utilising μ-piv and pressure measurements to determine the viscosity of a dna solution in a...

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Experimental Thermal and Fluid Science 30 (2006) 843–852

Utilising l-PIV and pressure measurements to determine the viscosityof a DNA solution in a microchannel

Damian M. Curtin *, David T. Newport, Mark R. Davies

Stokes Research Institute, University of Limerick, Limerick, Ireland

Abstract

There is currently considerable interest in the development of micro-scale polymerase chain reaction (PCR) systems. Smaller samplevolumes are required than for macro-scale systems, and faster process times are feasible. Although much attention has focused on theoutput of micro-PCR (l-PCR), little attention has been devoted to the detailed fluid mechanics of such devices. There are many technicalchallenges associated with systems of these length scales.

In this paper the effect of PCR on biofluid viscosity is examined. A theoretical expression for viscosity in PCR is derived. Transmissionelectron microscopy is used to determine the geometry of a 240 base pair segment of an Escherichia coli (E. coli) DNA molecule and theresults are used to predict the effect of PCR on biofluid viscosity.

Micro-particle image velocimetry (l-PIV) and pressure transducer measurements of water, amplified and unamplified E. coli DNAsolutions flowing in a polycarbonate microchannel are recorded. In a novel application of these established measurement techniques,the results are combined with curve fitting of a theoretical prediction for channel flow to estimate the viscosity of the E. coli solutions.The viscosity results are compared to the theoretical prediction for PCR viscosity and to measurements in a commercial viscometer.Viscosity measurements indicated no increase in fluid viscosity after PCR for a low molecular weight molecule.� 2006 Elsevier Inc. All rights reserved.

Keywords: Micro-particle image velocimetry; Viscosity measurement; Velocity measurement; DNA solution; Polymerase chain reaction; Transmissionelectron microscopy

1. Introduction

The polymerase chain reaction (PCR) has revolutionisedthe approach to microbiology and is a fundamental ele-ment in modern gene-research laboratories [1,2]. PCR isa powerful chemical amplification process. A PCR reactionis typically composed of 30 cycles or more. Each cycle hasthree steps. First double-stranded DNA is melted at 95 �C(denaturation), and then primers are annealed (hybridisa-tion) to the target site between 50 and 65 �C. Finally theprimers are extended along the single strand DNA with athermostable enzyme (Taq polymerase) between 72 and77 �C. The actual specific temperature depends on the

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doi:10.1016/j.expthermflusci.2006.03.014

* Corresponding author. Tel.: +353 061 213103; fax: +353 061 202393.E-mail address: [email protected] (D.M. Curtin).

primers used and DNA target of interest. Theoreticallythe process has an exponential yield.

At present there is considerable interest in the develop-ment of micro-electro mechanical systems (MEMS) withapplications in biochemistry and medical diagnostics[2–5]. The motivations in transferring existing laboratorytechniques and technology such as PCR to the micro-scaleare numerous. Among the advantages of smaller dimen-sions are increased surface-to-volume ratio, which vastlyimproves heating and cooling rates, reduction in reagentconsumption, increased reaction efficiency, portability andeconomics. Micro-PCR (l-PCR) systems can generally beclassified as static (well-based systems) [6–9] or dynamic(continuous-flow systems) [10–14]. Among the advantagesof a dynamic over a static l-PCR system are higherthroughputs and greater heating and cooling rates.

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Nomenclature

NA numerical aperture of objective lensN number of DNA particlesV volume (m3)bp base pair(s)dp seeding particle diameter (m)e PCR reaction efficiencyi PCR cycle numbern refractive indexu velocity in the x-direction (m/s)a shape factor

dzm measurement depth (m)u volume fractionko wavelength of imaged light in a vacuum (m)l dynamic viscosity (Pa s)h objective lens collection angle (�)

Subscripts

0 initial statep particle

Fig. 1. Schematic of channel cross-section and notation.

844 D.M. Curtin et al. / Experimental Thermal and Fluid Science 30 (2006) 843–852

Knowledge of the viscosity of biofluids such as blood,blood plasma, amniotic and synovial fluid has a significantrole in diagnostic, prognostic and preventative medicine[15]. A nanoliter viscometer for analysing biofluids, hasbeen reported previously [15]. The viscometer was capableof an accuracy of 3% when measuring the viscosity ofblood plasma samples; where accuracy was defined as therelative deviation from mean commercial viscometermeasurements or literature values. Viscosity measurementshave been used to determine DNA molecular weight andthe effect of solvents on conformation and structure [16].

