fluid dynamics of dissolved polymer molecules in confined geometries

FL43CH12-Graham ARI 19 November 2010 13:9

Fluid Dynamics of DissolvedPolymer Molecules inConfined GeometriesMichael D. GrahamDepartment of Chemical and Biological Engineering, University of Wisconsin-Madison,Madison, Wisconsin 53706; email: [email protected]

Annu. Rev. Fluid Mech. 2011. 43:273–98

First published online as a Review in Advance onAugust 25, 2010

The Annual Review of Fluid Mechanics is online atfluid.annualreviews.org

This article’s doi:10.1146/annurev-fluid-121108-145523

Copyright c© 2011 by Annual Reviews.All rights reserved

0066-4189/11/0115-0273$20.00

Keywords

microfluidics, DNA, diffusion, cross-stream migration, nanofluidics,multiscale simulation

Abstract

The past decade has seen a renaissance in the study of polymer solutionsflowing in confined geometries, the renaissance driven in part by advancesin visualization of large DNA molecules and the desire to manipulate DNAfor genomic applications. This article summarizes the features of the fun-damental polymer physics and fluid dynamics that are relevant to the flowof confined polymer solutions, then reviews the recent literature on thetopic. Experiments have clarified and extended prior work showing that dif-fusion of confined flexible polymers is substantially altered by confinementand that, during flow, polymers exhibit substantial cross-stream migration.Simulation methods have been developed that have the capability of captur-ing both polymer and fluid motion in confined geometries and yield resultsthat are in semiquantitative agreement with experiments in dilute solutions.Kinetic-theory treatments of simple polymer models have led to analyticallytractable models that qualitatively encompass the key phenomena observedin experiment.

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1. INTRODUCTION

The dynamics of solutions of long-chain polymer molecules during flow in confined geome-tries is important in fields ranging from enhanced oil recovery to coating processes to analyticaland preparatory separation techniques for macromolecules. Since the 1990s, another importantapplication of this topic has arisen: single-molecule approaches to the manipulation and character-ization of genomic DNA, using microfluidic and, more recently, nanofluidic devices (Dimalantaet al. 2004, Jo et al. 2007, Tegenfeldt et al. 2004b). Along with this important application havecome new experimental tools for studying polymer solution dynamics at the single-molecule level:Genomic double-stranded (ds)DNA is a large molecule [many micrometers in length, even forthe genome of a virus (Shaqfeh 2005)] that is amenable to staining with fluorescent dyes, so it canbe directly observed under a microscope (Perkins et al. 1995). This feature of DNA has led todramatic advances in the understanding of polymer dynamics in flow and in other nonequilibriumsituations such as electrophoresis, both in bulk solution and under confinement. Recent reviews(Larson 2005, Shaqfeh 2005) have addressed polymer dynamics, with an emphasis on DNA, inbulk solution. The present review focuses on flows of confined dilute polymer solutions, in whichthere is an intriguing and important interplay between polymer conformations and fluid dynamicsthat has only recently begun to be unraveled. Apart from their direct importance in applications,confined polymer solutions serve as an archetype for the dynamics of other confined complex fluidssuch as suspensions and emulsions. With the emergence of microfluidic technologies that involvecomplex and multiphase fluids (Squires & Quake 2005), it is important that the fundamentalmechanisms that underlie transport in these systems be understood.

What issues arise in a confined polymer solution? If the confinement has the same length scaleas the polymer molecules themselves, then the equilibrium conformations of the chains can beexpected to change substantially from their bulk behavior. Such a restrictive level of confinementallows for the possibility of velocity gradients that vary on the scale of the molecule, leading tochanges in the dynamics and conformations of the chain in flow. The boundary conditions onthe solvent restrict its motion in ways that may affect the polymer dynamics—relaxation time,diffusivity, transport processes in flow. Most generally, the presence of even one confining wallbreaks the translational invariance of the system, so that situations like simple shear that wouldnot generate inhomogenous polymer concentration fields in bulk flow may lead to concentrationgradients in confined ones.

