the use of microfluidics in rheology

Review

308

The Use of Microfluidics in Rheology

Xin Hu, Pouyan E. Boukany, Orin L. Hemminger, L. James Lee*

The combination of microfluidics and fluorescent-labeled DNAmolecules in rheology providesa unique opportunity to relate the microstructure of polymer molecules to their macroscopicproperties. For example, the direct visualization of amodel polymer system such as single DNAdynamics of individual DNA molecules, improves ourunderstanding of the rheological behavior of diluteand concentrated polymer solutions. This review sum-marizes the microfluidic rheology for synthetic polymerand DNA solutions, including both experimental andsimulation efforts, in investigating dilute and concen-trated polymer solutions.

Introduction

Rheology is the science of the deformation and flow of

materials. Colloids, emulsions, liquid crystals, synthetic

polymersandbiopolymers (DNA,proteins), foams, gels, and

membranes are widely used in our everyday life. The

rheological properties of these complex fluid systems are

oftencomplicatedbecauseof their structural viscoelasticity

as well as interactions among their fluid constituents.[1–5]

A central focus in the field of polymer rheology has been

constitutive relations derived from information gathered

from conventional rheometric measurements. With the

newly developed tools such as microfluidics-based rhe-

ometers and advanced fluorescence microscopy, we are

now able to directly image and understand the micro-

structures and dynamic behavior of long-chain molecules

in well-defined fluid fields and relate the visualization to

macroscopic rheological responses. The lengthscale studied

in conventional rheology is usually on the order of one

millimeter. Thus, conventional rheology or macrorheology

X. Hu, P. E. Boukany, O. L. Hemminger, L. J. LeeNanoscale Science and Engineering Center for AffordableNanoengineering of Polymeric Biomedical Devices (CANPBD)E-mail: [email protected]. E. Boukany, L. J. LeeDepartment of Chemical and Biomolecular Engineering, The OhioState University, Columbus, OH, 43210 USA

Macromol. Mater. Eng. 2011, 296, 308–320

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cannot provide significant direct information on the

molecular nature of fluids. For this reason microrheology,

i.e., rheology at the microscale, is receiving increased

attention in recent years.[6]

There are a number of advantages to usingmicrofluidics

in rheology. First, microfluidic devices can produce high

shear rates, high Weissenberg (Wi) and Deborah (De)

numbers at low Reynolds (Re) numbers, not achievable

by conventional rheometric measurements; where

Wi ¼ _gl, with _g being the local characteristic shear rate,

and l the dominant relaxation time, De¼ l/t, with t being

the characteristic flow time and Re¼ r _gw2hh _gð ÞðwþhÞ, with r being

fluid density, w the width and h is the height, and h _gð Þ theshear viscosity. Second, only a small amount of sample is

needed in microfluidics, thus making high-cost materials

and reagents such as DNA solutions affordable as a model

fluidspolymer inrheologicalmeasurements. Third, theflow

field in themicrofluidic device systems can also be directly

observed imaged by an inverted or confocal florescence

microscope and analyzed by the microparticle image

velocimetry (mPIV) technique[7] with the fluid viscosity

being estimated simultaneously by either mounting a

pressure transducer onto the micro-fabricated device or

applying non-contact force measurement techniques such

asusinganoptical tweezers.[8] Furthermore, different types

of flows can be easily generated in microfluidic-based

devices without any moving parts. For example, micro-

fluidic devices consisting of several microchannels have

been designed to produce extensional, mixed shear, and

library.com DOI: 10.1002/mame.201000246

L. James Lee is the Helen C. Kurtz Professor ofChemical Engineering at The Ohio State Univer-sity, and the Director of NSF Nanoscale Scienceand Engineering Center for Affordable Nano-engineering of Polymer Biomedical Devices. Hereceived a BS degree from National Taiwan Uni-versity and a Ph.D. degree from University ofMinnesota in 1979. His research covers polymerengineering, micro-/nanotechnology, and bio-technology. He has> 220 papers and 25 patents.Dr. Lee is a Fellow of Society of Plastics Engineers(SPE). He received the 2008 Malcolm E. PruittAward from Council of Chemical Research, 2008Engineering/Technology Award and 2010 Inter-national Award from SPE.

Xin Hu received the M.S. degree from PekingUniversity, China in 2000 and the Ph.D. degree inMechanical Engineering from The Ohio StateUniversity in 2006. He worked as a researchengineer in the center for Affordable Nanoengi-neering of Polymeric Biomedical Devices from2006 to 2010. His researchmainly focused on themesoscopic simulation of polymeric flows, FEMsimulation of electrokinetic micro/nanofluidics,Brownian dynamics simulation of long chainpolymers in microfluidics, and FEM simulationof cell electroporation in micro/nanodevices.Now he is a thermal and CFD scientist in UES,Inc. and works on simulation of thermal storagemanagement.

Pouyan E. Boukany obtained his M.S. degree inTextile Engineering with major in Textile Chem-istry and Fiber Science from the Isfahan Univer-sity of Technology, Iran. He received his Ph.D. inPolymer Science from University of Akron in2008. His doctoral work was to explore thenonlinear flow behavior of entangled DNA fluidsin Professor Shi-Qing Wang’s group. He is cur-rently a Postdoctoral Research Associateworkingwith Professor L. James Lee in the Center forAffordable Nanoengineering of Polymeric Bio-medical Devices at The Ohio State University.

The Use of Microfluidics in . . .