The addition of macromolecular solute increases theviscosity of a solution significantly [17]. The increase isdependant on concentration, size and structure of themacromolecule. In PCR the presence of the target DNAmacromolecule increases exponential with cycle numberbefore reaching a plateau at the end of the reaction.The effect of concentration, size and structure of targetDNA on fluid viscosity, with increasing cycle number, inl-PCR devices has not been explored. If solution viscosityincreases with cycle number, perhaps viscosity could bemonitored as an indicator of successful amplification. Fur-thermore, the implications of increasing solution viscositychallenges the legitimacy of employing constant viscositymodels in the design and simulation of l-PCR devicesand would also place an extra demand on micro-pumpsof the future.

In this paper the effect of PCR on biofluid viscosity isexamined. A theoretical expression for viscosity in PCRis derived. Transmission electron microscopy is used todetermine the geometry of a segment of an E. coli DNAmolecule and the results are used to predict the effect ofPCR on biofluid viscosity.

Micro-particle image velocimetry (l-PIV) and pressuretransducer measurements of water, amplified and unampli-fied E. coli DNA solutions flowing in a polycarbonatemicrochannel are recorded. In a novel application of theseestablished measurement techniques, the results arecombined with curve fitting of a theoretical prediction forchannel flow to estimate the viscosity of the E. coli solu-tions. The viscosity results are compared to the theoretical

prediction for PCR viscosity and to measurements in acommercial viscometer reported in [18].

2. Theory

The theoretical section of this paper introduces an ana-lytical solution for channel flow and presents an expressionfor viscosity of DNA solutions in a PCR.

2.1. Channel flow

The fully developed velocity distribution in a channel, ofdimensions shown in Fig. 1, can be expressed as [19]:

u1 ¼1

2ldPdx

b2 � y2 þ 4

b

X1n¼0

ð�1Þnþ1

N 3n

cosh Nnzcosh Nna

cos N ny

!

ð1ÞNn ¼ ð2nþ 1Þp=2b ð2Þ

For a known pressure gradient and velocity, the viscos-ity of a fluid can be determined from Eqs. (1) and (2). Thenext section describes the experimental process of measur-ing the velocity and pressure. A theoretical solution for

D.M. Curtin et al. / Experimental Thermal and Fluid Science 30 (2006) 843–852 845

velocity distributions in water and E. coli solutions is pre-sented in the results section and compared to velocitymeasurements.

2.2. Fluid viscosity

As observed in [20], the theoretical expression for theviscosity of solutions of macromolecules begins with thework of Einstein who determined the effective viscosity ofa dilute suspension of rigid hard spheres. The effective vis-cosity of a solution is given in the following.

l ¼ l0ð1þ auÞ ð3Þwhere l0, a and u are solvent viscosity, a numerical shapeconstant and particle volume fraction respectively. Eq. (3)is normally valid for suspensions of particles that can beapproximated by hard spheres for volume fractionsu < 0.02, however it has been applied up to u � 0.1 [21].When considering DNA macromolecules the particlevolume fraction may be written as:

u ¼ V pNi

Vð4Þ

where Vp is the DNA particle volume in suspension, Ni isthe number of DNA particles in PCR cycle number i,and V is the total volume. The amplification yield in theexponential phase of a PCR is expressed as [22]:

Ni ¼ N 0ð1þ eÞi ð5Þwhere N0 is the initial number of DNA template particlesbefore amplification, and e is the efficiency of the cycle.N0 is known from a PCR standard mix.

Substituting Eq. (5) into Eq. (4) and the result into Eq.(3) provides an expression for viscosity due to amplificationin a PCR reaction:

l ¼ l0 1þ aV pN 0ð1þ eÞi

V

� �ð6Þ

Assuming the solvent viscosity to be that of water, theunknowns are the shape factor, the particle volume, and

Fig. 2. Microchannel in polycarbonate substrate. Fine wire thermocou-ples are inserted in the channel for temperature measurements.

efficiency. The next section describes how these variablesare measured and results are presented in the followingsection.

3. Experimental techniques

In this section the microchannel device is described andthe preparation of the E. coli solutions is outlined. The pro-cedure to determine the geometry of the E. coli particle ispresented and the techniques used to measure pressureand velocities are also described.

