The Use of Microfluidics in Rheology - Wiley Online Library

Review

The Use of Microfluidics in Rheology Xin Hu, Pouyan E. Boukany, Orin L. Hemminger, L. James Lee*

The combination of microfluidics and fluorescent-labeled DNA molecules in rheology provides a unique opportunity to relate the microstructure of polymer molecules to their macroscopic properties. For example, the direct visualization of a model polymer system such as single DNA dynamics of individual DNA molecules, improves our understanding of the rheological behavior of dilute and concentrated polymer solutions. This review summarizes the microfluidic rheology for synthetic polymer and DNA solutions, including both experimental and simulation efforts, in investigating dilute and concentrated polymer solutions.

Introduction Rheology is the science of the deformation and flow of materials. Colloids, emulsions, liquid crystals, synthetic polymers and biopolymers (DNA, proteins), foams, gels, and membranes are widely used in our everyday life. The rheological properties of these complex fluid systems are often complicated because of 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 rheometers and advanced fluorescence microscopy, we are now able to directly image and understand the microstructures and dynamic behavior of long-chain molecules in well-defined fluid fields and relate the visualization to macroscopic rheological responses. The length scale 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. Lee Nanoscale Science and Engineering Center for Affordable Nanoengineering of Polymeric Biomedical Devices (CANPBD) E-mail: [email protected] P. E. Boukany, L. J. Lee Department of Chemical and Biomolecular Engineering, The Ohio State University, Columbus, OH, 43210 USA

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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 using microfluidics 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 _ with g_ being the local characteristic shear rate, Wi ¼ gl, and l the dominant relaxation time, De ¼ l/t, with t being _ 2h the characteristic flow time and Re ¼ hðrg_ gw ÞðwþhÞ, with r being fluid density, w the width and h is the height, and hðg_ Þ the shear 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 fluids polymer in rheological measurements. Third, the flow field in the microfluidic 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 as using an optical tweezers.[8] Furthermore, different types of flows can be easily generated in microfluidic-based devices without any moving parts. For example, microfluidic devices consisting of several microchannels have been designed to produce extensional, mixed shear, and

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DOI: 10.1002/mame.201000246

The Use of Microfluidics in . . . www.mme-journal.de

rotational flow patterns as in a ‘‘four-roll mill’’ type rheometer,by adjusting the relative flow rates in microchannels.[10,11] Since these microfluidic devices do not have any moving parts, the fabrication is much easier and thus less costly. Finally, microfluidic rheology conducted on ‘‘labon-a-chip’’ devices can be used to study non-Newtonian flow behavior responses of biological fluids for medical applications. Examples include serum flow in blood vessels, 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 polymer solutions. Section 3 discusses the use of DNA molecules 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 geometries. The flow fields of such polymer fluids in microfluidics can be characterized by using either micro-tracer particles or numerical simulation approaches. Experimental Observations of Flow Instabilities in Microfluidics Most polymers used in our daily life, from plastics to synthetic fibers to elastomers, are made from entangled long chains in the molten or solution state. By pushing or 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] Molecular mechanisms of these flow phenomena are still unclear despite decades of intensive experimental and theoretical investigation.[19] New insights in polymer rheology can be gained through the integration of microfluidics-based ‘‘rheometer-on-a-chip’’ and optical/confocal 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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L. James Lee is the Helen C. Kurtz Professor of Chemical Engineering at The Ohio State University, and the Director of NSF Nanoscale Science and Engineering Center for Affordable Nanoengineering of Polymer Biomedical Devices. He received a BS degree from National Taiwan University and a Ph.D. degree from University of Minnesota in 1979. His research covers polymer engineering, micro-/nanotechnology, and biotechnology. He has > 220 papers and 25 patents. Dr. Lee is a Fellow of Society of Plastics Engineers (SPE). He received the 2008 Malcolm E. Pruitt Award from Council of Chemical Research, 2008 Engineering/Technology Award and 2010 International Award from SPE. Xin Hu received the M.S. degree from Peking University, China in 2000 and the Ph.D. degree in Mechanical Engineering from The Ohio State University in 2006. He worked as a research engineer in the center for Affordable Nanoengineering of Polymeric Biomedical Devices from 2006 to 2010. His research mainly focused on the mesoscopic simulation of polymeric flows, FEM simulation of electrokinetic micro/nanofluidics, Brownian dynamics simulation of long chain polymers in microfluidics, and FEM simulation of cell electroporation in micro/nanodevices. Now he is a thermal and CFD scientist in UES, Inc. and works on simulation of thermal storage management. Pouyan E. Boukany obtained his M.S. degree in Textile Engineering with major in Textile Chemistry and Fiber Science from the Isfahan University of Technology, Iran. He received his Ph.D. in Polymer Science from University of Akron in 2008. His doctoral work was to explore the nonlinear flow behavior of entangled DNA fluids in Professor Shi-Qing Wang’s group. He is currently a Postdoctoral Research Associate working with Professor L. James Lee in the Center for Affordable Nanoengineering of Polymeric Biomedical Devices at The Ohio State University.

