ARTICLE pubs.acs.org/ac
Device for Rapid and Agile Measurement of Diffusivity in Micro- and Nanochannels Alessandro Grattoni,*,#,† Jaskaran Gill,#,† Erika Zabre,#,† Daniel Fine,#,† Fazle Hussain,#,‡ and Mauro Ferrari#,†,§ #
The University of Texas Health Science Center at Houston, 1825 Pressler Street Suite 537A, Houston, Texas, 77030, United States Department of Nanomedicine, Methodist Hospital Research Institute, 6670 Bertner Avenue, Houston, Texas 77030, United States
†
bS Supporting Information ABSTRACT: The lack of a viable theory for describing diffusivity when fluids are confined at the micro- and nanoscale [Ladero et al. Chem. Eng. Sci. 2007, 62, 666678; Deen AIChE J. 1987, 33, 14091425] has necessitated accurate measurement of diffusivity (D) [Jin and Chen Chromatographia 2000, 52, 1721; Nie et al. Science 1994, 266, 10181021; Durand et al. Anal. Chem. 2009, 81, 54075412], crucial for a host of micro- and nanofluidic technologies [Grattoni et al. Curr. Pharm. Biotechnol. 2010, 11, 343365]. We demonstrate a rapid and agile method for the direct measurement of diffusivity in a system possessing 104 to 105 precisely fabricated channels with characteristic sizes (β) ranging from micro- to nanometers. Custom chambers allowed us to measure the diffusivity in a closed unperturbed system using UV/vis spectroscopy. D was measured for rhodamine B (RhoB) in aqueous solution in channels of 200 and 1 μm, as well as 13 and 5.7 nm. The observed logarithmic scaling of diffusivity with β, in close agreement with prior experiments, but far from theoretical prediction, surprisingly highlights that diffusivity is significantly altered even at the microscale. Accurate measurement of D by reducing the size of the source reservoir by 3 orders of magnitude (from 150 μL to 910 nL) proves that a substantial reduction in measurement time (from 7 days to 40 min) can be achieved. Our design thus is ready for rapid translation into a standard analytical tool useful for multiple applications.
C
oncentration-driven molecular diffusion is a fundamental mass transfer phenomenon that is commonly described by Fick’s laws of diffusion.7 The diffusion coefficient or diffusivity, D, relates the mass flux to the concentration gradient and indicates the speed with which molecules diffuse through a medium. It is most often approximated by the StokesEinstein equation which assumes that the solute is a rigid sphere diffusing in a continuum of solvent at a low Reynolds number and infinite dilution: D¼
kB T 6πηRo
ð1Þ
where kB is Boltzmann’s constant, T is the temperature in kelvin, η is the absolute viscosity of the solvent, and Ro is the radius of the spherical solute.8 The StokesEinstein equation and its many variants for special conditions have thus far proven to be accurate for most diffusive problems and applications. However, numerous technologies, such as devices for the controlled release of drugs,911 employ micro- and nanofluidic systems that reveal increasing difference between the “measured” diffusivity and the calculated value with increasing fluid confinement.1214 In particular, significant deviations occur as the size of the system approaches the size of a single molecule, increasing the importance of moleculesurface interactions.1,15 r 2011 American Chemical Society
The determination of diffusivity is essential to the development of drug delivery as well as lab-on-a-chip micro- and nanofluidic devices for a variety of applications including proteomics,16,17 fluid mechanics,18 separation science,19 and the controlled delivery of molecules.6,15,20 A rapid, high-throughput device for the determination of diffusivity could reduce the time and cost of drug development.20 As such, several methods have been explored.3,5 The T-sensor is a microfluidic device that introduces two pressure-driven streams side by side to determine the analyte concentration in the sample stream.21,22 As the streams flow in parallel, the sample and receptor streams form an interdiffusion region that can be analyzed optically to determine concentration.23 The method described by Pappaert et al.,15 in which a gradient is fluorescently measured after dragging a rhodamine B “plug” into the observation field of a fluorescent microscope by shear flow in 2.95 μm deep microchannels, was shown to be effective for high velocity flows in channel depths down to 260 nm. Culbertson et al.42 used the cross chip design to statically extract the concentration profile of a rhodamine 6G (Rho6G) analyte plug from images taken over time and then used that profile to calculate diffusivity. Durand et al.13 measured Received: December 29, 2010 Accepted: February 24, 2011 Published: March 24, 2011 3096
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Figure 1. Schematics of the embodiments EA (A) and EB (B) comprising the diffusivity measuring devices.
