Intrinsic Viscosity of Polymers and Biopolymers Measured by

Benjamin A. Bircher , Luc Duempelmann , Kasper Renggli , Hans Peter Lang , Christoph Gerber , Nico Bruns , and Thomas Braun. Analytical Chemistry 2013...
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Anal. Chem. 2005, 77, 7137-7147

Intrinsic Viscosity of Polymers and Biopolymers Measured by Microchip Jinkee Lee and Anubhav Tripathi*

Biochemical Engineering Laboratories, Division of Engineering, Brown University, Providence, Rhode Island 02912

Intrinsic viscosity provides insight to molecular structure and interactions in solution. A new microchip method is described for fast and accurate measurements of viscosity and intrinsic viscosity of polymer and biopolymer solutions. Polymer samples are diluted with solvent in the microfluidic chip by imposing pressure gradients across the channel network. The concentration and flow dilutions of the polymer sample are calculated from the fluorescent signals recorded over a range of dilutions. The viscosities at various polymer dilutions are evaluated using mass and momentum balances in the pressure-driven microchannel flow. The technique is particularly important to many chemical, biological, and medical applications where sample is available in very small quantities. The intrinsic viscosity experiments were performed for three classes of polymer solutions: (a) poly(ethylene glycol), polymers with linear hydrocarbon chains; (b) bovine serum albumin, biopolymer chains with hydrophobic and hydrophilic amino acids, and (c) DNA fragments, biological macromolecules with double-stranded polymeric chains. The measured values of intrinsic viscosity agree remarkably well with the available data obtained using different methods. The data exhibit power law behavior for molecular weight as described by the Mark-Houwink-Sakurada equation. Experiments were performed to understand the effect of solvent quality and salt concentration on molecular conformations and the intrinsic viscosity of the polymers. This method offers a new way to study the conformational changes in proteins and DNA solutions in various buffer conditions such as pH, ionic strength, and surfactants. The effects of shear rate in the microchannel and mixing time on the accuracy and limitation of the measurement method are discussed. The physical properties of a polymer solution depend on solvent, temperature, and concentration. At low concentrations, the polymer chains are separated from each other, where each chain occupies a spherical volume of radius Rg. In this solution, the polymer-polymer interactions are small and the polymer coil volume is determined by polymer-solvent thermodynamic interactions. The hydrodynamic volume occupied by a given polymer mass is the intrinsic viscosity, [η], which is a parameter that can be determined by dilute solution viscosity measurements. Intrinsic * To whom correspondence should be addressed. Phone: (401) 863 3063. E-mail: [email protected]. 10.1021/ac050932r CCC: $30.25 Published on Web 10/12/2005

© 2005 American Chemical Society

viscosity probes the interaction of molecular structure with the solution. Several theories in polymer physics literature1,2 relate intrinsic viscosity to molecular properties of polymers such as molecular weight, overlap concentration, radius of gyration, and pore size of concentrated polymers. These properties are used in many applications of analytical chemistry, biotechnology, and separation science. For example, the properties of the sieving matrix in gel electrophoresis of DNA3-5 and protein (SDS-PAGE)6 are determined by measuring intrinsic viscosity values. These properties are critical for band resolution, dynamic range, and separation time calculations.7 The intrinsic viscosity values are also important in determining the solubility parameters of polymers in different solvents.8-10 Solubility parameters are very useful in determining the fundamental properties of materials11 and have been applied to drug-excipient interactions,12 development of transdermal patches,13 and drug permeation through the skin.14 Moreover, the degree of hydrophobic associations, hydrolysis, and size of micellar clusters can be determined from intrinsic viscosity measurements.15,16 Finally, the intrinsic viscosity values are important for probing biological macromolecular structure and interaction with solution.17 Biological polymer samples of nominal cost are usually restricted to nanoliter to microliter quantities, which essentially prohibits intrinsic viscosity measurement by large-volume conventional methods. (1) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (2) Yamakawa, H. Modern Theory of Polymer Solutions; Harper and Row: New York, 1971. (3) Shi, Y. N.; Simpson, P. C.; Scherer, J. R.; Wexler, D.; Skibola, C.; Smith, M. T.; Mathies, R. A. Anal. Chem. 1999, 71, 5354-5361. (4) Simpson, P. C.; Roach, D.; Woolley, A. T.; Thorsen, T.; Johnston, R.; Sensabaugh, G. F.; Mathies, R. A. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 2256-2261. (5) Tian, H. J.; Landers, J. P. Anal. Biochem. 2002, 309, 212-223. (6) Bousse, L.; Mouradian, S.; Minalla, A.; Yee, H.; Williams, K.; Dubrow, R. Anal. Chem. 2001, 73, 1207-1212. (7) Albarghouthi, M. N.; Barron, A. E. Electrophoresis 2000, 21, 4096-4111. (8) Karger, B. L.; Snyder, L. R.; Eon, C. J. Chromatogr. 1976, 125, 71-88. (9) Hanson, C. M. J. Paint Technol. 1967, 39, 104-117. (10) Bustamante, P.; Navarro-Lupion, J.; Escalera, B. Eur. J. Pharm. Sci. 2005, 24, 229-237. (11) Siddiqui, S. A.; Needles, H. L. Textile Res. J. 1982, 52, 570-579. (12) Rowe, R. C. Int. J. Pharm. 1988, 41, 223-226. (13) Minghetti, P.; Cilurzo, F.; Casiraghi, A.; Montanari, L. Int. J. Pharm. 1999, 190, 91-101. (14) Groning, R.; Braun, F. J. Pharmazie 1996, 51, 337-341. (15) Guo, L.; Tam, K. C.; Jenkins, R. D. Macromol. Chem. Phys. 1998, 199, 11751184. (16) Ng, W. K.; Tam, K. C.; Jenkins, R. D. Eur. Polym. J. 1999, 35, 1245-1252. (17) Harding, S. E. Progress Biophys. Mol. Biol. 1997, 68, 207-262.