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rotational flow patterns as in a ‘‘four-roll mill’’ type

rheometer,by adjusting the relative flow rates in micro-

channels.[10,11] Since thesemicrofluidic devices do not have

any moving parts, the fabrication is much easier and thus

less costly. Finally,microfluidic rheologyconductedon ‘‘lab-

on-a-chip’’ devices can be used to study non-Newtonian

flow behavior responses of biological fluids for medical

applications. Examples include serumflow inbloodvessels,

protein and DNA flow manipulation in bioseparation

devices, and controlled drug/gene delivery.

Polymer materials studied in rheology range from

solutions to melts. Due to space limitations, we only cover

the microfluidic rheology of polymer solutions here. This

review article is organized in the following order: Section 1

is a brief introduction to microfluidic rheology and its

advantages. Section 2 focuses on the experiments and

simulations of microfluidic rheology of conventional poly-

mer solutions. Section 3 discusses the use of DNAmolecules

as a model polymer for molecular imaging in microfluidic

rheology. In Section 4, we give several future directions

relevant to microfluidic rheology are proposed. Finally, a

summary and conclusions are provided in Section 5.

Microfluidic Rheology of Polymer Solutions

In this section, we focus on the dynamic response of

polymer solutions through micro-fabricated flow geome-

tries. The flow fields of such polymer fluids inmicrofluidics

can be characterized by using either micro-tracer particles

or numerical simulation approaches.

Experimental Observations of Flow Instabilities inMicrofluidics

Most polymers used in our daily life, from plastics to

synthetic fibers to elastomers, aremade fromentangled long

chains in themolten or solution state. By pushingor drawing

them through an extrusion shaping die or an injection

molding machine, these materials can be formed into

desirable shapes to develop useful products.[12,13] At high

deformation rates (De or Wi> 1.0), these entangled fluids

exhibit complex flow phenomena such as shear thinning,[14]

transient stress overshoot in start-up shear,[15,16] wall slip/

shear banding during shear,[17] necking during extension,[18]

rod-climbing, vortex formation in extrusion, spurt in

capillary flow, and extrudate (or die) swell and/or melt

fracture.[12,13] Molecularmechanisms of these flowphenom-

ena are still unclear despite decades of intensive experi-

mental and theoretical investigation.[19] New insights in

polymer rheology can be gained through the integration of

microfluidics-based ‘‘rheometer-on-a-chip’’ and optical/con-

focal microscopy, i.e. rheo-microscopy.

Flow through a contraction channel is one of the most

widely studied geometries in the field of polymer rheology

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Macromol. Mater. Eng. 2

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because of its importance in many polymer processing

operations. For example, the gross melt fracture of the

extrudate may be originated from the instability near the

entrance to a contraction geometry. Using a microfluidic

device, our group showed that the entry flow pattern in a

converging flow geometry was very different for two

aqueous polymer solutions (2% polyethylene oxide, PEO

and 1% hydroxyethyl cellulose, HEC) due to strain-hard-

eningvs. softening response in theextensionalflow(shown

in Figure 1 A–C). Local strain-hardening of PEO led to the

enhanced vortex formation near the inlet of a micro-sized

converging channel (with 368 convergence angle).[20]

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Figure 1. (A) Extensional response of PEO and HEC solutions. Entrance flow patterns for(B) PEO and (C) HEC solution. Reproduced with permission.[20] Copyright 2006, Springer.

Figure 2. (A) Summarizing 4:1 macroscopic contraction flow experiments on a Wi-Re sfluids. Summary of flow patterns in theWi-Re space for (B) different semi-dilute PEO solutEI¼Wi/Re¼ 2.8–68 at the same concentration of PEO solution (�0.075% PEO) inpermission.[21] Copyright 2005, Elsevier.

310 Macromol. Mater. Eng. 2011, 296, 308–320

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X. Hu, P. E. Boukany, O. L. Hemminger, L. J. Lee

Figure 2A shows regions of the Wi-Re

space for contraction flows containing

both inertia and elasticity in macroscale

experiments.[21] The highest achievable

Wi was less than 10 for Re less than one

in such cases. Using dilute aqueous

polymer solutions in contraction-

expansion microchannels, a similar flow

behavior over amuch broader range of Re

(0.44< Re< 64) andWi (0<Wi< 548) has

been achieved, which had previously

been unexplored because such conditions

were not accessible in the equivalent

macroscale experiments. We briefly

introduce several representative studies

here.

Rodd et al.[21] studied the flowbehavior

of semi-dilute PEO. solutions to in a

16:1:16 contraction-expansion flow geo-

metry. Both pressure drop and flow

visualization were used to characterize

pace for shear thinning viscoelastic and Bogerions (0.05–0.3% PEO), and (C) different ranges ofa contraction microchannel. Reproduced with

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Figure 3. Summary of flow patterns in theWi-Re space for four different DNA solutionsfrom weakly to highly entangled solutions in a 4:1 contraction microchannel.Reproduced with permission.[24] Copyright 2010, Elsevier.