the lateral dispersion of molecules exiting from a nanoslit into a microchannel. Finally, a method that has shown great potential is NMR.24,25 However, NMR measures a self-diffusion coefficient, related to the Brownian motion of a single molecule, which may significantly differ from the diffusivity of molecules in solutions, wherein D accounts for their collective behavior.26,27 The majority of the aforementioned techniques utilize fluorescence imaging, limited to fluorescent molecules. Molecules that do not present fluorescence signatures must be conjugated with a fluorophore. This may cause variation in the property of molecules and their diffusive transport. As an example, the conjugation of fluorescein isothiocyanate to bovine serum albumin (FITC-BSA) changes the effective charge of the molecule from 7 to 18 (at pH 7.4)28,29 and with it the extent of interactions with charged surfaces or other charged or polar analytes in solution. Additionally, fluorescence techniques may be limited by photobleaching of the dye30 and may require corrections in relation to background fluorescence.31 Finally, these methods require nonstandard, expensive equipment and have limitations on the accessible length scales of confinement or diffusive properties of the molecules. Absorption spectroscopy is a powerful analytical tool that is specific to a molecule’s unique absorption spectrum, delivers quantitative data, and is simple, inexpensive, and quick to perform, even for the nanoscale. Kievsky et al.32 outlined an innovative, albeit complicated, method to study diffusion in silica nanotubes of approximately 3 nm in diameter using regular absorption spectroscopy. In this paper, we demonstrate a novel device for quick, agile, and direct measurement of diffusivity at the nanoscale by measuring the timed release of molecules through precisely fabricated silicon micro- and nanochannels by means of custom UV/vis spectroscopy devices. In particular, we studied RhoB diffusivity in channels of 200 and 1 μm as well as 5 and 13 nm. A mathematical model of the silicon chip was also developed and used for comparison with the experimental data to yield diffusivity. We validated our results by comparison with literature data for RhoB and Rho6G. Finally, we further demonstrated our approach with a smaller scale diffusion device designed to reduce the volume of samples and time for measurement.
’ EXPERIMENTAL SECTION Diffusion Device Design. A custom designed device was developed for the analysis of nanoscale diffusivity through nanochannel membranes. Two embodiments, EA and EB, were
designed and fabricated (see Figure 1), both possessing a source and a sink reservoirs separated by a nanochannel membrane. An optical window in the sink reservoir allows measurement of the concentration of analytes by correlation with the UV-adsorption of the solution. In the case of EA, the sink reservoir is obtained by epoxying (OG116-31 UV-curing epoxy, Epoxy Technology, Inc., Billerica, MA) a 4.45 mL UV-transparent cuvette (BrandTech Scientific, Inc., Essex, CT) into a groove machined into the lower part of a hollow stainless steel (SS316L) body. The top reservoir of EA, presenting a maximum capacity 370 μL, is machined along the axis of a cylindrical body that is similarly composed of SS316L. The membrane is housed between the metal bodies by two silicone rubber O-rings (Apple Rubber, Lancaster, NY, USA) that are sealed once the two bodies are secured together by means of two SS316L M3 screws. In the case of EB, the 250 μL sink reservoir is obtained by epoxying UV-transparent windows (obtained by cutting square sections off of a UV-cuvette) onto the sides and bottom of a hollow SS316L body. The source reservoir, 910 nL in volume, was manufactured using a custom molded silicone elastomer (PDMS; Sylgard 184, Dow Corning, Midland, MI). The membrane is sealed between the source and sink reservoirs by means of an O-ring (bottom side) and the PDMS (top side). A 6061 aluminum alloy shell was used to homogeneously compress the PDMS body against the lower part of the device while tightening the screws. Both embodiments are tightly sealed to guard against evaporation. In EA, the analyte reservoir is hermetically capped with a silicone rubber plug (Mocap, Inc., St. Louis, MO), while in EB the PDMS body provides the waterproof sealing of the source reservoir. Membrane Description. For the experimental analysis, nanochannel membranes were microfabricated by means of a sacrificial layer technique using silicon-on-insulator (SOI) substrates as described by Fine et al.11 Nanochannels of 5 and 13 nm were produced in several different configurations of varying width and length parallel to the membrane surface and on top of the device layer of the SOI. Macrochannels and microchannels were etched perpendicular to the membrane surface to connect the nanochannels to the inlet and outlet sides of the membrane. Figure 2A shows a schematic of the membrane structure. In order to measure the diffusivity in the microchannels, a set of membranes were produced without the nitride nanochannel capping layer (Figure 2B). To obtain the diffusivity in the macrochannels, 3097
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Table 1. Characteristic Sizes of the Membrane and of the Micro- and Nanochannelsa width
length
depth
no. CH
membrane
6 mm
6 mm
702
macrochannels
200
200
670
161
inlet microchannels
3
30
414414
config μCH 1 config C1 inlet microchannels
8
1
30
144900
outlet microchannels
8
3
1.7
75348
nanochannels
8
1
5.7, 13 nm
144900
inlet microchannels
5
4
30
74060
outlet microchannels
5
5
1.7
40733
nanochannels
5
5
13 nm
74060
config C2
a
The units are in micrometers unless otherwise specified. The number of channels per membrane (no. CH) is also listed for each tested configuration.