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The intrinsic viscosity is commonly determined using one of three approaches described in the literature. In the first approach, the intrinsic viscosity is evaluated by measuring viscosities of the polymer solution over a range of concentrations. The viscosity is determined by either timing the flow of solution through a capillary tube or recording the force required to rotate two concentric surfaces separated by the solution. Subsequently, a plot of either the reduced (eq 1) or inherent (eq 2) viscosity versus macromolecular concentration is used to extract the intrinsic viscosity. Although instrumentations for “capillary”, “cone and plate”, and “concentric cylinder” viscometers have become automated, they require manual cleaning and loading of solution at each polymer concentration. While the capillary method suffers with difficult and lengthy procedures for calibration, cleaning, and temperature control, the viscometer technique suffers accuracy error at low torsions in dilute solution conditions. Moreover, these techniques require long preparation times and large volumes of samples of material for each polymer concentration. The second approach uses a differential viscometer,18-20 which is based on a fluid analogue of Wheatstone bridge electrical circuit. The differential pressure across a bridge of four fluid capillaries is measured to evaluate the relative viscosity of the polymer solution. Even though the differential method is highly sensitive and requires low sample volume, it again requires manual preparation of samples at different concentrations. The third approach uses light scattering and imaging techniques21 to infer the intrinsic viscosity. The mean square displacement of scattering (imaging) particles in the polymer solution is calculated using scattered intensity or particle position data. The intrinsic viscosity of solution is then evaluated using the Stokes-Einstein equation, which relates diffusivity and viscosity. Typically, large routines for processing intensity data or image analysis are required to evaluate the viscosity of the sample solution. The method is expensive and again requires manual preparation of different concentrations of polymer solution. In addition to these macroscopic methods, viscosity measurements have also been recently performed on microfluidics chips. Galambos and Forster22 developed a microfluidics viscometer with a symmetric T-shaped microchannel. In this technique, the viscosity of solution is inferred from the location of the dividing streamline between the merging reference fluid and sample solution. The method appears is likely to suffer with accuracy in dilute solution conditions. Srivastava et al.23 have recently developed a state-of-the-art, self-calibrating, nanoliter viscometer for liquids driven by surface tension. The viscosity of the liquid is calculated from the penetration velocity of the sample and reference liquid into an empty microchannel. The viscometer was tested with water and blood samples, and the results were found accurate within 10% for water and within 3% for blood plasma (18) Dutta, P. K.; Hammons, K.; Willibey, B.; Haney, M. A. Anal. Chim. Acta 1991, 249, 209-213. (19) Dutta, P. K.; Hammons, K.; Willibey, B.; Haney, M. A. J. Chromatography 1991, 536, 113-121. (20) Huang, C. Y.; Huang, S. J.; Yu, Y. C.; Lee, T. Y.; Shau, M. D. Angew. Makromol. Chem. 1999, 265, 25-30. (21) Wang, W. C.; Reinhall, P. G.; Yee, S. Measure. Sci. Technol. 1999, 10, 316322. (22) Galambos, P.; Forster, F. ASME, Anaheim, CA, 1998; pp 187-191. (23) Srivastava, N.; Davenport, R. D.; Burns, M. A. Anal. Chem. 2005, 77, 383392.

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samples. Measurement accuracy may suffer on reuse of the device due to dependence on wetting conditions. As with macroscopic methods, these microfluidic viscometers are developed for singlesample concentration loading. Hence, an experimental approach is needed that can offer accurate measurements of intrinsic viscosity without manual dilutions of polymers and biopolymers with multiple compositions. Moreover, the experimental approach should include calibration and be compatible with a wide variety of organic solvents commonly used in chemical, biological, and medical application. Such an approach will accelerate the study of conformational changes in proteins24,25 and DNA solutions in many buffer conditions such as pH, ionic strength, and surfactants. In this paper, we describe a new microfluidic-based method for fast and accurate measurements of viscosity and intrinsic viscosity of polymer and biopolymer solutions. The measurement procedure requires only that the sample be loaded onto the chip once. The polymer sample is diluted with a reference solution by imposing pressure gradients across the channel network in the microfluidic chip, which alters the contribution from each channel. The concentration of polymer in the microchannel is measured directly from the fluorescent intensity of a dye, which is diluted from a paired channel or contained in the polymer sample. The mass and momentum balance equations were derived to evaluate the viscosity of polymer solution at various dilutions. In the next section, we describe the new method for evaluation of intrinsic viscosity of polymers. The experimental details and microchip development are described after that followed by the Results and Discussion section. THEORY The viscosity measurement is one of the most frequently used approaches for characterizing macromolecular substances in solution. Each polymer coil in a solution contributes to viscosity. In very dilute solutions where the contribution is additive, the solution viscosity, η, increases above the solvent viscosity, ηs, linearly with polymer concentration, c. Hence, the solution viscosity can be described by the Huggins26 and Kraemer equations:27

(η - ηs) ) ηre ) [η] + KH[η]2c ηsc

(1)

ln(η/ηs) ) ηin ) [η] + KK[η]2c c

(2)

where KH and KK are the Huggins and the Kraemer coefficients, respectively. The intrinsic viscosity, [η], which represents the hydrodynamic volume occupied by a given polymer mass, is defined as the limiting value of either the reduced viscosity, ηre, or inherent viscosity, ηin, as both the shear rate and polymer (24) Keith, A. D.; Snipes, W. Science 1974, 183. (25) Ansari, A.; Jones, C. M.; Henry, E. R.; Hofrichter, J.; Eaton, W. A. Science 1992, 256, 1796-1798. (26) Huggins, M. L. J. Am. Chem. Soc. 1942, 64, 2716. (27) Kraemer, E. O. Ind. Eng. Chem. 1938, 30, 1200.

