift in the position of the RIG away from the optimal beam probing position in the microchannel. The effect is also more pronounced at the upstream detection position where the concentration gradient is sharper. Also, the viscosity effect is greater for the solutions at higher concentrations and high flow rate since diffusion has had less time to occur, which is what caused 1 ppth solutions at 50 µL/min to deviate significantly from the model. The 500 ppm samples still have some deviation from the model but only for the largest analytes. To explore viscosity effects further, a serial dilution of PEG 11840 was run from concentration of 10 ppth to 100 ppm at the flow rate of 50 µL/min. Figure 6B shows the upstream and downstream signals collected for the PEG 11840 solutions as a function of concentration in a log-log plot. The theory (eq 13) predicts these plots should be parallel with a slope of 1. Both log-log plots are essentially linear; however, the slopes are slightly less than 1, with the downstream signal plot having a slightly steeper slope (0.982) than the upstream signal plot (0.953). This difference in slopes is another way of noting that larger molecular mass analytes, such as the PEG 11840, increase the viscosity of the solution and alter the diffusion rate as concentration increases. In addition to the viscosity effect discussed above, the concentration dependence of the diffusion coefficient may also contribute to the deviation. Another way to view this is to simply plot the ratio R values for the data in Figure 6B as a function of concentration as shown in Figure 6C. In this figure, the black line represents the average ratio value for the first eight solutions with the 3σR error bar due instrument fluctuation noise throughout the work day and the error bars for each R value indicating the short-term runto-run noise. Since the ratio of the signal is ideally concentration independent (see eq 14), one would expect that the ratio value collected for all the concentrations would be the same value. Here one can clearly see the ratio value is constant within the longterm experimental error bar for all the concentrations less than ∼2 ppth and then the ratio value begins to increase. Indeed, the viscosity effect was not observed for all 500 ppm and 1 ppth PEGs Analytical Chemistry, Vol. 76, No. 10, May 15, 2004

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Figure 7. Measured diffusion coefficient from experimental data (eq 15) versus known diffusion coefficient (eq 17), at three different flow rates (the data sets with same symbols as in Figure 6A). The (+) data were collected with a flow rate of 50 µL/min at a concentration of 500 ppm, the 4 data were collected with a flow rate of 20 µL/min at a concentration of 1 ppth, and the b data were collected with a flow rate of 8 µL/min at a concentration of 1 ppth. The black line through the graph represents a line with a slope of 1 (not a best-fit line), which the data would fall on if their measured diffusion coefficients perfectly matched the known values.

Figure 6. (A) Ratio R versus molecular mass plots for PEGs with experimental data (data points) and theoretical curves (using eqs 14 and 17) overlaid at different flow rates: (0) 50 µL/min with solutions at a concentration of 1 ppth; (+) 50 µL/min with 500 ppm solutions; (4) 20 µL/min with 1 ppth solutions; (b) 8 µL/min with 1 ppth solutions. Data sets under the same conditions as 4 and b were also collected at 500 ppm, and the data points overlapped completely with the 1 ppth data. Thus, for clarity, only the 1 ppth data are shown at 8 and 20 µL/min flow rates. All simulation curves are plotted using the measured ro ) 79 µm and tA and tB determined for each flow rate. (B) Signal versus Concentration (log-log plots) for a serial dilution set of PEG 11840 solutions at both the upstream and downstream detection positions. Each concentration was injected three times with a 5-µL injection loop. Other conditions: flow rate 20 µL/min; probe beams had a radius of 79 µm. The two lines on the plot are leastsquares fits, where the fit for the upstream signal had a slope of 0.953 and the fit for the downstream data had a slope of 0.982. The uncertainty in the upstream slope was 0.005 and the uncertainty in the downstream slope was 0.007. (C) Ratio R versus concentration for a serial dilution of PEG 11 840 g/mol solutions (upstream and downstream data in part B). The black line represents the average ratio of the first eight points with a 3σ confidence interval of 0.036 indicated.

at the 8 and 20 µL/min flow rates (data sets in Figure 6A). Note, also, that the viscosity effect was not observed at even the 10 ppth 2732 Analytical Chemistry, Vol. 76, No. 10, May 15, 2004

solution for PEG 106 in previous work,13 because there is relatively little increase in the viscosity for the low molecular mass PEG. In general, the viscosity effect can be sufficiently minimized by using sufficiently low concentration and low flow rate. An emerging application for this detector is to use it as a means for determining unknown diffusion coefficients. A recently published paper describes the importance of making these measurements for the continued development of microfluidic technology.20 From eq 15 one can see that, from a given ratio R value and known or determined beam radius ro and tA and tB one should, in principle, be able to universally calibrate the sensor to determine unknown diffusion coefficients for a wide range of analytes. This potential is demonstrated in Figure 7 where three sets of data from Figure 6A have been plotted as the measured diffusion coefficient (calculated from experimental data using eq 15) versus the known diffusion coefficients (calculated from the known molecular masses using eq 17). Theory dictates this plot should show the three data sets at the three different flow rates all fall along a line with a slope of one. Indeed, Figure 7 shows that the µ-MMS makes excellent measurements for the diffusion coefficients, especially at the two slower flow rates where the data are completely overlapped and near the ideal line with a slope of 1 (not a best-fit line). There is some slight deviation for the 50 µL/min data for the higher molecular mass PEGs, consistent with the viscosity effect that has been previously discussed. The correlation of the measured diffusion coefficient with the known diffusion coefficient clearly shows the potential of this sensor to make fast and reliable diffusion coefficient measurements. CONCLUSIONS This work has introduced a rigorous theoretical description for the µ-MMS detection technique and provided experimental results that confirm the theory. The µ-MMS has been shown to be a sensitive, multidimensional detector that can, in principle, be applied to a number of different separation or flow injection (20) Culbertson, C. T.; Jacobson, S. C.; Ramsey, J. M. Talanta 2002, 56, 365373.

techniques. The detector has also been introduced as an instrument that can be used to measure unknown diffusion coefficients. One should be able to calibrate with one class of compounds and then apply the calibration to another class of compounds, which is extremely valuable for the development of microfluidics. Such a system can be readily implemented in µ-TAS to provide a fast, multidimensional detection technique that can be used to enhance the high-throughput capabilities of microfluidic devices. Continued development of this sensor will focus on its integration into µ-TAS as well as the investigation of new chemical analysis for which it is well suited.

ACKNOWLEDGMENT We thank the Center for Process Analytical Chemistry (CPAC), a National Science Foundation initiated University/Industry Cooperative Research Center at the University of Washington for financial support.

Received for review December 4, 2003. Accepted March 2, 2004. AC030405C

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