Figure 2. Schematics of the membrane: (A) configurations C1 and C2 of the complete membrane with nanochannels (nCH) and microchannels (μCH); (B) silicon microchanneled membrane (config μCH); (C) polished macrochannel (MCH) membrane.
the entire top portion of the membrane was removed by mechanical polishing (Figure 2C). Table 1 lists the structural details of the membrane. Prior to diffusion testing, each membrane was cleaned in piranha solution (30% H2O2:70% H2SO4) twice (see the Supporting Information), observed using light microscopy, and tested using convective nitrogen flow to assess its quality, as described elsewhere.33 The root mean squared surface roughness of the nanochannel surfaces was determined by atomic force microscopy (AFM), and the nanochannel depth was verified by analyzing cross sections of the device using tunneling and scanning electron microscopy (TEM, SEM) (see the Supporting Information). Experimental Setup. Prior to the experiments, all nondisposable parts of the diffusion devices were cleaned with soap and thoroughly rinsed with DI water before being autoclaved at 121 C for 30 min. RhoB powder (Sigma-Aldrich, St. Louis, MO) was mixed with Millipore deionized (DI) water at a concentration of 7.5 mg/mL. The membranes were wetted in isopropyl alcohol for 2 h and then rinsed in Millipore water for 14 h prior to diffusion testing. The sink reservoir of the device was carefully loaded with Millipore DI water to prevent the entrapment of bubbles to a liquid level that slightly protruded through the bottom O-ring. The wetted membrane was then clamped between the two bodies. Excess water was removed from the source reservoir and replaced with RhoB solution. In the case of EA, the device was capped with a silicone rubber cap pierced with a venting needle (25 gauge), to allow for the complete escape of air during the insertion of the cap. In EB, the loading was performed with two needles (30 gauge), one of which used to inject the solution and the other as a venting conduit. Table 2 lists the solution
volumes and the number of device replicates used for each experimental configuration. The experiment was performed in a controlled environment at 23 ( 0.2 C over a period of time required to exceed 98% of the released amount (approximately 2 h for EB and maximum 20 days for EA). Teflon-coated micro stir bars were employed to ensure solution homogeneity after verifying their effectiveness independently (see the Supporting Information). The absorbance of the solution in the sink reservoir of each device was measured several times with a UV/vis spectrophotometer (DU 730, Beckman Coulter, Inc.) at a peak wavelength of 401 nm (see the Supporting Information). Absorbance data was normalized and the corresponding analyte concentration determined using a standard curve. Diffusivity Calculation. The experimental mass release data were fitted with the following equation:34 mðtÞ ¼ ðCIN COUT Þ
VIN VOUT ð1 eλt Þ VIN þ VOUT
ð2Þ
where m(t) is the mass released over time, cIN and cOUT are the concentrations of RhoB in the source and sink reservoir at time = 0, and VIN and VOUT are the volume of the source and sink reservoirs, respectively. λ is given by: 1 VIN λ¼ 1þ ð3Þ Rtot VIN VOUT where Rtot is the total resistance of the fluidic network. The individual resistive components corresponding to each branch of the fluidic system are calculated as: Ri ¼
Li Dexpi Ai
ð4Þ
where Li and Ai are the length and cross-sectional area of the ith fluidic channel segment. The relationship between Rtot and Ri is reported in the Supporting Information. Dexpi is the diffusivity of the ith segment, which was extrapolated for each channel size from the experimental data by minimizing the sum of the quadratic differences between fitting curve and experimental values. First, by using eqs 13, Dexpi was extrapolated for the 3098