Figure 1. Schematic of the microfluidics chip. The chip consists of three reservoirs (A-C) connected via four microchannels (1-4). The mask widths of channels 1-4 are 40, 40, 20, and 40 µm, respectively. The depth of channels is 12 µm. Figure also shows magnified pictures of two nodes N1 and N2 and the cross section of isotropically etched channel.

concentration, c, approach zero.28 Thus, in mathematical terms, the intrinsic viscosity of polymer is expressed as

the coil size or radius of gyration Rg of the polymer in a dilute solution can be evaluated using30

[η] ) lim(ηin) ) lim(ηre)

[η] ≈ 6.2(Rg3NA/Mw)

cf0

cf0

(3)

The intrinsic viscosity depends on intrinsic and extrinsic properties of the solution such as molecular weight and structure of the dissolved polymer,1 ionic strength and hydrophobicity of solvent,1 and temperature of the system. Within a series of polymer homologues, [η] increases with the molecular weight Mw; hence, it is a measure of Mw. The intrinsic viscosity generally obeys power law behavior in molecular mass, which leads to the Mark-Houwink-Sakurada equation1,29

[η] ) KMaw

(4)

where parameters K and a depend on solvent quality, temperature, and coil size of the polymer. Usually, the exponent a ranges from 0.5 to 0.8 for flexible polymers, but it can be larger for stiff chain polymer molecules. In ordinary good solvents, the constants obtained are valid only within a rather limited range of Mw. A more detailed discussion on the estimation of Mark-HouwinkSakurada parameters can be found in the literature.30-33 The intrinsic viscosity is proportional to the reciprocal concentration of monomers in a volume of a polymer chain. Hence, (28) Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961. (29) Rubinstein, M.; Colby, R. Polymer Physics; Oxford University Press: New York, 2003. (30) de Gennes, P. G. Scaling concepts in polymer physics; Cornell University Press: Ithaca, NY, 1979. (31) Halabalova, W.; Simek, L.; Dostal, J.; Bohdanecky, M. Int. J. Polym. Anal. Charact. 2004, 9, 65-75. (32) Kuntman, A.; Baysal, B. M. Polymer 1993, 34, 3723-3726. (33) Rushing, T. S.; Hester, R. D. J. Appl. Polym. Sci. 2003, 89, 2831-2835.

(5)

where Mw is the average molecular weight of polymer chains. As the concentration increases, the polymer coils come closer and start to overlap each other. Since the number of polymer chains per unit volume is cNA/Mw, the concentration, c*, at which the overlap starts is estimated as

c* )

3Mw 1 1.5 ≈ 4πNA R 3 [η]

(6)

g

Although c*, estimated in eq 6, is an adequate criterion for the onset of molecular interactions when the macromolecules are only slightly extended by flow, highly deformed molecules might entangle or otherwise interact at concentrations much lower than this. MICROFLUIDIC METHOD In this section, we describe a microfluidic chip-based method for measurement of intrinsic viscosity of polymers. The method requires only few microliters (3 µL) of sample at a single concentration without manual dilution. Figure 1 shows the microfluidic chip design for the measurements of intrinsic viscosity of polymers. The channel network consists of four isotropically etched microchannels (denoted by 1-4) joined between three fluid reservoirs (denoted by A-C). The microchannels are initially primed with solvent of viscosity, ηs. Reservoir A is then loaded with the sample solution of polymer concentration, c0. Reservoir B is loaded with the solvent containing a trace amount of fluorescent dye and reservoir C remains empty. Under an imposed set of pressures PA, PB, and PC on reservoirs A-C, respectively, polymer samples from channel 1 and solvent from channel 2 are Analytical Chemistry, Vol. 77, No. 22, November 15, 2005

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mixed in channel 3. The width dimension of channel 3 is small to allow a rapid diffusional mixing of polymer molecules. A fluorescent detector reads the signal intensity of the mixed polymer solution in channel 4. The liquid flow rate Qi in the microchannel i (i ) 1, 2, 3, 4) can be evaluated by solving momentum equations:

channel 1:

PA - PN1 ) R1ηpQ1

(7)

channel 2:

PB - PN1 ) R2ηsQ2

(8)

channel 3:

PN1 - PN2 ) R3η(Q1 + Q2)

(9)

channel 4:

PN2 - PC ) R4η(Q1 + Q2)

(10)

Table 1. Experimental Protocol for Measurements of Intrinsic Viscosity sequence 1

load the chip

2

apply negative pressure in reservoir C, PA ) PB ) 0 apply pressure steps in reservoir A apply positive pressure in reservoir B

3 4

where PN1 and PN2 are unknown pressures attained at node points N1 and N2, respectively. η is the viscosity of liquid mixture in channels 3 and 4. and ηp is the viscosity of sample polymer solution in channel 1. The momentum equation for channels 3 and 4 assumes the instantaneous mixing of polymer molecules in the diluting solvent. The validity of the assumption will be discussed in the Results and Discussion section. The hydrodynamic resistance Ri of a isotropically etched microchannel i (i ) 1-4) is given by

Ri ) RiRrect ) i

[

1-

12 Ri Li widi3 192di π 5w i

∞

1 1 - exp(-nπwi/di)

n)1,3,5,...

n5 1 + exp(-nπwi/di)

∑

]