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the flow dynamics over a range of Re

(0.44<Re< 64) and Wi (0<Wi< 548) as

shown in Figure 2B. They reported that

the onset of elastic instabilities at the

contraction entry occurred at a criticalWicof 50.[21,22]

Rodd et al. also prepared solutionswitha

constant concentration of PEO but four

different glycerol-water mixtures to vary

the viscosity of the solvent. This allowed

them to investigate the entry flow

behavior in the Wi-Re space for four

different elastic numbers (El¼Wi/

Re¼ 2.8, 7, 19, and 68). Each solution

exhibited the same four flow regimes,

identified as Newtonian like flow, steady

viscoelastic flow, diverging flow, and

elastic corner vortex growth as shown in

Figure 2C. For the three lower El values, 2.8,

7, and 19, the Wi at which the transition

between two flow regimes occurred was

found to be aweak function of El, while the

same transition occurred at a higher Wi

Figure 4. Dye advection patterns for a cross-channel flow withtwo inputs and two outputs for (a) a Newtonian fluid, and (b) aPAA flexible polymer solution (De¼ l/t¼ 4.5), where the interfacebetween dyed and un-dyed fluid is deformed by flow instability.(c) and (d) Particle streak lines corresponding to (a) and (b)showing the symmetry-breaking instability. Reproduced withpermission. [27] Copyright 2006, American Physical Society.

when the El value was 68.[22]

Gulati et al. employed a semi-dilute DNA solution

containing 400mg/mL (0.04wt.-%) of l-DNA and investi-

gated its flow behavior through a 2:1 planar contraction

microchannel.[23] When 3.9<Wi< 193.3, a symmetric

growth of the corner vortex was observed. Recently, our

group used four different entangled DNA solutions with

concentrations ranging from 0.1 to 1.0wt.-% (with a wide

range of entanglements per chain Z¼ 7–55) to study the

flow of entangled fluids through a 4:1 contraction

microchannel. We achieved Wi higher than 20,000 with

Re less than 0.5, a regime not been reached in the past. For

weakly entangled solutions (Z< 30), the vortex flow was

dominant at high flow rates. However, for well entangled

DNA solutions (Z� 30), an unusual time-dependent shear

banding was observed near the contraction entrance

(shown in Figure 3).[24]

Microfluidics with complicated geometries such as

cross-slot, stagnation point, triangular (nozzle/diffuse)

shape, flow resistors and T-shaped geometries have

also been used to study flow instabilities of non-

Newtonian fluids.[25] Pathak and Hudson used a cross-

slot microfluidic device to investigate the extensional

rheology of a polymer solution. At high flow rates, a

central birefringent band representing a highly aligned

microstructure was observed near the stagnation point.[26]

Using the same geometry, Arratia et al. showed that

the velocity field of a polyacrylamide solution (PPA)

became strongly asymmetric with non-periodic fluctua-

tion at sufficiently high strain rates and low Re (shown

in Figure 4).[27]




Recently, Soulages et al. used a T-shapedmicrochannel to

investigate the instability of the steady planar stagnation

flow of PEO solutions. As shown in Figure 5, a strand of

highly oriented polymeric material was formed in the

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Figure 5. Viscoelastic (a–e) and Newtonian (f–j) flow patterns as a function of flow ratein a T-shaped microchannel. Reproduced with permission.[28] Copyright 2009, Elsevier.

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region of the strong planar extensional flow, resulting in an

additional symmetry-breaking transition at intermediate

Wi. A flow transition from a steady to an unsteady three-

dimensional flow was observed at a critical Wi for each

stagnation flow.[28]

Sousa et al. employed a microfluidic rectifier to produce

creeping flow conditions.[29] For viscoelastic fluids, the

pressuredropwas foundtobeconstant in theflowdirection

at low flow rates. However, increasing the flow rate led to

an anisotropic flow resistance in the forward and

backward flow directions at the same pressure drop, i.e.,

rectification effects emerged. The viscoelastic fluid flow

became unsteady in the forward direction due to the

emergence of elastic instabilities and the flow resistance

increased sharply as a function of the flow rate (shown in

Figure 6).

Figure 6. Flow patterns of a PAA fluid in the microchannel with a triangular shapeReproduced with permission.[29] Copyright 2010, Elsevier.


� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinh

Simulation of Viscoelastic Fluids inMicrofluidic Rheology

Most polymer fluids exhibit shear-rate

dependent viscosity and some solid-like

elastic properties. These two properties

are tightly inter-connected as viscoelas-

ticity, a critical property of many non-

Newtonian fluids. The governing

equations of non-Newtonian fluid flows

contain an unknown polymer stress

tensor, making the flow simulation a

non-trivial task. Typically, close-formed

constitutive equations (CEs) are used to

solve for polymer stresses. This macro-

scopic simulation approach only solves

the macroscopic variables such as poly-

mer stresses and the velocity field. Con-

ventional simulation methods such as

finite element (FE), finite difference (FD),

and finite volume (FV) methods are

applied to simulate viscoelastic fluids using various CEs

(for more information, please refer to.[2,3])

Several researchers have applied the macroscopic

simulation approach to microfluidic rheology. Poole et al.

investigated the stability of an upper-convected Maxwell

(UCM) fluid in a cross-slot microchannel,[30] while Soulages

et al. studied the instability of a Phan-Thien-Tanner (PTT)

fluid in a T-shaped microchannel.[28] Although some CEs

have considered the polymer configuration by treating

polymer molecules as bead-spring chains,[4,31–34] the CEs-

based simulation of polymer flows cannot represent the

true behaviors of polymers. For example, the polymer

chains near the wall behave differently from those in the

bulk flow due to the effects of hydrodynamic interaction

(HI) and repulsive forces from the wall. Since CEs do not

consider the existence of the channel wall, the effect of

polymer-wall interactions on polymer

flow cannot be exhibited in the macro-

scopic approach. The CEs-based simula-

tion also cannot achieve high Wi or De

numbers due to severe numerical

instability.