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Table 2. Experimental Details analyte
conc (mg/mL)
mem config
nCH size (nm)
device
source vol (μL)
sink vol (μL)
no. repl
RhoB
7.5
macrochannels
EA
50
2222
3
RhoB
7.5
config μCH
EA
50
4438
4
RhoB
7.5
config C1
EA
150
4438
3
RhoB
7.5
config C1
13
EA
150
4438
3
RhoB
7.5
config C2
13
EA
150
4438
3
RhoB
7.5
config C1
13
EB
250
1
5.7
0.91
macrochannels (i.e., DexpmCH). Second, this DexpmCH was used in eqs 13 to determine Dexpi in the microchannels (i.e., DexpμCH). These DexpmCH and DexpμCH terms were then used in eqs 13 to determine Dexpi in the nanochannels (DexpnCH). Equations 24 imply that a “pseudo steady state” and linear concentration gradient is established along the channels. To verify this assumption, we developed and employed a finite element model (FEM) of the nanochannel membrane (see the Supporting Information) and evaluated the maximum time required for a stable gradient to form within the membrane and the profile of the gradient along the channel length. The results corresponding to the “slowest” experimental configuration (configuration C1, 5.7 nm) show that a linear, pseudo steady state concentration gradient is established within 5 min, representing negligible—0.1% and 4.2%—of the measurement times with EA and EB, respectively. These results validated our assumptions.
’ RESULTS AND DISCUSSION Rhodamine B (RhoB) is neutrally charged at pH 7 and used extensively as a dye in studies of fluid transport. RhoB presents a size comparable to Rho6G (1.1 1.5 nm2)32 and exhibits both fluorescence and absorbance signatures of relevant intensity, which makes it an attractive choice for experimental testing. A comparison of RhoB with Rho6G was possible based on their identical molecular weight (479.01 g/mol) and similar molar absorptivity (10.5 104 and 10.7 104 L/(mol cm), respectively) that yields comparable sensitivity of the spectroscopically determined concentration. However, while RhoB is unchanged at pH 7 due to the presence of a free carboxyl group, Rho6G presents a positive charge at the same pH.35 The positive charge of Rho6G may significantly affect its diffusive transport in silica channels with β smaller than a few tens of nanometers due to confinement-enhanced interactions with the negatively charged oxidized surface (1 to 5 μC/cm2).36,37 In our analysis, β was defined as β = 4A/P, where A and P are the CS area and wetted perimeter of channels, respectively. This phenomenon can be neglected in microchannels where surface to volume ratios become orders of magnitude smaller. The experimental results of RhoB diffusion through 200 and 1 μm microchannels and 5.7 and 13 nm nanochannels are shown in Figure 3. Figure 3 shows the average of data obtained from a number of channels with length scales on the order of 102 for the macrochannels, 105 for the microchannels, and 104 for the nanochannels. Small standard deviations among experimental replicates may be ascribed to dimensional tolerances of the microfabrication process. The profiles of the percentage released amount over time are distinctly exponential, indicating a concentration-dependent diffusive transport well-described by Fick’s laws of diffusion. Unconstrained diffusive behavior was expected because of the large ratios between the smallest size of channel cross
Figure 3. Cumulative percentage release profiles over time for rhodamine B through 200 μm macrochannels, 1 μm microchannels, and 5 and 13 nm nanochannels with their respective Fickian model curves. Percentage standard deviations of data (STDEV) are listed in the graph legend.