This last issue, and specifically the possibility of the migration of polymer chains toward or awayfrom walls during flow, has long been recognized as significant and has been studied in a varietyof contexts. Agarwal et al. (1994) has reviewed much of the classical experimental and theoreticalliterature on this topic, so only a few key aspects of this work are discussed here. First, in simpleshear or pressure-driven flow of a polymer solution in which the flow geometry is much largerthan the molecular size, chains migrate away from walls at high-enough shear rates. This fact ismanifested macroscopically in two ways: first in an apparent slip boundary condition for the flow,with a slip length substantially larger than the molecular size, and second in an increased averagevelocity for polymer chains in capillary flow relative to the mean velocity of the solvent, especiallyat higher molecular weights. Exemplary studies of the latter phenomenon include Seo et al. (1996)and Sugarman (1988). In the former study, polyacrylamide solutions were pumped through a longcapillary, and the polymer concentration of the eluent was measured with a UV/Vis detector.At shear rates lower than 103 s−1, the polymer moved at the same velocity as the solvent, but athigher shear rates, the polymer eluted substantially faster, indicating that it had migrated towardthe center of the channel where the fluid velocity is higher than the average. The latter study wassimilar in principle, but the polymer used was dsDNA from the λ-phage virus, a molecule that has

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Persistence length:length scale beyondwhich orientations ofpolymer backbonesegments becomeuncorrelated

become the hydrogen atom of single-molecule polymer physics. In both cases, the wall shear rateγ at which migration began correlates well with the inverse of the longest relaxation time λ ofthe molecule in the solution: That is, migration arises when the Weissenberg number W i = λγ

exceeds about unity. The principles and mechanisms underlying these observations are a primarytheme of this review.

Diffusion of macromolecules in confined geometries has also long been studied (Happel &Brenner 1965, Teraoka 1996). Until relatively recently, however, the regime of a flexible polymerdiffusing in a confining geometry that substantially distorts the polymer has been inaccessible.Again, studies of genomic DNA in micro- and nanofabricated channels have clearly shown thefeatures of diffusion in this regime. These studies form the other major theme of this review.

2. CONFORMATIONS OF POLYMER CHAINS IN BULKAND CONFINED SOLUTIONS

2.1. Bulk Conformations

In this section, we briefly review the equilibrium conformations of polymer chains in bulk solution(Bird et al. 1987, de Gennes 1979, Doi & Edwards 1986) and then, as background for understandingthe dynamics of confined solutions, describe key results regarding how these conformations changein confined geometries. To begin, we consider a long linear polymer in bulk solution and define aunit vector u(s) tangent to the molecule’s backbone, where s is the position along the backbone. Inthe limit of long chain contour length L, the autocorrelation function 〈u(s ′) · u(s ′ + s )〉 of backboneorientation satisfies a relation of the general form

〈u(s ′) · u(s ′ + s )〉 ∼ e−s /L p , (1)

where Lp is a length scale called the persistence length of the polymer. For example, dsDNA can beviewed (at scales large compared to its diameter of 2 nm) as an inextensible string with a bendingmoment A, in which case the persistence length is given by the length scale at which bending andthermal energy balance:

Lp = kTA

, (2)

where k is Boltzmann’s constant and T is absolute temperature (Doi & Edwards 1986). Theexperimental value of Lp for dsDNA is approximately 50 nm (Shaqfeh 2005). For comparison,the widely studied drag-reducing polymer polyethylene oxide has Lp ≈ 0.15 nm (Mark & Flory1965). This review primarily considers flexible polymers, with N p = L/Lp � 1.

The backbone of a long flexible polymer chain can in principle collide with itself many times.Under so-called θ-solvent or ideal chain conditions, or in highly concentrated solutions, these self-intersections have a negligible effect on the equilibrium chain conformations and can be ignored.In this situation, the equilibrium spatial conformation of the chain is well approximated on lengthscales much larger than the persistence length by a classical random walk. In particular, the bulkradius of gyration Rbulk for the chain scales as Lp N 1/2

p . In a poor solvent, the chain collapses to acompact conformation; we do not consider poor solvent conditions here. In a good solvent, thesituation of primary interest here, the chain segments exhibit an effective repulsive interaction, asthey enthalpically prefer to be surrounded by solvent rather than other chain segments. Here theequilibrium chain conformations are self-avoiding random walks and Rbulk ∼ Lp N ν

p , where ν ≈3/5. Polymer concentration (number density of chains) c is often reported relative to the overlapconcentration, at which the chains are concentrated enough to begin to contact one another.