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-hardening vs. softening response in the extensional flow (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 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 contractionexpansion microchannels, a similar flow behavior over a much broader range of Re (0.44 < Re < 64) and Wi (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 flow behavior of semi-dilute PEO. solutions to in a 16:1:16 contraction-expansion flow geometry. Both pressure drop and flow visualization were used to characterize

Figure 2. (A) Summarizing 4:1 macroscopic contraction flow experiments on a Wi-Re space for shear thinning viscoelastic and Boger fluids. Summary of flow patterns in the Wi-Re space for (B) different semi-dilute PEO solutions (0.05–0.3% PEO), and (C) different ranges of EI ¼ Wi/Re ¼ 2.8–68 at the same concentration of PEO solution (0.075% PEO) in a contraction microchannel. Reproduced with permission.[21] Copyright 2005, 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 critical Wic of 50.[21,22] Rodd et al. also prepared solutions with a 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 (El ¼ Wi/ Re ¼ 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 Figure 3. Summary of flow patterns in the Wi-Re space for four different DNA solutions from weakly to highly entangled solutions in a 4:1 contraction microchannel. between two flow regimes occurred was Reproduced with permission.[24] Copyright 2010, Elsevier. found to be a weak function of El, while the same transition occurred at a higher Wi when the El value was 68.[22] Recently, Soulages et al. used a T-shaped microchannel to investigate the instability of the steady planar stagnation Gulati et al. employed a semi-dilute DNA solution flow of PEO solutions. As shown in Figure 5, a strand of containing 400 mg/mL (0.04 wt.-%) of l-DNA and investihighly oriented polymeric material was formed in the 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 1.0 wt.-% (with a wide range of entanglements per chain Z ¼ 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 nonNewtonian fluids.[25] Pathak and Hudson used a crossslot microfluidic device to investigate the extensional rheology of a polymer solution. At high flow rates, a central birefringent band representing a highly aligned Figure 4. Dye advection patterns for a cross-channel flow with microstructure was observed near the stagnation point.[26] two inputs and two outputs for (a) a Newtonian fluid, and (b) a Using the same geometry, Arratia et al. showed that PAA flexible polymer solution (De ¼ l/t ¼ 4.5), where the interface the velocity field of a polyacrylamide solution (PPA) between dyed and un-dyed fluid is deformed by flow instability. became strongly asymmetric with non-periodic fluctua(c) and (d) Particle streak lines corresponding to (a) and (b) tion at sufficiently high strain rates and low Re (shown showing the symmetry-breaking instability. Reproduced with permission. [27] Copyright 2006, American Physical Society. in Figure 4).[27]