section and molecule hydrodynamic diameter—ranging from 4.4 (the case of 5.7 nm) to 3.5 104 (the case of 200 μm). In previous work,11 we observed that a size ratio smaller than 3 was needed to transition from Fickian-like to saturated diffusion. Figure 3 also shows the Fickian modeling curves employed to extrapolate the diffusivity from the experimental data. The D values obtained from these experiments are listed in Table 3 together with experimental values obtained from the literature and found to be in close agreement. For a valid comparison with our results, the literature D values obtained in 100% aqueous solution were normalized with respect to the temperature (T = 23 C) and the viscosity of water at 23 C (μH2O = 0.937 mPa 3 s) by employing the StokesEinstein eq 1. A similar normalization was performed (T = 23 C and appropriate viscosity) for literature results obtained in ethanol (EtOH) and methanol (MeOH) solutions. The results obtained in aqueous solution and EtOH/MeOH were not directly compared because of the large difference in viscosity causing the normalization to ultimately be subject to significant error. In addition, different dipoledipole interactions between analyte and solvent may also cause significant differences in the diffusive transport.38 Figure 4 shows a plot of diffusion coefficient versus β for our experimental data and the normalized values from the literature for experiments performed in 100% aqueous solution, along with several theoretical predictions. To compare results obtained from channels presenting different geometric cross sections (CS) and aspect ratios, we considered β to be equal to the hydraulic diameter of the channels. By fitting our 100%-aqueous-solution data alone or together with literature values, logarithmic 3099
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3100
a
5.88
0.0029
18.55
18.55
2.95 700
0.0029 L
12.5 36
12.5 36
18.55
12.5 36
Rho6G
Rho6G
Rho6G
Rho6G
Rho6G
Rho6G
Rho6G
Rho6G Rho6G
Rho6G
Rho6G
Rho6G
RhoB
RhoB
RhoB
RhoB RhoB
RhoB
RhoB
RhoB
RhoB
6
6
2.88 10
6
2.68 106
2.71 10
6
2.69 10
4.90 10
15
2.80 10
25
25
25
25
25
25
23
2.50 106 6
25 23
25
25
25
25
20
2.70 106 2.90 106
4.59 10
6
4.14 10
2.90 10
14
2.95 10
6
2.77 10
50/50 v/v% H2O/MeOH
50/50 v/v% H2O/MeOH
50/50 v/v% H2O/MeOH
50/50 v/v% H2O/MeOH
75/25 v/v% H2O/EtOH
75/25 v/v% H2O/EtOH
50/50 v/v% H2O/MeOH þ NaCl
50/50 v/v% H2O/MeOH þ Na2B4O7 EtOH
H2O
H2O
H2O
H2O
PBS
1.534
1.534
1.534
1.534
1.57
1.57
1.65
1.534 1.14
0.896
0.896
0.896
0.896
0.968
1.65
1.65
1.65
1.65
1.69
1.69
1.65
1.65 1.14
0.937
0.937
0.937
0.937
0.915
0.937
4.36 10
2.66 106
2.48 106
2.50 10
6
2.48 10
6
4.52 10
15
2.58 106
2.50 106
2.49 106 2.90 106
6
3.93 10
6
2.75 10
14
2.80 10
6
2.84 10
6
4.06 106
6
0.896
4.27 106 H2O
1.36 106 1.29 106
1.36 10 1.29 106 25
1.30 106
1.30 10 6
3.86 10
5.90 107
0.937
6
0.937
5.90 107
H2O
6
(cm2/s)
Norm D
2.30 106
23 ( 0.2
(mPa 3 s)
μ at 23 C
2.30 106
3.86 10
3941
FLUO-detection length variation
FLUO-E-field potential
FLUO-stopped flow
FLUO-static imaging
UV-spectroscopy
FLUO-single molecule detection
FLUO-correlation spectroscopy
FLUO-single molecule detection FLUO-correlation spectroscopy
FLUO-E-field potential
FLUO-static imaging
UV-spectroscopy
FLUO-static imaging
FLUO-sheath flow T-Sensor
FLUO-static imaging
UV-spectroscopy EB UV-spectroscopy EA
UV-spectroscopy EA
UV-spectroscopy EA
UV-spectroscopy EA
UV-spectroscopy EA
Viscosity data for EtOH and MeOH solutions were found in the literature.