www.annualreviews.org • Fluid Dynamics of Dissolved Polymer Molecules 275

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Blob: term used inpolymer physics torepresentcharacteristic lengthscales intermediatebetween Lp and Rbulkthat arise in manysituations, includinggood solvents,semidilute solutions,and confinement

ELECTROKINETIC PHENOMENA IN DNA TRANSPORT

DNA is an acid, so it is negatively charged in solution, a fact that has long been used for its manipulation via variousforms of electrophoresis (Viovy 2000). The dynamics of a charged polymer in electric and/or flow fields can becomplex, as the motion of the polymer is coupled with the motion of the other ions in solution. All the experimentsdescribed in this review were performed in solutions with sufficiently high salt concentration (∼100 mM) thatany electrostatic interactions between segments of the DNA backbone were screened by the counterions, andthe chain backbone motions were sufficiently slow that the counterions could very rapidly move to stay with thebackbone. In this situation, the DNA can be treated as an uncharged polymer. During electrophoresis of DNA inbulk solution, the counterions move in the direction opposite to the chain backbone, and this electroosmosis leadsto screening of hydrodynamic interactions between the chain segments. A nontrivial consequence of this fact isthat the electrophoretic mobility of DNA in free solution is approximately independent of molecular weight. Innonuniform electric fields or in cases in which the DNA is not fully free to move under the influence of the field,the balance of electrophoresis of the chain and electroosmosis of the counterions is broken, and hydrodynamicinteractions are no longer fully screened.

This concentration, denoted c∗, is defined as the value of concentration at which c = ( 43 π R3

bulk)−1.Unless otherwise specified, we consider only the dilute regime c c ∗.

2.2. Conformations Under Confinement

Turning to the issue of confinement, and denoting the confinement length scale (e.g., the widthof a confining slit or tube) as W, three primary regimes can be identified (Figure 1).

2.2.1. Rbulk � W. In this weakly confined regime, the equilibrium conformational statistics ofthe polymer chain are largely unchanged from bulk values. There will nevertheless be a depletionlayer of thickness Ld ≈ Rbulk near solid surfaces, as it is entropically unfavorable for a polymerchain to have a conformation such that its center of mass is closer than about Rbulk from a wall.During flow, when the chain is extended by a substantial fraction of its contour length L, the ratioL/W may become important (Hernandez-Ortiz et al. 2006b).

2.2.2. L p � W � Rbulk. In this moderately confined regime, the polymer chain is confinedsubstantially but still behaves as a flexible chain. In ideal solvent conditions, the chain would becompressed in the confined directions, but the chain statistics in the unconfined directions wouldbe unchanged (Casassa 1967). In the more usual situation of a good solvent, confinement of thechain in one or two directions leads to expansion of the chain in the unconfined dimensions.Daoud & de Gennes (1977) (see also Hsieh & Doyle 2008, Teraoka 1996) presented predictionsof the scaling of chain expansion in the unconfined dimensions by generalizing the concept ofa blob that had been previously developed in the context of concentrated solutions or polymermelts (de Gennes 1979)—loosely speaking, a blob is a subchain of connected segments withcorrelated behavior. These authors argued that excluded volume interactions would lead to thechain behaving essentially as a string of blobs, each of diameter W, as illustrated by the translucentspheres in the center image of Figure 1. For scales smaller than W, the chain’s backbone behaves asa simple three-dimensional (3D) self-avoiding random walk as in the bulk, but on larger scales theconfinement alters this behavior. A blob with diameter W has g ∼ (W /Lp )5/3 segments and thusthere are N b ∼ N p/g ∼ N p (Lp/W )−5/3 blobs in the chain. The overall average chain extension

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Weak confinement

Moderate confinement

Strong confinement

W

W

W

2Rbulk

x

y

z

2R ≈ W

Figure 1Regimes of confinement for flexible polymer chains in solution. In the center image, the translucent spheresindicate blobs of size W of polymer segments.

is then determined by assuming that the chain undergoes a self-avoiding random walk, with steplength W in the unconfined direction(s). In the quasi-2D case of a slit, the root-mean-squaredend-to-end distance or radius of gyration Rslit scales as WN 3/4

b . Nondimensionalizing this resultusing the bulk chain size, it becomes

Rs lit/Rbulk ∼ (W /Rbulk)−1/4. (3)

In the quasi-1D case of a tube, the chain cannot execute a random walk on length scales greaterthan W because it would have to turn back on itself, an energetically highly unfavorable situation.Therefore, the end-to-end distance in this case, Rtube, will simply scale as WN 1

b , or

Rtube/Rbulk ∼ (W /Rbulk)−2/3. (4)

These scaling results are in good agreement with simulations (Chen et al. 2004, Jendrejack et al.2003a,b, Wall et al. 1978) and form a starting point for predicting the scaling of dynamic propertiessuch as diffusivity or relaxation time, as discussed below.