3.1. Microchannel device

A 219 lm wide by 233 lm deep serpentine channel wasmilled in a 0.5 mm thick polycarbonate substrate. A posi-tive and negative pressure tap channel of equal dimensionswas also machined at 90� to the serpentine channel to facil-itate pressure measurements. The channel length betweenpressure taps was measured to be 281.52 mm. Four Preci-sion Fine Wire Thermocouples (Omega Engineering Inc.)with a wire diameter of 76 lm were fitted in the channel.The thermocouples were introduced through the back ofthe substrate at four locations, and glued in place justbelow the base of the channel to minimise flow disturbance.The substrate was fitted with micro-fluidic ports (UpchurchScientific) for sample delivery and pressure measurement.The channel was sealed using EasySealTM sealing film(Hampton Research). The substrate was fastened betweenan aluminium and brass plate with screws to ensure thesubstrate remained flat. A segment of the serpentine chan-nel and one fitted thermocouple are shown in Fig. 2. Thesubstrate with brass and aluminium plates is shown inFig. 3.

3.2. Solution preparation

Two reaction mixtures were prepared. One intended forPCR and the other to remain unamplified. A 3000 basepair (bp) pGEM 5Zf(+) E. coli vector (Promega UK)was used as the starting DNA molecule. The primers usedwere: M13 Reverse primer (�29) 5 0 cag gaa aca gct atg acc3 0 (18 mer) and M13 Forward primer (�43) 5 0 agg gtt ttccca gtc acg acg tt 3 0 (23 mer) (MWG Biotech). The sizeof the PCR product was 240 bp in length.

Each reaction mixture contained 250 ll of PCR Jump-Start Taq Ready Mix, 10 ll of each of the oligonucleotideprimers, 220 ll of sterile water and 10 ll of the DNA stan-dard mix containing 10 pg/ll of template DNA. The totalvolume of each reaction mixture was 500 ll.

One reaction mixture was amplified via PCR along witha negative control using the following thermocycling proto-col: Initial denaturation at 94 �C for 120 s, 40 cycles ofdenaturation at 94 �C for 15 s, hybridisation at 60 �C for60 s, extension at 72 �C for 60 s.

Successful PCR was confirmed by agarose gel electro-phoresis. Based on an approximate molecular mass of

Fig. 3. Assembled microchannel device with a €1 coin for scale.


330Da for one nucleotide [23], the initial number of DNAtemplate particles before amplification, N0, was estimatedto be 3.8 · 107. Typical reaction efficiency following Light-Cycler fluorescent intensity monitoring was approximately80% in the exponential phase.

3.3. Particle geometry

Transmission electron microscopy (TEM) was used todetermine the approximate geometry of the 240 bp E. coli

DNA molecule. DNA samples were deposited on thecarbon surface of 400 mesh copper electron microscopeCarbon/Formvar grids (Agar Scientific). The samples werestained with uranyl acetate. The samples were examinedusing a JEOL JEM-2010 electron microscope at magnifica-tions ·26,000–670,000. A sample image is presented in theresults section.

3.4. Pressure measurements

A diaphragm DP15 Validyne pressure transducer and aValidyne CD15 carrier demodulator were used to measurepressure. A Grant 1000 Series Squirrel Logger was con-nected to the carrier demodulator to record measurementsover time. The transducer was calibrated using a variablehead of water at constant temperature as a pressurestandard. The calibration was linear in the operating pres-sure range.

Fluid was pumped through the channel using a HarvardApparatus PHD 2000 programmable syringe pump. Waterand the amplified and unamplified E. coli solutions werepumped through the channel at various flow rates; at20 �C. Pressure readings were continuously recorded andare presented in Section 4.

3.5. Optical velocity measurements

The development of l-PIV has enabled measurement ofthe velocity fields in liquid micro-flows. This facilitatesbetter understanding of the fluid transport phenomena, opti-mal design, and validation of numerical modelling of fluidbased MEMs. The principle and techniques of PIV are dis-cussed in greater detail in [24–26]. l-PIV differs from

macro-PIV in that volume illumination is used, rather thanplanar, and the images are acquired through a microscope.The measurement plane is defined through the depth of focusof the microscope objective lens [27]. The use of fluorescentseeding particles eliminates surface reflections. The firstmicron-resolution particle image velocimetry system wasdeveloped to measure instantaneous and ensemble-averagedflow fields in micron-scale fluidic devices [28]. A spatial res-olution in the velocity vector field of 6.9 lm in the plane par-allel to the flow direction and 1.5 lm in the depth directionwas achieved for a Hele-Shaw flow. The development andapplications of l-PIV are reviewed in detail in [29].