An alternative way to simulate viscoe-

lastic fluids is the so-called mesoscopic

approach. This approach simulates poly-

mer molecules in a coarse-grained way

and calculates polymer stresses by an

ensemble average. For example, the

CONNFFESSIT (Calculation of Non-New-

tonian Flow: Finite Elements andStochas-

tic Simulation Technique) method is a

mesoscopic simulation method devel-

oped by Laso and Ottinger.[35] In this

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method, the flow domain is first discretized into a finite

element mesh. Polymer molecules are simplified as bead-

spring or bead-rod dumbbells or chains. There are two

differentmethods to calculate the configuration of polymer

chains. One is the Lagrangian method by throwing dumb-

bells or chains into the flow field and tracking their

positions and conformation; the other is the Eulerian

method by associating dumbbells or chains with the nodal

point and only calculating their configuration changes. The

latter method is called the Brownian configuration field

(BCF) method,[36] as shown in Figure 7.

The CONNFFESSIT method uses the iteration method to

calculate the steady state velocity field. At the beginning, a

Newtonian velocity field is used to calculate the config-

uration of polymer chains in the flow field. Then polymer

stresses are calculated with the Kramer’s expression.

Finally the velocity field is updated by adding the

contribution of polymer stresses. This process is repeated

until the numerical solution is converged.[37]

The CONNFFESSITmethod has been successfully applied

to simulate diverse viscoelastic fluid flows (steady or

transient) such as in 4:1 abrupt contraction,[38,39] journal

bearing flow,[40] passing single cylinder or cylindrical

arrays,[39,41] die exit,[42] gas assisted injection molding,[43]

and others. The application of CONNFFESIT inmicro-scaled

viscoelastic flows is straightforward. For example, our

group studied the vortex formation in a channel with

microfeatures.[44] We used the FENE dumbbell in CON-

NFFESSIT and obtained similar results to those shown by

experiments.

Although the CONNFFESSIT-based simulation can reach

a higher Wi number than CEs-based simulation, it still

cannot achieve the same Wi numbers observed in the

microfluidics experiments, which could be as large as

20,000 with concentrated polymer solutions. One way to

solve this problem is to replace the bead-spring dumbbell

with the bead-spring chain in the simulation.[45,46] In the

future, a finer-grained simulation approach such as

molecular dynamics (MD) and the multi-scaled simulation

methods combining MD, BDS and FEM are expected to

emerge as a solution for modeling viscoelastic flows,

particularly for microfluidic rheology.

Figure 7. A schematic of CONNFFESSIT with the BCF method.




DNA as a Model Polymer for MolecularImaging

Macroscopic properties of polymer solutions are directly

related to the dynamics of individual polymer molecules

and their interactions. Once the behaviors of individual

polymer molecules and molecule-molecule interactions

canbeanalyzed, the reasonwhyapolymersolutionshowsa

particularproperty inacertainflowfieldcanbeunderstood.

However, conventional polymermolecules are too small to

be directly observed. Also macroscopic experimental

techniques such as light scattering and birefringence

cannot detect changes in microscopic configurations of

polymer molecules during flow.

This was the situation until in 1994 when Steven Chu’s

group first used pre-stained long chain DNA as a model

polymer to investigate the rheological properties of dilute

polymer solutions at the molecular level.[47] DNA

molecules are much larger than typical polymer molecules

(> 10M vs.< 1� 106 gmol�1 in molecular weight). For

example, the contour length of a naked l-DNAmolecule (48

kbps, molecular weight � 31.5� 106 gmol�1) is around

16mm.Alsowhen labeledwithfluorescent dye, DNA canbe

directly observed with the fluorescence microscope. The

brilliant idea of using DNA as a model polymer directly

connects microscopic dynamics to macroscopic properties

of polymer solutions and opens a new door to the field of

polymer rheology.

Experiments and Simulations of DNA Dynamics inDilute Solutions

Free Relaxation of DNA Molecules

Perkins, et al. first studied the free relaxation of individual

fluorescent-dyed and highly-extended DNA molecules in

dilute solutions.[48] The relationship between the visual

length of DNAmolecules and timewas recorded in order to

study the free relaxation process. Figure 8 shows the visual

length vs. time for three types of DNA molecules with

contour lengths of 7.7, 21.1, and 39.1mm. Inset shows the

relaxationprocessof a tetheredDNAmoleculewithoneend

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attached to a polystyrene latex micro-

sphere, which was trapped by the optical

tweezers.

From this data, the longest relaxation

time can be calculated. Also DNA mole-

cules with different lengths were used to

investigate the scaling law on the longest

relaxation time t. They found that t follows

a scaling law on the polymer length L,

t� L3y, where 3n is the scaling exponent.

Here 3y ¼ 1:66� 0:10 or 1:65� 0:13

depending on the data analysis methods

heim313

Figure 8. Relaxation of initial fully stretched tethered DNA mol-ecules with three different contour lengths. Inset is the relaxationprocess of a DNA molecule. Reproduced with permission.[47]

Copyright 1994, American Association for the Advancement ofScience.