(11)

action/pressure control

detail prime channels with buffer; place polymer solution (∼3%) in reservoir A and buffer (with trace of dye) in reservoir B measure initial buffer signal Smin and the steady-state signal Smix measure corresponding signals Smix measure 100% dye signal Smax

polymer sample in reservoir A can be evaluated by using eqs 7, 8, and 12 as

R2 y ηp ) ηs R1 1 - y

(

S - Smin Q2 )y) Smax - Smin Q1 + Q2

(12)

where Smin is the signal measured initially with channel containing clean buffer and Smax is the measured signal when 100% of the fluid in channel 2 flows into channel 4; i.e., Q4 ) Q2. It is assumed that fluorescent dye molecules attain homogeneous concentration across the width and depth of the microchannel. The dye molecules are also assumed not to alter the conformation and structure of polymer molecules. Let us first consider the microchip flow when reservoirs A and B are set at zero pressure (PA ) PB ) 0) and reservoir C is set at vacuum pressure PC. In this case, the unknown viscosity of 7140

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(13)

The above measurement is independent of the hydrodynamic resistances in channels 3 and 4 and the applied pressure PC. In addition, the value of y depends only on the rapid mixing of small fluorescent dye molecules. Hence, the extent of mixing of polymer molecules in channels 3 and 4 does not affect the accuracy of measured polymer viscosity. Next, the polymer sample is diluted in channels 3 and 4 by imposing set of different pressures in reservoirs A and C. The pressure in reservoir B is set at zero for simplicity. The polymer concentration, c, in the detection channel, channel 4, which depends on the dilution from reservoir A, is determined as

c ) c0(1 - y) where wi, di, and Li are the width, depth, and length of the microchannel i, respectively. The correction factor Ri, which is multiplied by the hydrodynamic resistance of rectangular channel, accounts for the isotropic shape of the channel. Figure 1 shows a schematic of an isotropic shape of depth d1 and mask width w1. The essential feature of the new method is to measure the steady-state signal intensity in channel 4 under a set of known pressures applied to three reservoirs. The pressures at the three reservoirs of the chip are set such that liquid flows from reservoirs A and B to reservoir C. The measured fluorescence signal S in channel 4 is related to the relative dilution of dye from reservoir B as

)

(14)

where c0 is the known concentration of polymer in loaded sample. The viscosity of polymer solution at different concentrations can then be evaluated as

R2 P C - P A R1 P C η ) ηs y - ηp (1 - y) (15) R 3 + R 4 PA R3 + R 4 P A where the hydrodynamic resistances and the reservoir pressures are known. The polymer viscosity ηp of polymer sample and the normalized signal intensity, y, are evaluated using eqs 13 and 12, respectively. Table 1 summarizes the sequence of experimental steps for measurement of the fluid viscosity. This viscosity measurement technique evaluates polymer solutions at userselected dilutions by simply imposing a series of pressures on reservoirs A and C and then measuring the fluorescent intensities in the dilution channel. Once the viscosity and concentration values are known, the intrinsic viscosity is automatically evaluated using eq 3. It is important to emphasize that the method does not require any prior knowledge of viscosity of the polymer sample. The initial polymer concentration c0 and the reference solvent viscosity ηs are assumed to be known. MATERIALS AND EXPERIMENTAL SETUP Microchip Fabrication. Microfluidic chip fabrication was accomplished using the Brown University Microfabrication Facili-

ties. Microfluidic devices were fabricated out of glass substrates using a protocol based on standard microlithographic techniques. The glass substrate (Corning, NY, 1-mm thickness) was coated with chrome (∼500-Å thicknesses) using chemical vapor deposition after which a layer of photoresist (Microposit S1813, Shipley, Marlborough, MA) was deposited. After UV printing of the features onto the photoresist layer (∼1.3-µm thickness) from the mask, the microchannels were etched using a calibrated chemical etching versus time curves for buffered hydrofluoric acid. The wet etching resulted in isotropic channels with shape shown in Figure 1. The mask widths of channels 1, 2, and 4 were 40 µm each and channel 3 was 20 µm. Channel 3 and channel 4 (see Figure 1) were 10 and 49 mm long, respectively. The side channels were 16 mm long and all channels were 12 µm deep. The correction factors Ri (eq 11) were evaluated numerically (R1 ) R2 ) R4 ≈ 1.13, R3 ≈ 1.18) using the computational software CFDRCACE (ESI-CFD Group, Huntsville, AL). A second glass wafer was prepared with reservoir holes, fabricated using a glass drill mounted on a computer numeric control machine. The etched glass wafer with the microchannel geometry was then bonded to the capping glass wafer using a controlled thermal bonding procedure. This fabrication procedure is routinely accomplished with excellent precision and yield. The channels can also be coated with silane coating (Gelest Inc., Morrisville, PA) to suppress undesirable adhesion to the channel walls. There are many advantages to using glass chips for intrinsic viscosity measurements of biomolecules. For example, unlike traditional poly(dimethylsiloxane) microfluidic chips, the glass microfluidic chips are free from solvent adsorption, swelling, and other detectionrelated problems, which makes them ideally suited for detection assays, such as intrinsic viscosity of macromolecules. Microchip Control System. The dilutions and fluid flow were regulated with a programmable control system designed and developed in the laboratory. This control system has two components. First, it contains three independent pressure ports that are capable of imposing and measuring pressure profiles in any microchannel network consisting of fluids. Peristaltic pumps (Watson-Marlow Bredel Inc., Wilmington, MA), driven by a dc motor, regulate air pressure over the chip reservoirs, which in turn pressurize the liquid samples in the reservoirs. Accurate differential pressure sensors (GE Novasensor, Billerica, MA) monitor the air pressures. Control electronics convert an external control voltage into a precise air pressure through negative feedback regulation. Second, a dedicated LABVIEW software program (National Instruments Corp., Austin, TX) provides automatic calibration and programming features for the user. Unlike conventional systems where fluid is driven through syringes and tubes, this air-only system is contamination free, which is important for intrinsic viscosity measurements at high dilutions. A Nikon TE 2000U inverted microscope (Nikon Inc., Japan) with epifluorescence, phase contrast, and dual-channel photometer measured the fluorescence intensity in the microchannels. The microchannel detection area was placed in the alignment path of the microscope system. The microfluidic chip reservoirs where the fluid and polymer samples were located were joined together using a cartridge manifold, which was connected to the pressure tubes from the control system. Figure 2 shows a schematic of