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Simulation of Viscoelastic Fluids in Microfluidic Rheology Most polymer fluids exhibit shear-rate dependent viscosity and some solid-like elastic properties. These two properties are tightly inter-connected as viscoelasticity, a critical property of many nonNewtonian 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 macroscopic simulation approach only solves Figure 5. Viscoelastic (a–e) and Newtonian (f–j) flow patterns as a function of flow rate the macroscopic variables such as polyin a T-shaped microchannel. Reproduced with permission.[28] Copyright 2009, Elsevier. mer stresses and the velocity field. Conventional simulation methods such as finite element (FE), finite difference (FD), and finite volume (FV) methods are applied to simulate viscoelastic fluids using various CEs region of the strong planar extensional flow, resulting in an (for more information, please refer to.[2,3]) additional symmetry-breaking transition at intermediate Wi. A flow transition from a steady to an unsteady threeSeveral researchers have applied the macroscopic dimensional flow was observed at a critical Wi for each simulation approach to microfluidic rheology. Poole et al. stagnation flow.[28] investigated the stability of an upper-convected Maxwell (UCM) fluid in a cross-slot microchannel,[30] while Soulages Sousa et al. employed a microfluidic rectifier to produce [29] creeping flow conditions. For viscoelastic fluids, the et al. studied the instability of a Phan-Thien-Tanner (PTT) fluid in a T-shaped microchannel.[28] Although some CEs pressure drop was found to be constant in the flow direction at low flow rates. However, increasing the flow rate led to have considered the polymer configuration by treating an anisotropic flow resistance in the forward and polymer molecules as bead-spring chains,[4,31–34] the CEsbackward flow directions at the same pressure drop, i.e., based simulation of polymer flows cannot represent the rectification effects emerged. The viscoelastic fluid flow true behaviors of polymers. For example, the polymer became unsteady in the forward direction due to the chains near the wall behave differently from those in the emergence of elastic instabilities and the flow resistance bulk flow due to the effects of hydrodynamic interaction increased sharply as a function of the flow rate (shown in (HI) and repulsive forces from the wall. Since CEs do not Figure 6). consider the existence of the channel wall, the effect of polymer-wall interactions on polymer flow cannot be exhibited in the macroscopic approach. The CEs-based simulation also cannot achieve high Wi or De numbers due to severe numerical instability. An alternative way to simulate viscoelastic fluids is the so-called mesoscopic approach. This approach simulates polymer molecules in a coarse-grained way and calculates polymer stresses by an ensemble average. For example, the CONNFFESSIT (Calculation of Non-Newtonian Flow: Finite Elements and Stochastic Simulation Technique) method is a Figure 6. Flow patterns of a PAA fluid in the microchannel with a triangular shape mesoscopic simulation method devel¨ ttinger.[35] In this Reproduced with permission.[29] Copyright 2010, Elsevier. oped by Laso and O

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method, the flow domain is first discretized into a finite element mesh. Polymer molecules are simplified as beadspring or bead-rod dumbbells or chains. There are two different methods to calculate the configuration of polymer chains. One is the Lagrangian method by throwing dumbbells 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 configuration 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 CONNFFESSIT method 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 in micro-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 CONNFFESSIT 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 Molecular Imaging 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 can be analyzed, the reason why a polymer solution shows a particular property in a certain flow field can be understood. However, conventional polymer molecules 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 (> 10 M vs. < 1 106 g mol1 in molecular weight). For example, the contour length of a naked l-DNA molecule (48 kbps, molecular weight 31.5 106 g mol1) is around 16 mm. Also when labeled with fluorescent dye, DNA can be 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 in Dilute 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 DNA molecules and time was 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 39.1 mm. Inset shows the relaxation process of a tethered DNA molecule with one end attached to a polystyrene latex microsphere, which was trapped by the optical tweezers. From this data, the longest relaxation time can be calculated. Also DNA molecules 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


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Coil-Stretching Transition of DNA Molecules in a Pure Extensional Flow

Figure 8. Relaxation of initial fully stretched tethered DNA molecules with three different contour lengths. Inset is the relaxation process of a DNA molecule. Reproduced with permission.[47] Copyright 1994, American Association for the Advancement of Science.

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 3n ¼ 2 for the freely draining chain; Zimm model predicts that 3n ¼ 1.5 for the ‘‘theta’’ solvent by considering the hydrodynamic interactions; and 3n ¼ 1.8 for a ‘‘good’’ solvent.[31,50] Thus, none of the current theoretical models can reliably predict the scaling law in the relaxation process of DNA molecules. 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.

Compared to the relaxation process, the transition of DNA configuration from the equilibrium state to the nonequilibrium state is a rather interesting topic of study. One of the widely used methods to drive DNA molecules 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 and mixed flows are their combinations. Chu’s group investigated dynamics of untethered l-DNA molecules in a nearly 2D pure extensional flow generated in a cross-channel.[51,52] They observed the stretching of DNA molecules when the De number was larger than 0.4, slightly less than the theoretical value of 0.5. With the aid of fluorescent microscopy, Perkins et al. studied the coilstretch 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 a DNA molecule was dependent on its initial conformation. DNA molecules with different initial conformations showed different coil-stretch transition process. Figure 9(A) shows the coil-stretch transitions of DNA molecules with four initial conformations: dumbbell, half dumbbell, kinked and folded, while the averaged extension vs. residency time for different initial conformations is shown in Figure 9(B). Such strong heterogeneity due to the ‘‘individualism’’ of polymer molecules in the extensional flow cannot be obtained by macroscopic measurements such as light scattering and birefringence.