18.55
18.55
12.5 36
18.55
12.5 36
12.5 36
0.0029
0.0029 L
100000
1240.60
825 2500
¥
18.55
12.5 36
100000
0.026 0.026
0.013 (C1) 0.013 (C2)
¥
0.026
0.013 (C1)
16.4 100000
0.010
0.0057 (C1)
10 45 ¥
1.5
200
1 3 (μCH)
solvent
200 200 (mCH)
6
T (C)
μ D (cm2/s) (mPa 3 s)
technique
size β (μm)
molecule
characteristic
channel
size (μm)
Culbertson (2002)42
Kievsky (2008)32
Nie (1994)4
Fister (1998)43 Hansen (1998)44
Culbertson (2002)42
Kievsky (2008)32
Pappaert (2005)15
Munson (2005)31
Culbertson (2002)42
our data
source
Table 3. Summary of Literature Values for Rhodamine B and 6G Diffusivity Obtained from a Variety of Techniques in Various Buffers and Confinement Length Scalesa
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Figure 4. Normalized diffusivities for rhodamine B and 6G in 100% aqueous solutions for both our experimentally determined values and those found in the literature (T = 296.13 K).
relationships between diffusivity and β were found that are described by the relationships DO = 2.98 107 ln(β) þ 2.27 106 and DA = 3.04 107 ln(β) þ 2.37 106, respectively. These curves present a maximum difference in slope of 2% in the considered range of β. Both relationships indicate negligible diffusivity values when the nanochannel approaches the molecule size and present plateau values of approximately 4 106 cm2/s in free solution. Surprisingly, significant reductions of the diffusivity were observed even at the microscale (approximately 30% and 40% lower values as compared to D in free solution, for β = 1.5 and 5.88 μm, respectively). This result contrasts with the idea that diffusivity is substantially affected by confinement only when the sizes of the channels and diffusing molecules are within 2 orders of magnitude.4547 While the diffusion coefficients determined with the fluorescent technique described by Pappaert et al.15 showed close agreement with the logarithmic fitting curve, an average difference of approximately 27% was observed with the values measured by Culbertson et al. (ranging from 4.14 106 to 4.59 106 cm2/s)42 using a different fluorescence-based method. Errors in these higher values of D may be related to the photobleaching of RhoB and Rho6G. A lower value of D was extrapolated by Munson et al.31 also employing fluorescence imaging, in this case of sheath flow in a T-sensor.21,22 The technique allowed Munson to prevent the results from being affected by background fluorescence signals coming from adsorbed fluorophore on the channel surface. However, the extrapolation of the diffusion coefficient required a significant number of corrections to account for effects in the flow development which ultimately may have affected the final value of D. Significant deviation from the logarithmic fit can be seen at lengths scales in the low single digit nanometers for both our data and the results obtained by Kievsky et al.32 This variation may be ascribed to confinement enhancement resulting from molecular interactions with the silica surface, adsorptiondesorption dynamics and, in the case of Kievsky et al., single file diffusion in nanochannels with a diameter of only 2.3 times larger than the Rho6G.
Figure 5. Normalized literature data of diffusivities for rhodamine 6G in methanol and ethanol solutions at T = 296.13 K.
Figure 5 shows the comparison of literature data obtained in EtOH and MeOH solutions. The D values are substantially lower than observed for aqueous solutions. This is mostly related to the 43% and 45% larger viscosities of H2OMeOH (η = 1.65 mPa 3 s for 50:50 v/v% H2O/MeOH)40 and H2OEtOH (η = 1.69 mPa 3 s for 75:25 v/v% H2O/EtOH),39 respectively, as compared to H2O at 295.15 K. Despite this difference and the smaller amount of available data, Figure 5 shows a similar trend of diffusivity scaling with β. Finally we compared the experimental data with theoretical predictions. By considering a rhodamine of radius R0 = 6.3 Å, the StokesEinstein predictions (eq 1; D = 3.65 106 cm2/s and 2.07 106 for aqueous and EtOH/MeOH solutions, respectively) underestimate the diffusivity in free diffusion by approximately 10% and 21% for 100% aqueous and EtOH/ MeOH solutions, respectively. Similarly, a difference of 16% is 3101
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Figure 6. Comparison of results obtained with devices presenting different source and sink volumes.