Figure 2. Schematic of the microchip and detection assembly. (a) Air pressure lines from pressure controller, (b) reservoir A, (c) reservoir B, (d) cartridge, (e) glass microfluidics chip, (f) microscope plate, and (g) microscope optics.

the microchip and detection assembly. The input-output control system was connected to a computer via a data acquisition board. The software LABVIEW was used to control the input-output commands via the data acquisition board. The program automatically calculates and applies the appropriate pressures to each of the microfluidic chip reservoirs. Table 1 summarizes the sequence of experimental steps for measurements of the fluid viscosity. Although the current study is performed using research-grade instruments, an accurate yet inexpensive version of an intrinsic viscometer can be accomplished using a single photodetector and two inexpensive air pressure controls. Off-chip control measurements were performed on a rheometer (TA Instruments, New Castle, DE, model AR2000N) using a double concentric cylinder geometry under imposed shear rates. Materials. Microchip experiments were performed using poly(ethylene glycol) (PEG), bovine serum albumin (BSA), and shortchain DNA molecules. Samples of PEG of three average molecular weights (Mw ) 4000, 10000, 35000 g/mol; polydispersivity index 1.49) were supplied from Sigma-Aldrich. BSA (grade >98%, 6.7 < pH < 7.3) was obtained from Fisher Scientific. Filtered 18-MΩ DI water (Elga Classic, Lowell, MA) was used to prepare master solutions. A 10-base palindromic DNA sequence (5′-ATC GTA CGA T-3′, 5′-functionalized with 6-FAM) was custom synthesized by Integrated DNA Technologies (Coralville, IA). The DNA was heated to 95 °C for 2 min and cooled to room temperature to form labeled double-stranded DNA sequences. A filtered buffer (solvent) solution consisting of 10 mM Tris-HCl (Sigma-Aldrich) and 1 mM EDTA (VWR International) was used for polymer solutions at 100 mg/mL. Solvent pH was measured using a Twin pHmeter (Horiba, Japan) and adjusted to pH 8. RESULTS AND DISCUSSION (a) Accuracy and Error Analysis. We now discuss the accuracy of the measured viscosity (and intrinsic viscosity) of the polymer solutions, which primarily depends on three experimental features: (i) the accuracy of imposed pressure control, (ii) the Analytical Chemistry, Vol. 77, No. 22, November 15, 2005

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Figure 3. Numerically evaluated concentration profiles in channel 3. Simulation were performed for Q3 ) 0.68 nL/s, Diffusivity BSA is 60 µm2/s. The results show ∼95% mixing in 5 s.

rate of diffusional mixing of polymer molecules in channels 3 and 4, and (iii) the flow-induced conformation change of polymer molecules. By differentiating eq 14 with respect to PA, the relative magnitude of error due to pressure fluctuations is evaluated as

(

)

R2 ηs δη δPA ) 1+ y η PA R3 + R 4 η

(16)

where δPA denotes the pressure fluctuations in the pressure port A. The errors due to pressure fluctuations were minimized in the chip by designing a small hydrodynamic resistance ratio R2/(R3 + R4) ≈ 0.01 and exploiting a full range of controllable pressure PA. The pressure fluctuations in the pressure controller used our experiments was measured to be δPA ) 0.01 psi. Hence, the percent error in the measured viscosity due to pressure fluctuations was less than 5%. The second source of error arises from the finite-time mixing of polymer molecules in channel 3 even though the dye molecules diffuse almost instantaneously across the microchannel. The diffusional mixing decreases as the magnitude of the flow increases relative to the diffusional flow across the channel. The convective and diffusional components of the molecules can be described using a modified Peclet number, defined as

Pe )

2DL3 L3/U3 convective time ) 2 ) 2 diffusion time w3 /2D w3 U3

(17)

where U3 the average velocity in the channel. For Pe > 1, the molecular diffusion is fast and the polymer composition in channel 7142

Analytical Chemistry, Vol. 77, No. 22, November 15, 2005

3 is nearly homogeneous across its width. To estimate the exact mixing time and concentration profiles of polymer molecules in the isotropic 3D channels, a computational software CFDRC-ACE (ESI-CFD Group, Huntsville, AL) was used. The accuracy of the software code was first validated by comparing 2D simulations with known analytical expressions. Figure 3 shows 3D simulation results for the concentration profile of BSA molecules across channel 3. The plot shows 95% complete mixing of BSA molecules at the exit of channel 3 in 5 s. The percent mixing is estimated by calculating the ratio of the difference in concentrations at two channel walls at the mixing time and at time t ) 0. It is important to note that the 2D approximation (tdiff ∼ w32/2D, where w3 is the width of channel 3 and D is the molecular diffusivity of the polymer) leads to higher predictions for mixing time. For example, BSA is 95% mixed in 5 s in an isotropic 3D channel (channel 3) while 2D approximate analytical expressions predict only 93% mixing in 5 s. Here, 2D values are evaluated using the width of channel as the average of top and bottom width of the microchannel. Based on above analysis and numerical results, channel 3 was designed with mask width w3 ) 20 µm and length L3 ) 10 mm. When polymer concentration is not homogeneous across the width of the channel, eq 9 is not expected to accurately describe the momentum balance of a 100% mixed polymer solution in channel 3. The concentration inhomogeneity is substantial, especially for large macromolecules with slow diffusivities. As a result, unacceptable errors may be introduced into the evaluation of intrinsic viscosity of polymers. Under these circumstances, the viscosity of the polymer solution will be η ( ∆ηerror instead of η, which characterizes the viscosity of a completely mixed polymer solution. The percent error in the measured viscosity due to