In this investigation 0.49 lm diameter polystyrene parti-cles, of approximate neutral buoyancy, were used to seedthe flow. The particles were internally labelled with a fluo-rescent dye that absorbs at a peak wavelength of 542 nm(green) and emits at a peak wavelength of 612 nm (red).The l-PIV system consisted of an IX50 Olympus inverted-stage epi-flourescent microscope fitted with a constant illu-mination mercury lamp and fluorescent filters to view thefluorescent particles. The microscope was mounted on avibration isolation table. A CCD camera with a 768 · 484pixel array was attached to the side port of the microscopeto capture flow images. Velocity measurements wererecorded at a flow rate of 0.6 ll/min (1 · 10�11 m3/s). Theexperimental set-up for pressure and velocity measurementsin the channel is shown in Figs. 4 and 5. An image of theseeded flow is shown in Fig. 6.

The measurement depth, dzm, for an intensity thresholdof one-tenth of the maximum in focus intensity, is given [27].

dzm ¼3nk0

NA2þ 2:16dp

tan hþ dp ð7Þ

In the present investigation the measurement depth of thel-PIV system was calculated to be 31 lm.

A total of 30 image pairs were recorded in a straightsection of the microchannel, at a location 2.7 mm down-stream of a bend, for steady flow of the three solutions at20 �C. The time between image pairs was 33.33 ms. Theimages were interrogated and validated using Insight 5software (TSI Inc.).

The following analysis and validation procedurewas applied. An interrogation region of 32 · 32 pixels in

Fig. 4. Microchannel test-section sitting on an inverted microscope. The syringe pump delivers fluid to the device and the datalogger records temperatureand pressure data.

Fig. 5. Experimental set-up showing the inverted microscope, syringe pump and pressure transducer sitting on a vibration isolated optical table. The PIVsystem is on the right, while the camera is attached to the left side-port of the microscope (covered beneath syringe pump).


cross-correlation mode with 50% overlap and zero windowoffset was used. The channel wall area of the image fieldwas excluded from interrogation by masking.

TSI’s Direct Correlator with a gaussian peak locator, anintensity threshold setting of 10% of the in-focus intensityand a minimum signal-to-noise ratio of 1.1 was used toproduce the vector fields. Considering that flow directionat the measurement location is predominately in the posi-tive x-direction a velocity range in the x-direction less than0 lm/s and greater than +600 lm/s was excluded to elimi-nate spurious vectors. A velocity range in the y-directionless than �100 lm/s and greater than +100 lm/s was alsoexcluded. A median criterion using a 3 · 3 neighbourhood/kernel size with a tolerance of 200 lm/s was applied.

Vectors that did not satisfy the specified criteria wereremoved and interpolated using a 3 · 3 neighbourhood/kernel size with a tolerance of 200 lm/s. No smoothingfunction was applied. An ensemble average of the 30 vectorfields was produced for each of the three flows. The resultsare presented in Section 4.

4. Results

Results are presented for particle geometry measure-ments and predicted viscosity for E. coli DNA solutionunder-going PCR. Pressure measurements for water andE. coli solutions flowing through the microchannel are alsopresented. An example of the full velocity vector field from

Fig. 6. A still image of the 0.49 lm red fluorescing microspheres in themicrochannel. Flow is from bottom to right. The frame is over-exposed tocreate a particle streaking effect, a stagnation zone is observed at the elbowwhere the particles are clearly defined. The l-PIV measurements wererecorded at a location 2.7 mm downstream of the bend to avoid flowdevelopment effects. (For interpretation of references in color in this figurelegend, the reader is referred to the web version of this article.)

0

50

100

150

200

250

0 5 10 15 20 25 30 35 40PCR Cycle Number

Visc

osity

Cha

nge

/ (%

)

240bp E. coli Standard Undiluted10 fold Dilution100 fold Dilution

Volume Fraction Limit

Fig. 8. Predicted viscosity increase with PCR cycle number for various 10-fold dilutions of the 240 bp E. coli DNA standard from Eq. (6). The initialviscosity was assumed to be that of water at 20 �C. The data-points abovethe dashed line represents viscosity calculated for a volume fraction, u,greater than 0.1.


l-PIV measurements is shown and measured velocity pro-files are presented with their theoretical counterparts.