2.2.3. W � L p � Rbulk. In this strongly confined case, arguments based on the conformationof flexible chains break down completely and must be replaced by the direct consideration of the


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ƒi

ri

Figure 2A bead-spring chain model of a flexible polymer molecule.

Hydrodynamicinteraction: the effectthat the fluid motiongenerated by onemoving object in fluidhas on other objects inthe fluid

backbone of the chain (Odijk 2008, Reisner et al. 2005, Tegenfeldt et al. 2004a). Flow in thisregime has not been extensively studied because of the very large pressure drops required to driveflow at such small scales (�100 nm in the case of dsDNA). So although this case is important forDNA nanotechnology, in which case electrophoresis can be used as an alternate to pressure-drivenflow, it is beyond the scope of this review (see Hsieh & Doyle 2008 for a discussion of statics anddynamics in this case).

3. LOW–REYNOLDS NUMBER FLOWS DRIVENBY PARTICLE MOTIONS

The most common coarse-grained model of a flexible polymer in a solvent is the bead-springchain model (Bird et al. 1987, Larson 2005), illustrated schematically in Figure 2. In this model,collections of polymer segments are treated as spherical beads connected by springs representingthe coarse-grained entropic forces associated with deformation of the random walk of the chain’sbackbone. Hydrodynamic drag, Brownian, and excluded volume forces (i.e., the repulsions be-tween polymer segments in a good solvent) complete the force balance on each bead of the chain.The flow driven by the motion of each bead affects the motions of other beads, a phenomenoncalled hydrodynamic interaction. Confinement alters this flow and thus the dynamics of polymerchains in flow. Before discussing the recent literature on the dynamics of confined polymers, it istherefore useful to review some fundamental issues associated with particle motions in flow.

3.1. The Mobility Tensor for a System of Point Particles

If we treat bead i of a bead-spring chain as a sphere of radius a that experiences Stokes drag as itmoves through the flow, then the relationship between the bead velocity vi and the force it exertson the fluid is given by f i = ζ (vi −v(ri )), where ζ is the Stokes’ law friction coefficient ζ = 6πηa ,v(ri) is the fluid velocity at the bead position, and η is the fluid viscosity. (The finite size of the beadonly arises in the friction coefficient—for all other purposes it is considered to be a point particle.)There are three contributions to this fluid velocity: (a) the externally imposed fluid velocity fieldv∞ (e.g., a Poiseuille profile), (b) the motion driven by the forces exerted by the other beads inthe system, and (c) the correction to the velocity experienced by the bead due to the presence ofconfining walls.

We consider first a two-bead chain (a dumbbell) in an unbounded domain. Here the forcebalance (neglecting particle inertia) shows that the fluid velocities v(ri) experienced by bead i are

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Long-rangedinteraction:interaction betweenobjects that decayssufficiently slowly withdistance that thecumulative effect ofmany distant objects isnot negligible

given by (v1

v2

)=

(v∞(r1)v∞(r2)

)+

(1ζδ G∞(r1 − r2)

G∞(r2 − r1) 1ζδ

)·(

f 1

f 2

), (5)

where G∞(r) = 18πηr (δ + rr/r2) is the Oseen-Burgers tensor, or Stokeslet. Letting V = (v1, v2)T ,

and so on, this can be succinctly rewritten:

V = V∞ + M · F . (6)

The tensor M is called the mobility tensor. In simulations of coarse-grained models of polymerdynamics, the singularity of the Stokeslet must be regularized—the most common choice is the so-called RPY (Rotne-Prager-Yamakawa) tensor (Bird et al. 1987, Rotne & Prager 1969, Yamakawa1970).