Table 2. Molecular Properties of Tested Polymers molecular weight diffusivity relaxation polymer (g/mol) (µm2/s) time (s) PEG BSA DNA

Figure 4. The parameter space for measuring intrinsic viscosity of polymers using microchips. The shaded portion shows the operating space for accurate measurements.

incomplete mixing in channel 3 can be estimated using eqs 7-10 as

∆ηerror × 100 ) η

R3 × PA - PC y R1ηp R2ηs - R1ηp PA - PB 1-y

[(

)(

(

) )

]

100 (18) Here we have assumed a flow of completely mixed polymer solution in channel 4. Using standard values for operating pressure, geometrical resistances, and polymer and solvent viscosity, the BSA dilution error was found to be less than 2%. The third source of error arises from flow- or shear-induced conformational change of the polymer molecule. The intrinsic viscosity value is defined as the relaxed configuration of the polymer. The polymer conformation is altered (stretched) at high values of shear rate γ˘ , which is usually the case in microchannel flows (γ˘ ) 200 s-1). The conformational change of polymer solutions is usually described by the Deborah number De ) λγ˘ , where λ ) [η]ηsMw/RT is the relaxation time of the polymer. For De < 0.5, the polymer molecules are relaxed and likely to be in the original conformational state. Hence, the intrinsic viscosity measurements are performed at low Deborah number flows. To summarize the above error analysis, a plot of average velocity in channel 3 versus molecular weight of the polymer molecule is shown in Figure 4. The dividing lines De ) 0.5 and Pe ) 1 split the parameter space into four regions. Here λ ∼ Mw1.8 and D ∼ Mw-0.6 is assumed. The intrinsic viscosity measurements should be performed in the operating space, shown as shaded portion, where De < 0.5 and Pe > 1. On-chip microchannel mixing is incomplete for high molecular weight polymers. As shown in Figure 4, the reduction in the width of the microchannel can shift the Pe ) 1 line to increase the operating space of the measurement technique. Low molecular weight polymers and biopolymers such as low molecular weight polymers, proteins, antibodies, and enzymes lie in the operating space shown in Figure 4. The intrinsic viscosity

4000 10 000 35 000 65 000 7128

172.6 127.1 83.7 60 111.5

1.88 × 10-7 2.26 × 10-8 4.48 × 10-9 4.15 × 10-5 5.33 × 10-8

Deborah no. (De)

modified Peclet no. when fluid speed is 1 mm/s (Pe)

3.13 × 10-5 3.77 × 10-6 7.48 × 10-7 0.01 8.88 × 10-6

3.37 2.48 1.63 1.17 2.18

of high molecular weight polymers can also be evaluated by measuring the viscosities of manually diluted polymer samples on the chip. Since viscosity measurements do not require concentration homogeneity of polymer in channel 3. Hence, the microchip method is advantageous for precious DNA samples. Table 2 summarizes the dimensionless parameters for DNA, BSA, and PEG molecules. (b) Intrinsic Viscosity Measurements. Microchip experiments were performed using three classes of polymer solutions: (a) PEG consisting of polymers with straight aliphatic chains; (b) BSA consisting of protein chains with hydrophobic and hydrophilic amino acids; and (c) short-chain DNA consisting of doublestranded polymeric chains. All the experiments were conducted at 23 °C. Samples were loaded once without any subsequent replenishment. Poly(ethylene glycol). Intrinsic viscosity measurements were carried out for PEG samples of three molecular weights (Mw ) 4000, 10 000, 35 000). PEG has unique solubility properties, dissolving in water in all proportions at moderate temperatures over a very wide range of polymerization states. PEG has been extensively investigated for various biological purposes, such as drug conjugation, as an embedding material for cells,34 and as an inducer for bundling phenomenon of F actin.35 Both the Deborah number and Peclet number (see Table 2) satisfy the criterion (Figure 4) for accurate measurements using the microfluidic chip. The relaxation time of PEG molecules is small (λ ) [η]ηsMw/RT ∼ 2 × 10-7s) and the molecular diffusivity is large (∼150 µm2/ s).36 A 3-µL sample of PEG with initial concentration of 3 wt % was loaded on the chip, and the procedure described in Table 1 was implemented. Figure 5a shows pressure profiles measured in three reservoirs for all dilution steps. The pressure ramp time was rapid at 0.55 psi/s with a good signal-to-noise ratio of 125. Figure 5b shows the empirical fluorescence intensity for a 10kDa PEG sample as different dilution steps were imposed. The signal change resulting from each dilution step is easily discernible above the noise (∼0.01 unit) of the system, which was consistently below (0.01 psi for the chosen pressure controllers. For each dilution step, the fluorescence intensity approached a steady state after a short transition time (∼0.5 s) between dilutions. Similar results were observed for other PEG polymer samples tested for viscosity measurements. The shear rate in the microchannel was 150-250 s-1, and total experimental run time was 10 min. The PEG viscosity, µp, was evaluated using eq 13 (PA ) PB ) 0) and was found to be 0.0265 ( 0.0003 P for the 35-kDa PEG. Note that this measurement did not require concentration (34) Wolosewick, J. J. J. Cell Biol. 1980, 86, 675-681. (35) Hosek, M.; Tang, J. X. Phys. Rev. E. In press. (36) Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975.

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Figure 5. (a) Measured pressure profile for dilution experiments for PEG of Mw ) 10 000 at T ) 23 °C. (b) The measured fluorescence intensity for PEG of Mw ) 10 000 at T ) 23 °C.