A TEM image of amplified E. coli DNA is shown inFig. 7. Fig. 8 illustrates predicted viscosity change withPCR cycle number. Typical pressure transducer readings

Fig. 7. TEM image of amplified E. coli DNA at a magnification of·40,200. The E. coli DNA is stained with uranyl acetate and appears blackagainst the grey background of the sample carbon formvar plate. Theaverage characteristic length was found to be approximately 15 nm andthe geometry was assumed spherical (i.e. shape factor of 2.5). The effectivehydraulic radius is two-thirds of the radius of gyration [16], giving ahydraulic diameter of 5 nm.

for fluid flow through the microchannel with step-increasesin flow rate are shown in Fig. 9. Steady state pressure trans-ducer readings for flow of water, unamplified and amplifiedE. coli solutions through the microchannel at various flowrates are presented in Fig. 10. Fig. 11 illustrates an Ensem-ble-Averaged l-PIV vector field of 30 vector fields for flowof amplified E. coli solution in the microchannel. l-PIVvelocity profiles for water and E. coli solutions comparedto theoretical velocity profiles for water and E. coli areshown in Fig. 12. Normalised l-PIV velocity profiles arepresented in Fig. 13. A theoretical velocity prediction,using Eq. (1), for flow of the E. coli solutions in the micro-channel is shown in Fig. 14. A theoretical solution was alsocalculated corresponding to the measurement depth ofl-PIV measurements across the microchannel using Eq.(1). The theoretical velocity distribution consists of 50velocity point values at regular intervals and 30 velocitypoint values in the depth wise direction as shown in

0

50

100

150

200

250

0 500 1000 1500 2000 2500 3000Time/(s)

Pres

sure

/(Pa)

Pressure Transducer Flow R

ate/(μl/min)

0.6

1.2

1.8

2.4

3.0

3.6

4.2

4.8

Fig. 9. Typical pressure transducer readings for flow of water through themicrochannel with step-increases in flow rate over time (1 ll/min =1.67 · 10�11 m3/s). Capacitance effects are evident with a change in flowrate.

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

3.0E-04

3.5E-04

4.0E-04

4.5E-04

-0.125 -0.075 -0.025 0.025 0.075 0.125

Channel Width/(mm)

Velo

city

/(m/s

)

uPIV WateruPIV E. coli UnamplifieduPIV E. coli 40 cyclesPredicted E. coli Velocity for Viscosity = 0.00125 Pa sTheory Water

Fig. 12. l-PIV velocity profiles for water and E. coli solutions comparedto a theoretical solution for water and fitted curve for E. coli. The fittedcurve for the E. coli solutions is the average of the velocity profiles shownin Fig. 15.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5x*

u*

uPIV WateruPIV E. coli UnamplifieduPIV E. coli 40 cyclesPredicted E. coli Velocity for Viscosity = 0.00125 Pa sTheory Water

Fig. 13. Normalised l-PIV velocity profiles for water and E. coli solutionscompared to a theoretical solution for water and fitted curve for E. coli.

0

50

100

150

200

250

300

3500.

0

0.6

1.2

1.8

2.4

3.0

3.6

4.2

4.8

5.4

Flow Rate/(μl/min)

Pres

sure

/(Pa)

Theory WaterPressure Transducer Water

Pressure Transducer E. coli UnamplifiedPressure Transducer E. coli 40 Cycles

Fig. 10. Steady state pressure transducer readings for flow of water,unamplified and amplified E. coli solutions through the microchannel atvarious flow rates.

Fig. 11. Ensemble-averaged l-PIV vector field of 30 vector fields for flowof amplified E. coli solution in the microchannel. Flow is from left to right.The vector resolution is 4.86 lm. The Reynolds number, based on theaverage velocity of 1.95 · 10�4 m/s, is 0.044.