Figure 9. (A) Coil-stretch transition for DNA molecules with four different types of initial conformation near the stagnation point of a pure extensional flow (inset); (B) ensemble average extension for different initial conformation vs. residency time. Reproduced with permission.[51] Copyright 1997, American Association for the Advancement of Science.

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For l-DNA, the effect of HI could be neglected in the simulation without affecting the overall stretching amount as long as the effective persistence length was adjusted to match the relaxation data in experiments. However, for a longer DNA with contour length 1.3 mm, which is 62 times longer than l-DNA, the HI effect could not be neglected any more. Hysteresis phenomenon was observed by Schroeder 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 De number, or strain rate. At the certain Figure 10. Extension vs. residence time, or strain for DNA molecules with different initial range of strain rate, both coiled and conformations under (a) De ¼ 2; (b) De ¼ 48. Reproduced with permission.[52] Copyright stretched DNA molecules co-existed in 1998, American Association for the Advancement of Science. the buffer solution. In fact, this is the first order coil-stretch transition proposed by de Gennes and this ‘‘bistable equilibrium’’ is due to The stretching of single DNA molecules under different the strong ‘‘deformation-dependent drag’’ induced by HI for extensional rates, or De numbers was also studied. It was such long DNA molecule.[59] The conformation hysteresis of found that both the amount of stretched DNA molecules and the ensemble averaged extension increased at a larger long DNA molecules was simulated using BDS.[60] In the De number. Figure 10 shows that most of coiled DNA simulation, DNA molecules with two different initial molecules maintained their coiled status at smaller De configurations (stretched and coiled) were considered. numbers, while all were stretched at a larger De number. Quantitative agreement was achieved between the experiThis indicates that DNA molecules experienced larger ment and BDS. deformation with a larger extensional rate, or a larger Tumbling of DNA Molecules in Simple Shear & Mixed stretching force. Flows The coil-stretch transition of DNA in the extensional flow has been successfully recaptured by the Brownian DNA dynamics in a simple shear flow was also studied by dynamics simulation (BDS). Larson et al. showed that the Chu’s group.[61] They found that DNA molecules displayed a coil-stretch transition of l–DNA molecules can be simutumbling movement shown by the schematic in lated with BDS using the worm-like chain (WLC) model.[50] Figure 11(a). A DNA molecule can be stretched to be nearly They did not consider the effect of hydrodynamic interacaligned to the flow direction (here it is along the x-axis direction). Then if the y-component of Brownian force tion (HI) in the simulation. However, by adjusting the slightly rotates the DNA molecule, two ends of the DNA effective persistence length, they obtained almost the same molecule might be exerted by external forces in the results as experiments done by Chu’s group. opposite direction and move towards their center-of-mass. The HI effect on polymer dynamics in bulk has been Thus, DNA molecule will be retracted to a coiled shape. This investigated by many researchers.[54,55] The HI velocity, as a is the so-called ‘‘coil-stretch-tumbling-recoil’’ movement of perturbation to the bulk flow, is calculated through a DNA molecules in the simple shear flow. In experiments, mobility tensor such as the Oseen-Burger tensor. However, this tensor is non-positive definite when two bead locations Smith et al. observed that there were three typical behaviors of DNA molecules in the shear flow as shown in Figure 11(b). are close. Thus, the Rotne-Prager-Yamakawa (RPY) tensor is In the first row, the DNA molecule showed a ‘‘coil-stretchchosen to replace the Oseen-Burger tensor in the calculatumbling-recoil’’ movement. In the second row, the DNA tion. Heish and Larson applied the RPY tensor as the molecule was stretched gradually to a certain length and mobility tensor in the BDS of dynamics of Polystyrene (PS) maintained its shape. In the third row, the stretching of and l-DNA molecules in the extensional flow with HI.[56,57] DNA molecule was weak, but its conformation changed Their results showed that the HI effect was more important greatly and the ‘‘tumbling’’ movement was observed. for the dynamics of short PS molecules, and it could be neglected for the dynamics of l-DNA since the HI velocity BDS has been successfully applied to simulate DNA dynamics in the simple shear flow.[62] Hur et al. investidecreased rapidly with the increased distance between two beads. gated the averaged stretch amount vs. the Wi number using