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used in their experiments. This agrees well with the

experiments by intrinsic viscosity measurements[48] and

dynamic light scattering,[49] but disagrees with the

analytical theories in that the Rouse theory gives 3n¼ 2

for the freely draining chain; Zimm model predicts

that 3n¼ 1.5 for the ‘‘theta’’ solvent by considering the

hydrodynamic interactions; and 3n¼ 1.8 for a ‘‘good’’

solvent.[31,50] Thus, none of the current theoretical models

can reliablypredict the scaling law in the relaxationprocess

of DNAmolecules. Effects of monomer-solvent interaction,

hydrodynamic interaction, excluded volume force and the

even nonlinear elastic force of DNA chain should be added

into consideration in the theory.

Figure 9. (A) Coil-stretch transition for DNA molecules with four differextensional flow (inset); (B) ensemble average extension for difpermission.[51] Copyright 1997, American Association for the Advance



Coil-Stretching Transition of DNA Molecules in a PureExtensional Flow

Compared to the relaxation process, the transition of DNA

configuration from the equilibrium state to the non-

equilibrium state is a rather interesting topic of study.

One of thewidely usedmethods to drive DNAmolecules to

the non-equilibrium state is by using different types of

hydrodynamic flows. Since the velocity gradient can be

decomposed into symmetric and anti-symmetric parts,

there are typically two types of flow patterns: the

extensional flow related to the symmetric part of the

velocity gradient and the pure rotational flow related to the

anti-symmetric part of the velocity gradient. Other flow

patterns such as the simple shear andmixed flows are their

combinations.

Chu’s group investigated dynamics of untethered l-DNA

molecules inanearly2Dpureextensionalflowgenerated in

a cross-channel.[51,52] They observed the stretching of DNA

moleculeswhen theDenumberwas larger than0.4, slightly

less than the theoretical value of 0.5. With the aid of

fluorescent microscopy, Perkins et al. studied the coil-

stretch transition of DNA molecules with seven typical

initial conformations: dumbbell, half dumbbell, kinked,

folded, uniform, extended, and coil [53]. They found that the

conformation change of aDNAmoleculewas dependent on

its initial conformation. DNA molecules with different

initial conformations showed different coil-stretch transi-

tion process. Figure 9(A) shows the coil-stretch transitions

of DNA molecules with four initial conformations: dumb-

bell, half dumbbell, kinked and folded, while the averaged

extension vs. residency time for different initial conforma-

tions is shown in Figure 9(B). Such strongheterogeneity due

to the ‘‘individualism’’ of polymer molecules in the

extensional flow cannot be obtained by macroscopic

measurements such as light scattering and birefringence.

ent types of initial conformation near the stagnation point of a pureferent initial conformation vs. residency time. Reproduced withment of Science.

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Figure 10. Extension vs. residence time, or strain for DNA molecules with different initialconformations under (a) De¼ 2; (b) De¼ 48. Reproduced with permission.[52] Copyright1998, American Association for the Advancement of Science.


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The stretching of single DNA molecules under different

extensional rates, or De numbers was also studied. It was

found that both the amount of stretched DNA molecules

and the ensemble averaged extension increased at a larger

De number. Figure 10 shows that most of coiled DNA

molecules maintained their coiled status at smaller De

numbers, while all were stretched at a larger De number.

This indicates that DNA molecules experienced larger

deformation with a larger extensional rate, or a larger

stretching force.

The coil-stretch transition ofDNA in the extensional flow

has been successfully recaptured by the Brownian

dynamics simulation (BDS). Larson et al. showed that the

coil-stretch transition of l–DNA molecules can be simu-

lated with BDS using the worm-like chain (WLC) model.[50]

They did not consider the effect of hydrodynamic interac-

tion (HI) in the simulation. However, by adjusting the

effectivepersistence length, theyobtainedalmost the same

results as experiments done by Chu’s group.

The HI effect on polymer dynamics in bulk has been

investigatedbymany researchers.[54,55] TheHIvelocity, as a

perturbation to the bulk flow, is calculated through a

mobility tensor such as the Oseen-Burger tensor. However,

this tensor isnon-positivedefinitewhentwobead locations

are close. Thus, the Rotne-Prager-Yamakawa (RPY) tensor is

chosen to replace the Oseen-Burger tensor in the calcula-

tion. Heish and Larson applied the RPY tensor as the

mobility tensor in the BDS of dynamics of Polystyrene (PS)

and l-DNAmolecules in the extensional flowwith HI.[56,57]

Their results showed that theHI effectwasmore important

for the dynamics of short PS molecules, and it could be

neglected for the dynamics of l-DNA since the HI velocity

decreased rapidly with the increased distance between

two beads.



� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

For l-DNA, the effect of HI could be

neglected in the simulation without

affecting the overall stretching

amount as long as the effective persis-

tence lengthwasadjusted tomatch the

relaxation data in experiments. How-

ever, for a longer DNA with contour

length � 1.3mm, which is 62 times

longer than l-DNA, the HI effect could

not be neglected any more. Hysteresis

phenomenonwas observed by Schroe-

der et al. with such long DNA in the

extensional flow.[58] They found that

there are two different stretching

states of DNA molecules for the same

Denumber, or strain rate.At thecertain

range of strain rate, both coiled and

stretched DNAmolecules co-existed in

the buffer solution. In fact, this is the

first order coil-stretch transition pro-

posedbydeGennesand this ‘‘bistable equilibrium’’ is due to

the strong ‘‘deformation-dependentdrag’’ inducedbyHI for

such longDNAmolecule.[59] The conformationhysteresis of

long DNA molecules was simulated using BDS.[60] In the

simulation, DNA molecules with two different initial

configurations (stretched and coiled) were considered.