Fig. 15. The actual theoretical velocity profile is calculatedby averaging through the 31 lm depth of this distribution,producing a single velocity profile consisting of 50 velocitypoints across the width of the channel. Fitting a 5th orderpolynomial to these velocity points results in a continuousprofile. This continuous velocity profile can then be com-pared to measurements at any position across the widthof the channel. Inserting the measured pressure into Eq.(1) and adjusting the value of viscosity results in a best fitof the continuous theoretical velocity profile to experimen-tal velocity measurements across the entire width of themicrochannel, giving the viscosity of the solution. This

results in a predicted viscosity of 0.00125 Pa s for theE. coli solutions.

Fig. 14. Theoretical prediction for flow of E. coli solutions in the micro-channel for a measured pressure loss of 39.2 Pa. The distribution is formedusing Eq. (1) and represents a 50 · 50 velocity array.

Fig. 15. Theoretical prediction corresponding to the depth of focus of l-PIV measurements for flow of E. coli solutions in the microchannel for ameasured pressure loss of 39.2 Pa. The average of this distribution isshown in Fig. 12. The theoretical velocity prediction is fitted to themeasured velocity profile by inserting the measured pressure into Eq. (1)and incrementing the value of viscosity until a best fit with measurementsis obtained. This results in a predicted viscosity of 0.00125 Pa s for flow ofthe E. coli solutions in the microchannel. The distribution represents a50 · 30 velocity array.


5. Discussion

Fig. 10 illustrates that both of the E. coli solutions areNewtonian as the pressure difference increases linearly withflow rate. For a given flow rate the pressure difference inthe E. coli solutions is greater than that of water. Thereforethe viscosity of both E. coli solutions is greater than that ofwater. There appears to be no difference in viscositybetween the amplified and unamplified solutions as themeasured pressure difference is the same for both solutions.Fig. 8 suggests a 19.3% change in amplified solution viscos-

ity by cycle number 36 at the volume fraction limit, yet theviscosity of both solutions was 0.00125 Pa s. A viscosity of0.001244 Pa s was reported in [18] for measurements of anunamplified E. coli solution in a commercial viscometer.This represents a difference of 0.5%.

Eq. (5) describes an unsustainable exponential growth inthe concentration of E. coli DNA and also assumes a con-stant efficiency. In the initial cycles of PCR the reaction effi-ciency can be assumed to be constant but it reduces as thereaction progresses due to the fact that the enzyme (TaqDNA polymerase) is exhausted [30]. Therefore, the expo-nential growth in viscosity, as portrayed by Fig. 8, doesnot continue for the full duration of the reaction and shouldbe accompanied by a plateau phase towards the end of thereaction. The reaction may have entered the plateau phasebetween cycle 30 and 35, in which case the change in visco-sity would be insignificant according to Fig. 8. Another pos-sible reason for no viscosity change may be due to a lowerthan expected reaction efficiency.

The major implication for design and simulation ofl-PCR devices, in the context of low molecular weight mol-ecules, is that constant viscosity models can be employedlegitimately. However, an increase in fluid viscosity withPCR and its implications in micro-devices merits furtherinvestigation. This is especially prevalent in the case ofhigher initial concentrations and also in PCR of DNA mol-ecules with a much higher molecular weight. Larger mole-cules would certainly have a non-spherical geometry [31] insolution which would imply a larger shape factor and hencea larger viscosity.

The l-PIV data in Fig. 12 exhibits a very slight asymme-try. This is attributed to a slight rotational misalignment ofthe CCD camera with the flow field and is not due to flowdevelopment effects. Velocity measurements were recordeda sufficient distance downstream from a bend to avoid flowdevelopment effects. The development length was calcu-lated to be a mere 6.46 lm for water at the flow rate atwhich l-PIV data was recorded, using the equation givenby [32]. The measurements were recorded at a location2.7 mm downstream of the bend.

Fitting a 5th order polynomial to the theoretical velocitypoints results in a continuous profile with a standard curve fiterror of 0.02%, as a percentage of the mean. The theoreticalflow rates calculated from the theoretical velocity profiles are�0.55% and +0.65% of the experimental flow rate, 0.6 ll/min (1 · 10�11 m3/s), for water and E. coli solutions respec-tively. The standard error of the measured velocity from thetheoretical velocity profile for water, unamplified E. coli

solution and amplified E. coli solution are 6.51%, 4.45%and 3.18% respectively. The combined error is 7.08%,5.12% and 3.85% for water, unamplified and amplifiedE. coli solutions respectively.