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It was reported by Fang et al. that the DNA molecules near the wall (1 mm 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 concentration near the wall was much lower than that in bulk. There was a depletion 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 calculate HI with wall. The simplest way is to use the Stokeslet-based method or Figure 11. (a) ‘‘Coil-stretch-tumbling-recoil’’ movement of DNA molecule in shear flow; Green’s function method to modify the (b) Dynamics of three DNA molecules with different initial conformations. Reproduced Oseen-Burger or RPY tensor obtained in the with permission.[61] Copyright 1999, American Association for the Advancement of bulk flow so that it can be used to calculate Science. HI with walls.[70–72] However, this method works only for simple geometries such as microchannels with flat surfaces. Calculation of HI in more general geometries is much FENE dumbbell, Kramer’s bead-rod chain, and WLC. They more complicated. For example, the ‘‘smoothed profile also studied the power spectral density (PSD) at different Wi method’’ (SPM),[73] a direct simulation for particulate flows, numbers. Shear flow can be conisdered as a simple mixed flow since could be used to calculate DNA dynamics with HI in it is combined by half amount of extensional flow and half confined geometries. However, the computation time is amount of rotational flow. Shaqfeh’s group has carried out a very large by a direct simulation method. To save series of experiments and simulations of DNA molecules in computation time in calculating DNA dynamics in confined linear mixed flows.[63,64] The velocity gradients in these geometries with HI, researchers are more interested in combining BDS with finer-scaled simulation techniques flows are constant, but the extensional part or extension such as lattice Boltzmann method (LBM)[74] and stochastic rate competes with the rotational part or vorticity. Their work provided valuable insights to understand the polymer rotation dynamics (SRD).[75] Such multi-scaled simulation dynamics in more complicated flows. methods are more efficient in computation than direct simulation. DNA Dynamics in Other Complex Flows The effect of excluded volume (EV) from intra-/interpolymers and polymer-wall interaction is also important. DNA dynamics was investigated in other complex flows such as hydrodynamic focusing[65] and entrance flow from The EV force is calculated through the specified EV potentials. Typical EV potentials are the truncated Lenthe reservoir to the microchannel.[66] The flow gradients in nard-Jones (L-J) potential,[76,77] hard core repulsive potensuch complex microfluidic devices are not constant; thus, DNA molecules experience different flow gradients when tial,[78] and repulsive narrow Gaussian potential.[79] All the they are moving with the flows. DNA dynamics in such BDS in confined geometries have considered the EV effect complex flows is time/location-dependent. between two beads and between beads and wall. DNA dynamics in different hydrodynamic flows have been throughly studied by many researchers. Since we DNA Dynamics in Concentrated Solutions cannot cover all the experiments and simulations in this topic due to the page limitations of this review, please refer If the concentration of a polymer solution is higher than the to comprehensive reviews by Larson[67] and by Shaqfeh.[68] ‘‘overlapping concentration’’, the effect of chain entanglement must be considered. Chain entanglement is extremely DNA Dynamics in Confined Geometries complicated and difficult to model. Edwards simplified this problem into the ‘‘tube’’ model that assumes that an DNA dynamics in confined geometries are important in entangled polymer (either linear or cross-linked) is confined many biomedical applications such as DNA separation, in a tube-like region by surrounding polymers.[80] The gene mapping, and gene delivery. Generally speaking, when considering DNA dynamics in a confined geometry, motion of polymer inside this ‘‘tube’’ is called ‘‘reptation’’, we need to consider all the interactions including wallwhich is proposed by de Gennes.[81] He considered the polymer, intramolecular, and intermolecular forces. diffusion of polymer chains through the entangled net-