Quantitative agreementwas achieved between the experi-

ment and BDS.

Tumbling of DNA Molecules in Simple Shear & MixedFlows

DNA dynamics in a simple shear flow was also studied by

Chu’s group.[61] They found thatDNAmolecules displayeda

tumbling movement shown by the schematic in

Figure 11(a). A DNAmolecule can be stretched to be nearly

aligned to the flow direction (here it is along the x-axis

direction). Then if the y-component of Brownian force

slightly rotates the DNA molecule, two ends of the DNA

molecule might be exerted by external forces in the

opposite direction andmove towards their center-of-mass.

Thus, DNAmoleculewill be retracted to a coiled shape. This

is the so-called ‘‘coil-stretch-tumbling-recoil’’ movement of

DNA molecules in the simple shear flow. In experiments,

Smithetal. observedthat therewere threetypicalbehaviors

ofDNAmolecules in theshearflowasshowninFigure11(b).

In the first row, the DNA molecule showed a ‘‘coil-stretch-

tumbling-recoil’’ movement. In the second row, the DNA

molecule was stretched gradually to a certain length and

maintained its shape. In the third row, the stretching of

DNA molecule was weak, but its conformation changed

greatly and the ‘‘tumbling’’ movement was observed.

BDS has been successfully applied to simulate DNA

dynamics in the simple shear flow.[62] Hur et al. investi-

gated theaveraged stretchamountvs. theWinumberusing

315

Figure 11. (a) ‘‘Coil-stretch-tumbling-recoil’’ movement of DNA molecule in shear flow;(b) Dynamics of three DNAmolecules with different initial conformations. Reproducedwith permission.[61] Copyright 1999, American Association for the Advancement ofScience.

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FENE dumbbell, Kramer’s bead-rod chain, and WLC. They

also studied thepower spectral density (PSD) atdifferentWi

numbers.

Shearflowcanbeconisderedasasimplemixedflowsince

it is combined by half amount of extensional flow and half

amountof rotationalflow. Shaqfeh’s grouphas carriedouta

series of experiments and simulations of DNAmolecules in

linear mixed flows.[63,64] The velocity gradients in these

flows are constant, but the extensional part or extension

rate competes with the rotational part or vorticity. Their

workprovidedvaluable insights tounderstand thepolymer

dynamics in more complicated flows.

DNA Dynamics in Other Complex Flows

DNA dynamics was investigated in other complex flows

such as hydrodynamic focusing[65] and entrance flow from

the reservoir to the microchannel.[66] The flow gradients in

such complex microfluidic devices are not constant; thus,

DNA molecules experience different flow gradients when

they are moving with the flows. DNA dynamics in such

complex flows is time/location-dependent.

DNA dynamics in different hydrodynamic flows have

been throughly studied by many researchers. Since we

cannot cover all the experiments and simulations in this

topic due to the page limitations of this review, please refer

to comprehensive reviews by Larson[67] and by Shaqfeh.[68]

DNA Dynamics in Confined Geometries

DNA dynamics in confined geometries are important in

many biomedical applications such as DNA separation,

gene mapping, and gene delivery. Generally speaking,

when considering DNA dynamics in a confined geometry,

we need to consider all the interactions including wall-

polymer, intramolecular, and intermolecular forces.


� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Wein

It was reported by Fang et al. that the

DNA molecules near the wall (1mm

distance) of a microchannel were mainly

in coiled shapes in a shear flow,while DNA

molecules were stretched when they were

away from the wall. The DNA concentra-

tion near the wall was much lower than

that inbulk. Therewasadepletion layer for

DNA molecules at high flow rates[69]

because the symmetry of HI was cut off

due to the existence of the wall.

In simulation, there are different ways

to calculateHIwithwall. The simplestway

is to use the Stokeslet-based method or

Green’s function method to modify the

Oseen-Burger or RPY tensor obtained in the

bulk flow so that it can be used to calculate

HI withwalls.[70–72] However, thismethod

works only for simple geometries such as

microchannels with flat surfaces.

Calculation of HI in more general geometries is much

more complicated. For example, the ‘‘smoothed profile

method’’ (SPM),[73] a direct simulation for particulate flows,

could be used to calculate DNA dynamics with HI in

confined geometries. However, the computation time is

very large by a direct simulation method. To save

computation time in calculatingDNAdynamics in confined

geometries with HI, researchers are more interested in

combining BDS with finer-scaled simulation techniques

such as lattice Boltzmann method (LBM)[74] and stochastic

rotation dynamics (SRD).[75] Such multi-scaled simulation

methods are more efficient in computation than direct

simulation.

The effect of excluded volume (EV) from intra-/inter-

polymers and polymer-wall interaction is also important.

The EV force is calculated through the specified EV

potentials. Typical EV potentials are the truncated Len-

nard-Jones (L-J) potential,[76,77] hard core repulsive poten-

tial,[78] and repulsive narrow Gaussian potential.[79] All the

BDS in confined geometries have considered the EV effect

between two beads and between beads and wall.