All other variables being equal the percentage error invelocity measurement is equivalent to a percentage errorin viscosity measurement. Hence, the uncertainty in the vis-cosity prediction of 0.00125 Pa s for the unamplified andamplified E. coli solutions is 5.12% and 3.85% respectively.


This is comparable to the 3% error for blood plasmameasurements in a nanoliter viscometer reported in [15].

Error due to Brownian motion is intrinsic in the calcu-lated uncertainty. Seed particles chosen for l-PIV experi-ments must be small enough to faithfully follow the flow.They must also be large enough to image and to dampenthe effects of Brownian motion without blocking the test-section. An approximation for the relative error due toBrownian motion in velocity fields is given in [28]. For acharacteristic velocity of 1.95 · 10�4 m/s the error due toBrownian motion in velocity measurements in water was0.19%. For the amplified and unamplified E. coli solutionsthe error was found to be 0.16%.

The TEM measurements of the E. coli DNA geometrywere recorded on dry samples. The effect of fluid flow onmolecule conformation requires assessment. The effect ofshear and elongation on DNA macromolecules of 48,502base pairs in a microchannel flow was observed in [31]. Avariety of molecule conformations from coiled to stretchedwere recorded using fluorescent microscopy. At a theoreti-cal wall shear stress of 0.2 N/m2 the majority of moleculesremained coiled. The onset of molecule elongation wasnoted at 0.4 N/m2. Chain scission was suggested to haveoccurred at 0.6 N/m2. The presence of elongation floweffects was unlikely to have occurred in the investigatedflows, as the fluid was not subjected to acceleration in thechannel. However, wall shear stress should be considered.The maximum measured wall shear stress was 9.67 ·10�3 N/m2. This level of shear stress is substantially lessthan the level reported in [31] at which the majority of mol-ecules remained coiled. Molecule concentration was alsofound to have an effect on individual molecule geometry[33]. For similar flow conditions, molecules in a concen-trated DNA solution (28.15 pg/ll) were observed to stretchless in comparison to a dilute solution (0.3 pg/ll). Inter-preting this finding, the authors suggested that a greaterconcentration of molecules in solution dissipates theapplied stress more effectively. In the case of the unampli-fied E. coli solution the concentration was 10 pg/ll. Theprojected concentration in the amplified solution at the vol-ume fraction limit in Fig. 8 is 16.38 lg/ll. This predictionsuggests that as a PCR progresses the solution moves froma semi-dilute towards a concentrated state. A combinationof the low wall shear stress in both solutions and the largenumber of molecules present in the amplified solutionwould indicate a tendency for the E. coli molecules toremain coiled. Thus the spherical geometry chosen tocalculate the change in viscosity due to amplification inEq. (6) is most appropriate for the applied flow rate in thiscase.

It may be argued that the use of a costly PIV system torecord such data would hinder the widespread implementa-tion of the technique described. However, emerging tech-nology such as on-chip tunable dye lasers [34], in situmicroflow sensors using flowing thermal lenses [35], micro-channel integrated hotwire systems [36] and pressure trans-ducers will provide a more affordable and fully integrated

means of employing the viscosity measurement techniquedescribed.

6. Conclusions

A theoretical expression for viscosity in PCR was pre-sented. Viscosity measurements indicated no increase influid viscosity after PCR for a low molecular weight mole-cule. The major implication for design and simulation ofl-PCR devices, involving low molecular weight molecules,is that constant viscosity models can be employed legiti-mately. Further investigation is merited. This is most rele-vant in the case of large initial concentrations and also inPCR of DNA molecules with a high molecular weight.

Using l-PIV and pressure transducer measurements ina microchannel to infer fluid viscosity represents a novelimplementation of existing measurement techniques. As itdoes not require the extraction of a fluid sample it can beclassified as a zero-volume viscosity measurement tech-nique. An uncertainty of 3.85% in viscosity measurementshas been demonstrated. Viscosity measurements to within0.5% of a commercial viscometer have been achieved.

Acknowledgements

The authors would like to acknowledge the financialsupport and facilities provided by The Stokes ResearchInstitute and the Mechanical and Aeronautical Engineeringdepartment, University of Limerick. Thank you to AngelaMorris and Dr. Fiona Gilchrist, Stokes Research Institute,for solution and TEM sample preparation. Thank you toDavid Tanner, MSSI, University of Limerick, for TEMmeasurements.

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utilising μ-piv and pressure measurements to determine the viscosity of a dna solution in a...

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