observed between aqueous data and the prediction of the semiempirical WilkeChang48 modification to StokesEinstein (D = 3.34 106 cm2/s) calculated as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! φMW sol 8 ð5Þ D ¼ 7:4 10 T ηVsolute 0:6 Where φ represents the association factor of the solvent (2.6 for H2O),49 MWsol is the solvent molecular weight and, Vsolute is the molar volume of the solute. Other studies have developed hindrance factors to take into account the effect of confinement on molecular diffusion. Such corrections, applicable to specific cases,2 are shown to overestimate the diffusion coefficient of RhoB and Rho6G in the range of β considered in this study. Specifically, we considered the StokesEinstein diffusivity values corrected by a hindrance factor H45,46 for neutral spheres in nanoslits: H ¼ ð1 λÞ½1 1:004λ þ 0:418λ3 þ 0:21λ4 0:169λ5 þ 0ðλ6 Þ
ð6Þ
where λ = 2R0/h and h is the depth of the nanoslits. Finally, we considered the Renkin’s factor R47 for the correction of D: R ¼ ð1 λÞ2 ½1 2:104λ þ 2:09λ3 0:95λ5 Þ
ð7Þ
Both corrections predict D variations smaller than 1% for β larger than 500 nm. The substantial discrepancy between theories and experimental data highlights the importance of an accurate experimental measurement of confined diffusivity. To analyze the speed and volume limits of our approach, we developed the device EB with source and sink volumes of 0.91 and 250 μL. This second prototype was developed to shorten the highly accurate but lengthy measurement times needed for EA as compared to fluorescence techniques. In the case of EA, our measurements reached 95% of the released amount in a time ranging from approximately 7 h (the case of 200 μm macrochannel membranes) to 7 days (the case of 5.7 nm membranes). As expected, we were able to greatly reduce measurement time from 7 days to approximately 40 min (at 95% of released amount) when performing the experiment with a 13 nm membrane. Figure 6 shows the comparison of the percentage
cumulative release of RhoB through 13 nm membranes obtained with EA and EB. Despite the drastic reduction of in volume, the measurement provided an accurate measurement of D = 1.36 106 cm2/s with a variation of only 5.1%. This experiment demonstrated that our approach can be used for rapid measurements of D with a minimal amount of sample (in the nanoliter scale), increasing its appeal for the analysis of expensive analytes. A further reduction of the volume of the reservoirs would result in higher uncertainty in D. Our device, combined with the micronanochanneled silicon membrane, provides a tool for agile, reproducible, and direct measurement of the effective diffusivities of molecules and particles by means of UV/vis/NIR absorption spectroscopy. Regardless of the presence of a fluorescence signature, absorption allows the measurement of the effective diffusivities of molecules and particles in a wide range of β (from 2 nm to a few millimeters), as well as the derivation of their scaling function that accounts for analyte interactions with the walls and hindrance effects. High statistical relevance is achieved through averaging over 104105 parallel channels that present tight dimensional tolerances and low surface roughness (root mean squared roughness = 0.15 and 0.35 nm for top Si3N4 and bottom SiO2 surface, respectively).11 Experimental analysis did not require sampling of the sink solution, which allowed the measurement of D within an unperturbed system that minimizes experimental error. The inexpensive testing device EA can accommodate a range of disposable cuvette sizes, from microcuvettes (V = 70850 μL) to macrocuvettes (V = 4.5 mL), and is compatible with automatic measurements using standard spectrophotometers. Given the reduced transmittance of disposable cuvettes in the wavelength range from 190 to 240 nm, quartz cuvettes can also be used for molecules with absorbance spectra in this range at increased expense. This flexibility of design allows for a minimization of the total experimental cost and measurement time as compared to fluorescence microscopy techniques and a significant reduction in complexity of the measurement setup that does not require specific expertise. Cost could be reduced even further with a completely disposable design in which additional sensors such as pH meters and pressure transducers could be implemented.