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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 does not consider the retraction of polymer chains along the ‘‘tube’’. Based on the mechanism of both repation and retraction, Doi and Edwards developed the DoiEdwards theory and formulated it to the constitutive equation for concentrated Figure 12. (a) Schematic diagram of a single DNA molecule stretched between two polymer solutions and melts.[82] optically trapped micro-spheres in a concentrated solution of entangled DNA. (b) The ‘‘tube’’ theory of entangled polyReptation models postulate that collective intermolecular interactions give rise to a tubelike confining field (dashed lines)). (c) Average force induced by a displacement Dy mers plays an important role in polymer at 13 mm/s (gray) vs. a displacement Dx at 65 mm/s (black). Arrows mark the maximum physics and rheology. However, the displacements. Inset: displacement profiles. Reproduced with permission.[84] Copyright ‘‘reptation’’ of polymer molecules has 2007, American Physical Society. not been directly observed in experiments until Chu’s group used DNA molecules to study the entangled polymer solutions.[47] as a velocity and molecular conformation tracer may They compared the relaxation of a stretched stained DNA provide new opportunities to discover new physical molecule in concentrated unstained DNA solutions with phenomena necessary for a full theoretical picture of that in pure solvent and found that the former took a much nonlinear deformation and flow behavior of entangled longer time to recoil. The self-diffusion of entangled DNA in polymer solutions.[85,86] concentrated DNA solutions was also investigated by Smith [83] et al. and they found that the self-diffusion scaling exponent a ¼ 1.8 was close to the value of 2 predicted 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 was measured in response to a parallel displacement, Dx (black) as shown in Figure 12.[84] Recently, we integrated a commercial rheometer with a confocal fluorescent microscope (CFM) to directly image the conformational changes of stained DNA molecules in the non-linear response regime of entangled DNA solutions with simultaneous velocimetric and rheometric measurements (see Figure 13A). When the Wi > 1.0, the change of 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 Figure 13. (A) Schematic depiction of rheometric molecular imaging setup. (B) The stress growth as a function of time at conformations were measured at different positions along g_ app ¼ 0.5 s1 where the inset indicates no-slip prior to the stress the sample thickness. Molecules were disentangled only overshoot. (C) Time-dependent conformational changes of DNA in the first monolayer where adsorbed chains were in the slip layer. (D) The conformation of DNA across the gap stretched and the molecules everywhere else remained during steady slip at g_ app ¼ 0.5 s1 in steady state, (t ¼ 100 s). coiled and essentially entangled, as evidenced by the small Reproduced with permission.[85] Copyright 2010, American Physical Society. bulk shear rate (Figure 13D). Using stained DNA molecules



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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 electrophoretic 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 compared with experiments by our and other research groups.[90–94] The use of electric field brings more physics into rheology. A recent work showed that rheological properties of a nonNewtonian 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 to measure the shear viscosity of nonNewtonian 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 the major 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

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nanofluidic devices, a recent review by Hsieh and Doyle 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 interactions (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 DNA molecules in a nano-confinement with the purpose of developing better understanding necessary to develop useful biomedical 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 information was used to verify the use of de Gennes 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 were enriched on the same side of the nanochannel.[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 independent 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 DNA molecules were electrophoretically driven from a nano-slit into a nanochannel.[100] This enabled accurate measurement of molecular length, and DNA stretching was 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 microwells.[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 higher mobility because their larger size gave them a higher probability to escape the entropic traps. Next,




Craighead et al. created a nano-pillar array and used an electric field to drive biological molecules into the array.[102] Upon removal of the electric field the molecules that did not fully enter the array recoiled to the higher-entropy region outside of the nano-pillar array. Smaller molecules that passed completely into the nano-array were separated from larger ones that recoiled. The recoil was believed to be due to confinement rather than stretching and offered 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 the microstructure of a polymer material 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 observed macroscopic flow behaviors. The use of DNA as a model polymer allows us to further understand molecular dynamics in dilute and concentrated polymer solutions. In dilute DNA solutions, the ‘‘individualism’’ 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 chains has a large impact on their future behaviors. Relative to dilute solutions, individual molecular behavior in concentrated solutions has not been well studied, partially due to the challenges associated with simulating chainchain 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 the National Science Foundation sponsored Nanoscale Science and Engineering Center for Affordable Nanoengineering of Polymeric Biomedical Devices (Grant number: EEC-0425626).

Received: July 2, 2010; Revised: November 22, 2010; Published online: January 28, 2011; DOI: 10.1002/mame.201000246 Keywords: flow instability; microfluidics; rheology; rheometry


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