DNA Dynamics in Concentrated Solutions

If the concentration of apolymer solution is higher than the

‘‘overlapping concentration’’, the effect of chain entangle-

mentmustbeconsidered.Chainentanglement isextremely

complicated and difficult tomodel. Edwards simplified this

problem into the ‘‘tube’’ model that assumes that an

entangledpolymer (either linear or cross-linked) is confined

in a tube-like region by surrounding polymers.[80] The

motion of polymer inside this ‘‘tube’’ is called ‘‘reptation’’,

which is proposed by de Gennes.[81] He considered the

diffusion of polymer chains through the entangled net-

heim www.MaterialsViews.com

Figure 12. (a) Schematic diagram of a single DNA molecule stretched between twooptically trapped micro-spheres in a concentrated solution of entangled DNA. (b)Reptation models postulate that collective intermolecular interactions give rise to atubelike confining field (dashed lines)). (c) Average force induced by a displacement Dyat 13mm/s (gray) vs. a displacement Dx at 65mm/s (black). Arrows mark the maximumdisplacements. Inset: displacement profiles. Reproduced with permission.[84] Copyright2007, American Physical Society.


www.mme-journal.de

work. His ‘‘repation’’ model obtained a

scaling law stating that the longest

relaxation time of a linear polymer is

proportional to the cube of its molecular

weight. However, the ‘‘repation’’ model

doesnot consider the retractionofpolymer

chains along the ‘‘tube’’. Based on the

mechanism of both repation and retrac-

tion, Doi and Edwards developed the Doi-

Edwards theory and formulated it to the

constitutive equation for concentrated

polymer solutions and melts.[82]

The ‘‘tube’’ theory of entangled poly-

mers plays an important role in polymer

physics and rheology. However, the

‘‘reptation’’ of polymer molecules has

not been directly observed in experiments

until Chu’s group used DNA molecules to

study the entangled polymer solutions.[47]

Figure 13. (A) Schematic depiction of rheometric molecular ima-ging setup. (B) The stress growth as a function of time at_gapp ¼0.5 s�1 where the inset indicates no-slip prior to the stressovershoot. (C) Time-dependent conformational changes of DNAin the slip layer. (D) The conformation of DNA across the gapduring steady slip at _gapp ¼0.5 s�1 in steady state, (t¼ 100 s).Reproduced with permission.[85] Copyright 2010, AmericanPhysical Society.

They compared the relaxation of a stretched stained DNA

molecule in concentrated unstained DNA solutions with

that in pure solvent and found that the former took amuch

longer time to recoil. The self-diffusion of entangledDNA in

concentratedDNAsolutionswasalso investigatedbySmith

et al.[83] and they found that the self-diffusion scaling

exponenta¼�1.8was close to thevalue of�2predicted by

the ‘‘reptation’’ theory. Thus the work done by Chu’s group

verified the validation of the ‘‘reptation’’ theory. Later,

Robertson and Smith used optical tweezers to measure the

intermolecular forces acting on a single DNA chain by

surrounding entangled chains. The tube-shaped confining

field was quantified by measuring the confining force per

unit length in response to an imposed displacement. The

force increased linearly with small displacement in the

perpendicaular direction, Dy (gray). In cotrast, the neglible

force wasmeasured in response to a parallel displacement,

Dx (black) as shown in Figure 12.[84]

Recently, we integrated a commercial rheometer with a

confocal fluorescentmicroscope (CFM) todirectly image the

conformational changes of stained DNA molecules in the

non-linear response regime of entangled DNA solutions

with simultaneous velocimetric and rheometric measure-

ments (seeFigure13A).WhentheWi> 1.0, thechangeof the

boundary condition from no-slip to slip produced a stress

overshoot as shown in Figure 13B. Specifically, adsorbed

DNA chains remained unperturbed till after the stress

maximum when the molecules started to stretch and

elongate at the surface (shown in Figure 13C). The DNA

conformations were measured at different positions along

the sample thickness. Molecules were disentangled only

in the first monolayer where adsorbed chains were

stretched and the molecules everywhere else remained

coiled and essentially entangled, as evidenced by the small

bulk shear rate (Figure 13D). Using stained DNAmolecules




as a velocity and molecular conformation tracer may

provide new opportunities to discover new physical

phenomena necessary for a full theoretical picture of

nonlinear deformation and flow behavior of entangled

polymer solutions.[85,86]

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Future Research Directions

We briefly describe two directions for rheology research at

small scales. One is related to the effect of electric fields on

microfluidic rheology, while the other is the rheology in

nanofluidics.

Microfluidic Rheology Under Electric Field

The driving force in most microfluidic rheology studies is

the hydrodynamic pressure, which is usually created using

syringe pumps. However, considerable effort has been

expended to create microflows using electrokinetic forces

because they are particularly relevant to biofluids in

medical and biological applications.[87] Our group has

demonstrated that different electrokinetic flows such as

simple shear, pure extensional and rotational flows can be

generated by combining electro-osmotic flow (EOF) and

electrophoresis (EP) within a single micro-device.[88,89] The

use of electrokinetics in microfluidics eliminates the need

for pumps or tubing so the sample volume can be

substantially reduced.

Currently, there is increased interest in DNA electro-

phoretic dynamics since DNA is an electrolyte of very

important biomedical value. BDS simulation of DNA

molecules in microfluidic devices under applied electric

fields has been carried out and comparedwith experiments

by our and other research groups.[90–94]

Theuseofelectricfieldbringsmorephysics into rheology.

A recent work showed that rheological properties of a non-

Newtonian solution could be quantified by its electrical

properties. In this experiment, electrochemical impedance

spectroscopy was used to record the response of a blood

sample in the AC electric field. It was reported that the

electrical resistance was a function of the shear rate or

viscosity of the blood.[95] This technique may serve as an

alternative approach tomeasure the shear viscosity of non-

Newtonian fluids in micro/nanoscale-based devices.