’ CONCLUSIONS We have demonstrated a rapid and agile system for determining the diffusivity in fluidic systems with characteristic lengths from as low as a few nanometers to as high as hundreds of micrometers. The self-contained custom diffusion chambers allow the measurement of diffusion dynamics without the perturbations related to sink reservoir sampling by measuring molecular release externally using UV/vis spectroscopy. Diffusion chamber operation is demonstrated with, although not limited to, silicon based fluidic channel membranes produced using the precision and accuracy of silicon microfabrication. These membranes not only possess tight production tolerances but allow better statistical sampling as well due to their high channel densities. The diffusion chambers are characterized using rhodamine B, a commonly used fluorescent dye that allows for an easy comparison with the literature. The diffusivity determined by both our techniques and those found in the literature scale logarithmically with β, a result that demonstrates significant deviation from classical diffusion theory even at the microscale. An understanding of this phenomenon will be crucial to realize the full potential of micro- and nanofluidic technologies. 3102
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’ ASSOCIATED CONTENT
bS
Supporting Information. Details of various steps of the experimental analysis presented in the manuscript. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*Mailing address: 6670 Bertner Ave, R7-119 Houston, TX, 77030. E-mail:
[email protected]. Fax: (713) 441-7438. Present Addresses ‡
Department of Mechanical Engineering, University of Houston, 4800 Calhoun Road N207D, Houston, Texas 77204, United States § Department of Bioengineering, Rice University, 6500 Main Street, Suite 135, Houston, Texas 77005, United States
’ ACKNOWLEDGMENT The first two authors contributed equally to this work. The authors are grateful to Arturas Ziemys for his intellectual contribution in the development of the device. This project has been supported with federal funds from NASA (NNJ06HE06A and NNX08AW91G) and the Department of Defense (DODW81XWH-09-1-0212), as well as funds from State of Texas Emerging Technology Fund, NanoMedical Systems (NMS). ’ REFERENCES (1) Ladero, M.; Santos, A.; Garcia-Ochoa, F. Chem. Eng. Sci. 2007, 62, 666–678. (2) Deen, W. M. AIChE J. 1987, 33, 1409–1425. (3) Jin, W.; Chen, H. Chromatographia 2000, 52, 17–21. (4) Nie, S.; Chiu, D. T.; Zare, R. N. Science 1994, 266, 1018–1021. (5) Durand, N. F.; Dellagiacoma, C.; Goetschmann, R.; Bertsch, A.; M€arki, I.; Lasser,, T.; Renaud, P. Anal. Chem. 2009, 81, 5407–5412. (6) Grattoni, A.; Fine, D.; Ziemys, A.; Gill, J.; Zabre, E.; Goodall, R.; Ferrari, M. Curr. Pharm. Biotechnol. 2010, 11, 343–365. (7) Fick, A. Ann. Phys. Chem. 1855, 170, 59–86. (8) Einstein, A. Investigations on the Theory of the Brownian Movement, 1st ed.; Dover Publications: USA; 1956. (9) Martin, F.; Walczak, R.; Boiarski, A.; Cohen, M.; West, T.; Cosentino, C.; Shapiro, J.; Ferrari, M. J. Controlled Release 2005, 102, 123–133. (10) Grattoni, A.; Shen, H.; Fine, D.; Ziemys, A.; Gill, J. S.; Hudson, L.; Hosali, S.; Goodall, R.; Liu, X.; Ferrari, M. Pharm. Res. 2010, 29 (2), 292–300. (11) Fine, D.; Grattoni, A.; Hosali, S.; Ziemys, A.; De Rosa, E.; Gill, J.; Medema, R.; Hudson, L.; Kojic, M.; Milosevic, M.; Brousseau Iii, L.; Goodall, R.; Ferrari, M.; Liu, X. Lab Chip. 2010, 10, 3074–3083. (12) Strychalski, E. A.; Levy, S. L.; Craighead, H. G. Macromolecules 2008, 41, 7716–7721. (13) Durand, N. F.; Bertsch, A.; Todorova, M.; Renaud, P. Appl. Phys. Lett. 2007, 91, 203106–203106-3. (14) De Santo, I.; Causa, F.; Netti, P. A. Anal. Chem. 2010, 82, 997–1005. (15) Pappaert, K.; Biesemans, J.; Clicq, D.; Vankrunkelsven, S.; Desmet, G. Lab Chip. 2005, 5, 1104–1110. (16) Cooper, J. M.; Johannessen, E. A.; Cumming, D. R. S. Bridging the gap between micro and nanotechnology: Using Lab-on-a-chip to enable nanosensors for genomics, proteomics, and diagnostic screening. Network And Parallel Computing, Proceedings, 3222, 2004; pp 517 521. (17) Marko-Varga, G.; Nilsson, J.; Laurell, T. Electrophoresis 2003, 24, 3521–3532. (18) Obeid, P. J.; Christopoulos, T. K. Crit. Rev. Clin. Lab. Sci. 2004, 41, 429–465.
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