Figure 6. Measured steady-state viscosity as a function of concentration of PEG with left y-axis and BSA with right y-axis at T ) 23 °C. The microchip measured values show excellent agreement with those measured using rheometer.

homogeneity of polymer in channel 3. Next, the polymer viscosities were evaluated using eq 13 at different dilution steps. The respective concentrations of PEG molecules were calculated using eq 12. Figure 6 shows a plot of measured viscosity versus 7144 Analytical Chemistry, Vol. 77, No. 22, November 15, 2005

Figure 7. (a) Reduced and inherent viscosity for PEG of Mw ) 35 000 at T ) 23 °C. (b) The reduced and inherent viscosity for PEG of Mw ) 10 000 and Mw ) 4000 at T ) 23 °C. (c) Intrinsic viscosity versus molecular weight of PEG. The solid line shows the MarkHouwink-Sakurada equation, where K is 0.0224 and a is 0.73.

concentration of 35-kDa PEG. The data show that accurate measurements were possible for polymer concentration as low as 0.5 wt %. Also shown in Figure 6 are the results of macroscale rheometer viscosity measurements, which are in excellent agreement with those performed on the microchip. The error bars are drawn by calculating the standard deviation of viscosity values

Table 3. Molecular and Solvability Parameters Derived from Intrinsic Viscosity Measurements polymer PEG

BSA DNA

molecular weight 35 000 10 000 4000 35 000 10 000 10 000 65 000 7129

solvent DI water CaCl2 10 mM DI:methanol ) 1:1 DI water TE

intrinsic viscosity [η] (mL/g)

relaxation time (ns)

Huggins constant KH

Kraemer constant KK

radius of gyration Rg (nm)

overlap concn c* (g/mL)

44.6 18.8 9.3 36.7 11.9 22.3 3.6 3.9

187 22.6 4.5 154 14.3 26.8 28.1 16.5

0.18 0.49 0.22 0.25 0.44 0.16 0.22 0.07

-0.19 -0.08 -0.25 -0.18 -0.09 -0.26 -0.24 -0.39

7.51 3.71 2.16 7.04 3.18 3.93 3.99 1.55

0.03 0.08 0.16 0.04 0.13 0.07 0.42 0.38

obtained in five experimental runs. The rheometer measurements were performed under an imposed shear rate of γ˘ ) 200 s-1. The viscosity of the solution was found to be independent of shear rate for shear rates up to γ˘ ) 1000 s-1. Following sample preparation, data collection for polymers of three different molecular weights required 10 min on the microchip compared to 2 h on the rheometer. Additionally, at least 8 mL of each test sample was required to measure viscosity on the rheometer. Figure 7a shows a plot of reduced and inherent viscosity versus PEG concentration. The inherent and reduced viscosities were evaluated using eqs 2 and 1, respectively. The Huggins and Kraemer equations provide almost an exact fit to the data. The error bars are drawn by calculating the standard deviation of data obtained in multiple runs. As the concentration goes to zero (eq 3), both the reduced and intrinsic viscosities converge to a single value [η] ) 44.6 ( 1.5 mL/g. The microchip viscosity agrees remarkably well with the value obtained from rheometer data and with comparable polymers in the literature.20 Figure 7b show the reduced and inherent viscosities at various concentrations of PEG of molecular weights 10 000 and 4000. Once again, the Huggins and Kraemer equations provide excellent fits to data. The viscosity µp was evaluated to be 0.017 ( 0.001 and 0.013 ( 0.0008 P for 10 000 and 4000 PEG, respectively. The intrinsic viscosity decreases with the molecular weight as expected. These values agree well with the rheometer results. Table 3 summarizes the results of intrinsic viscosity measurements on different molecular weight solutions of PEG. The overlap concentration, c*, was also evaluated using eq 6 and is included in Table 3. Note that the starting concentrations of polymer solutions were much lower than the polymer overlap concentration values. In the present work, the parameters K and a for the MarkHouwink-Sakurada relation (eq 4) were estimated from the intercept and the slope of straight line fitting the bilogarithmic plot of [η] versus Mw. Figure 7c shows a plot of the intrinsic viscosity [η] versus Mw on a log-log scale. The data fit remarkably well on a straight line. The values of the constants K ) 0.0224 mL mol-0.73/g1.73 and a ) 0.73 are in good agreement with other reported results.20 The value 0.73 suggests that PEG polymer chains can be considered as flexible chains that expand to bring about many contact points with solvent molecules.30 The radius of gyration Rg of 35-kDa PEG was estimated to be 7.5 nm. Table 3 tabulates the estimated radius of gyration values for other tested polymer samples. The effect of solvent quality was also explored in the microchip measurements of intrinsic viscosity. Solvent quality is a qualitative characterization of the polymer-solvent interaction. A solution of a polymer in a “better” solvent is characterized by a higher

Figure 8. Effect of solvent conditions on reduced and inherent viscosities of PEG of Mw ) 10 000.

value of the thermodynamic interactions1,33 than a solution of the same polymer in a “poorer” solvent. Flory’s work1 describes these thermodynamic interactions in terms of enthalpic and entropic contributions during mixing of polymer and solvent molecules. The ions in solvent and the pH concentration influence polymer configuration. Polymers, with hydrophobic and hydrophilic parts, alter the hydrophilic interaction of polymer with solvent molecules upon addition of a salt. This results in polymer shrinkage and a reduction in the intrinsic viscosity value. These effects can easily be explored using a microfluidic approach. In addition to deionized water, two solvent types were explored: (a) 1:1 deionized water and methanol mixture and (b) 10 mM CaCl2/water solution. Figure 8 shows a plot of reduced and inherent viscosities of 10kDa PEG in solvent a and solvent b. When PEG was dissolved in solvent a, the polymer chains were swollen to increase number of contacts with solvent molecules resulting in an increase in average intrinsic viscosity value from 18.8 to 22.3 mL/g. In other words, the radius of gyration was increased from 3.7 to 3.93 nm. When polymer chains were dissolved in solvent b, the chains were contracted by steric interactions with CaCl2 molecules resulting in a decrease in the intrinsic viscosity value from 18.8 to 11.9 mL/ g. In other words, the radius of gyration was reduced from 3.7 to 3.18 nm. Similar effects of solvent quality were observed for 35kDa PEG. The Huggins and Kraemer constants were also evaluated from slopes, which are of opposite signs of linear fits of data using eqs 1 and 2. The Huggins constant for PEG was evaluated to be 0.44 in 10 mM CaCl2 system while it was 0.49 in DI water solvent. The Huggins constant can be used to explain the conformation Analytical Chemistry, Vol. 77, No. 22, November 15, 2005