Nanofluidic Rheology

When the confinement size gets down to the nanoscale, the

polymer-wall and polymer-polymer interactions become

dominant. The confinement effect from the surrounding

environment will definitely affect the dynamics of the

polymer chains. Because of extremely high pressure build

up, the adoption of nanofluidic devices prohibits the use of

hydrodynamic pressure to drive the flow of concentrated

polymer solutions. Other surface forces such as surface

tension need to be used to drive highly viscous polymer

fluids in the flow. This is themajor challenge in nanofluidic

rheology. Compared to conventional polymers, DNA

molecules are negatively charged and thus they can be

driven by electrophoresis. For single DNA dynamics in



nanofluidic devices, a recent reviewbyHsieh andDoyle has

given very detailed summary on the confinement effect on

DNA dynamics, theories on the scaling laws on diffusivity

and relaxation time on nanoscales, and progress in

experiments of DNA in different confinements.[96]

Simulation of polymer behavior in nanofluidic devices

will require multi-scale simulation techniques since the

polymer-wall and polymer-polymer interactions are very

strong. Also the solvent-solute and solvent-wall interac-

tions (such as slip or no-slip condition) can play an

important role at the nanoscale. The coarse-grained

simulation methods may not be able to capture these

interactions.

Experimentally, nanochannel devices have been utilized

for both fundamental studies of the behavior of DNA

molecules confined in a nanochannel and practical

applications such as DNA separations, DNA sequencing,

and sensors. Many studies have been conducted on the

stretching, conformation, and dynamics of DNAmolecules

in a nano-confinement with the purpose of developing

better understanding necessary to develop useful biome-

dical nanofluidic devices. For example, nanochannels were

used to study the dynamics of l-DNA by measuring the

contour length and extension of the molecules.[97] This

informationwas used to verify the use of deGennes scaling

theory for self-avoiding walks. Pu et al. studied ion

transport across a nanochannel under an electric field

and found that both positively and negatively charged ions

wereenrichedonthesamesideof thenanochannel.[98] They

developed a simple model to qualitatively describe the

mechanisms of this effect using double-layer overlap. Stein

et al. studied the pressure driven transport of DNA in both

micro- and nanofluidic channels and found that the

behavior of DNA molecules is dominated by the statistical

properties of polymer coils.[99] They found that the

pressure-driven mobility of the molecules increases with

molecular length in large channels, but remains indepen-

dent of length in channels that are small compared with

molecular coil size. Reccius et al. studied the conformation,

length, speed, and intensity of single DNA constrained in a

nanochannel. The DNAmolecules were electrophoretically

driven from a nano-slit into a nanochannel.[100] This

enabled accurate measurement of molecular length, and

DNA stretchingwas found to increase with applied electric

field, which was estimated to be 56 times higher in the

nanochannel than in the nano-slit based on device

dimensions due to focusing of the electric field.

In 2000, Han and Craighead created a nanofluidic

separation device using nano-slits to connect micro-

wells.[101] Under an electric field DNA molecules were

separated based on their size as they crossed this entropic

trap array. Surprisingly, they found that larger molecules

had a highermobility because their larger size gave them a

higher probability to escape the entropic traps. Next,

011, 296, 308–320



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Craighead et al. created a nano-pillar array and used an

electric field todrivebiologicalmolecules into thearray.[102]

Upon removal of the electric field themolecules that didnot

fully enter the array recoiled to the higher-entropy region

outside of the nano-pillar array. Smaller molecules that

passedcompletely into thenano-arraywereseparated from

largerones that recoiled. The recoilwasbelieved tobedue to

confinement rather than stretchingandoffered insight into

how entropy decreases with confinement. Another device

utilized nano-pillar arrays inside microfluidic channels to

enable size-fractioning of DNA after passing through the

nano-pillar array under an electric field.[103]

Summaries and Conclusions

Microfluidic rheology has gained more attention due to its

ability to connect themicrostructure of a polymermaterial

with its macroscopic properties and has many advantages

which cannot be achieved by conventional rheology.

Polymer flow in microfluidics is characterized by high

Weissenberg numbers and shear rates, valuable for many

industrial and engineering applications.

The microscale also facilitates molecular imaging of

individual polymer molecules, enabling experimental

verification of the physical behavior of polymer molecules

that leads to the observedmacroscopic flow behaviors. The

use of DNA as a model polymer allows us to further

understandmolecular dynamics indilute and concentrated

polymer solutions. In dilute DNA solutions, the ‘‘individu-

alism’’ of DNA dynamics in the flow demonstrates that the

response of polymers to external flow is highly dependent

on their initial configuration. Thus, the history of polymer

chainshas a large impact on their futurebehaviors. Relative

to dilute solutions, individual molecular behavior in

concentrated solutions has not been well studied, partially

due to the challenges associated with simulating chain-

chain interactions of concentrated solutions. However;

improvements in simulation and experimental techniques

could lead to ground-breaking discoveries regarding the

understanding of the physics behind viscoelastic flow

behaviors.

Acknowledgements: This work was partially supported by theNational Science Foundation sponsored Nanoscale Science andEngineering Center for Affordable Nanoengineering of PolymericBiomedical Devices (Grant number: EEC-0425626).

Received: July 2, 2010; Revised: November 22, 2010; Publishedonline: January 28, 2011; DOI: 10.1002/mame.201000246

Keywords: flow instability; microfluidics; rheology; rheometry




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