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Figure 9. Reduced and inherent viscosity for BSA at T ) 23 °C.

of polymer. A decrease in Huggins constant indicates an expansion of polymer chains. The Huggins and Kraemer constants for all test samples are tabulated in Table 3. The intrinsic viscosity measurements of PEG samples clearly demonstrate the accuracy and validity of the microfluidic measurements. The measurements consume considerably less time and reagents in producing accurate measurements. Proteins. The conformational state of proteins, folded or unfolded, significantly influences functionality. Since the intrinsic viscosity provides information on the overall size and shape of the molecule, it is a sensitive and simple indicator of denaturation. The major advantage of the microfluidics method is in exploration of the dependence of buffer conditions on protein conformation and in measurements of intrinsic viscosity of proteins, which are often available only in nanogram quantities. Experiments were performed to measure intrinsic viscosity of BSA as a model using the microchip method. BSA is one of the most widely studied proteins and most abundant plasma protein with typical concentrations at 35-50 mg/mL. Its molecular weight is 65 000 and is reported to have low intrinsic viscosity values between 3.6 and 4.2 mL/g.17,37-39 Figure 6 shows a plot of viscosity versus concentration of BSA. The microchip experiments show an excellent agreement with these performed on the rheometer. Figure 9 shows a plot of inherent and reduced viscosity of BSA for various concentrations. The data clearly show the microchip method is capable of accurately capturing the small changes in the inherent and reduced viscosities with protein concentration. The data converge to the intrinsic viscosity of 3.6 mL/g, which agrees well with the literature values.17,37-39 The Huggins constant was evaluated to be 0.22. The radius of gyration is calculated to be 4 nm. DNA. Proof-of-concept viscosity measurements with DNA in dilute solution were performed on the microchip. Intrinsic viscosity measurements were performed with a synthesized doublestranded DNA of size 10 base pairs (bp) that was covalently functionalized with the fluorescent dye, 6-FAM, at the 5′-ends. The palindromic DNA sequence was heated to 95 °C for 2 min and (37) Peters, T. J. Adv. Protein Chem. 1985, 37, 161-245. (38) Kunts, I. D.; Kauzmann, W. Adv. Protein Chem. 1974, 28, 239-345. (39) Monkos, K. Int. J. Biol. Macromol. 1996, 18, 61-68.

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Figure 10. Reduced and inherent viscosity for 10-bp doublestranded DNA at T ) 23 °C.

cooled to room temperature to form labeled double-stranded DNA. The small fragment size satisfies the operating conditions of De < 0.5 and Pe > 1 (see Table 3). Figure 10 shows a plot of inherent and reduced viscosity of DNA at four dilutions. Data converge to the intrinsic viscosity value of 3.9 mL/g. Due to a lack of literature data, the intrinsic viscosity value is compared with theoretical predictions. A 10-bp DNA molecule with helical secondary structures is short enough so that bending effects are negligible and it can be regarded as a straight, rigid, rodlike particle. The hydrodynamic properties of rodlike molecules in dilute solution have been the subject of much theoretical work and computational work.40-42 From these works, the intrinsic viscosity of short-chain rodlike DNA fragment of length L, diameter d, and aspect ratio p ) L/d can be expressed as either40

[η] )

3 ν πNAL 4p2 Mw

or43

[η] )

πNAL3 2 45(ln p - β) Mw

(19)

where ν ) 2.77 + 1.647 (ln p)2 - 1.211 (ln p)3 + 0.6124 (ln p)4 and β ) 0.9 + 1.4/p - 8.9/p2 + 8.8/p3. Both the expressions in eq 19 predict very close values for the intrinsic viscosities. The aspect ratio of a 10-bp fragment (length of 10 bp is 3.4 nm and diameter of double helix is ∼2 nm) was calculated to be p ≈ 2. From eq 19, the theoretical value of intrinsic viscosity of 10-bp DNA fragment, which is in close agreement with the experimentally measured value, is 3.5 mL/g (see Figure 10). Here, the aspect (40) Ortega, A.; de la Torre, J. G. J. Chem. Phys. 2003, 119, 9914-9919. (41) Avalos, J. B.; Rubi, J. M.; Bedeaux, D. Macromolecules 1993, 26, 25502561. (42) Brenner, H. Int. J. Multiphase Flow 1974, 1, 195. (43) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986.

ratio calculations included the increase in size due to the FAM functionalization. Cost constraints prevented comparison with a conventional rheometer experiment. CONCLUSIONS A microchip-based method is developed for measurements of viscosity and intrinsic viscosity of polymer and biopolymer solutions. Fluorescent intensities provide a direct measure of the channel dilution ratio. Since the pressures over the reservoirs of the chip are known, the viscosities are calculated from momentum balance equations for each channel of the chip. The precise microchip geometry, accurate pressure control for dilution, and calibrated fluorescence signal are key components of this powerful tool for intrinsic viscosity measurements. The measured values

of intrinsic viscosity agree remarkably well with the available data obtained using different methods. The method offers a new way to study the conformational changes in proteins and DNA solutions in various buffer conditions such as pH, ionic strength, and surfactants. ACKNOWLEDGMENT We acknowledge the support of the Brown Startup Funds for this research. We thank Jacob Rosenstein and Matt Kerby for their help in experiments. Received for review May 27, 2005. Accepted September 6, 2005. AC050932R

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