Because the Stokeslet changes in a confined geometry, the mobility tensor changes as well.Without loss of generality, the Stokeslet for a confined geometry can be written ( Jendrejack et al.2003b) as

G(ri , r j ) = G∞(ri − r j ) + GW (ri , r j ), (7)

and the pair mobility becomes

M =(

1ζδ + GW (r1, r1) G∞(r1 − r2) + GW (r1, r2)

G∞(r2 − r1) + GW (r2, r1) 1ζδ + GW (r2, r2)

). (8)

The contributions of GW to the diagonal elements of M reflect the change in mobility of eachbead individually because of the confinement, whereas the changes in the off-diagonal elementsreflect the effect of confinement on the hydrodynamic interactions between beads. The latter arethe most important in determining the qualitative changes to polymer dynamics brought on byconfinement.

3.2. Hydrodynamic Interactions in Unconfined Stokes Flow Are Long Ranged

An important issue in understanding hydrodynamic confinement effects is the nature of hydro-dynamic interactions between particles in Stokes flow. Let us consider a suspension of particles,homogeneously distributed in an unbounded d-dimensional domain, so that on large scales thesystem can be considered to have a constant concentration c. If the particles interact through a fieldthat decays with distance as r−p, then a naive scaling estimate E of the effect on one test particleof all the others is given by

E = limL→∞

∫ L

ac r−prd−1 dr . (9)

This integral diverges when d ≥ p , in which case the interactions are said to be long ranged. Thisdivergence indicates that the number of particles in the shell of size dr a distance r away fromthe test particle grows faster with r than the interaction between particles decays with r. Becausethe Stokes velocity field driven by a system of point forces or dipoles in three dimensions decayswith p = 1 or p = 2, respectively, the hydrodynamic interactions among a set of particles in anunbounded fluid are long ranged.

In polymer solution dynamics, the long-ranged nature of bulk hydrodynamic interactionshas the important consequence that the diffusivity Dbulk of a flexible chain in solutions obeysZimm scaling, Dbulk ∼ R−1

bulk (Bird et al. 1987). Hydrodynamic interactions act cooperatively tocouple distant chain segments to one another and thus reduce the resistance to motion relative touncoupled segments. If the hydrodynamic interactions between chain segments were ignored, then


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Screening: thepresence of physicaleffects that act to makefinite the range ofnominally long-rangedinteractions

the diffusivity would satisfy Rouse scaling, Dbulk ∼ N −1p . As shown below, the long-ranged nature

of hydrodynamic interactions changes substantially under confinement, leading to significantchanges in polymer dynamics.

3.3. Properties of Point Force Flows in Bounded Domains

3.3.1. Half-space. We consider first a point force at position x = (0, y, 0) above a plane no-slip wall y = 0. Blake (1971) showed that the exact solution in this case can be written as aStokeslet plus a simple sum of images located below the wall at position x = (0, −y, 0), namely aStokeslet corresponding to a force of opposite sign, a source dipole and a Stokeslet doublet. (Theseimages make up GW .) Given the solution for a point force above a plane wall, it is straightforwardto determine the solution for a point force dipole. As further described below, this solution isparticularly important because it plays a dominant role in the dynamics of a deformable particleor macromolecule suspended in a flow near a wall.

3.3.2. Slit. Now we consider the quasi-2D case of flow driven by a point force in a slit with heightW. In this case an exact solution, due to Liron & Mochon (1976), is available, but not in a simpleform, so here we simply point out some key aspects of the solution (Diamant 2009). For a forceperpendicular to the walls, the velocity field decays exponentially rapidly. For a force parallel tothe walls, the velocity field has a parabolic form in the wall-normal ( y) direction and decays as 1/r2.Specifically, the far-field solution has the form of a 2D (or, more precisely, Hele-Shaw) sourcedipole. This structure has interesting consequences for pair hydrodynamic interactions of colloidalsuspensions confined to a slit. In bulk solution, the motion of one particle pulled in the positivex direction, for example, drives a Stokeslet flow that moves all other particles in the positive xdirection as well, regardless of their position relative to the pulled particle. In the slit, this is nottrue—some particles will actually be moved in the negative x direction, an effect dubbed antidrag(Diamant et al. 2005). The nature of the hydrodynamic interactions for a confined polymer insolution is addressed in detail below.

3.3.3. Pore. In a quasi-1D pore geometry (e.g., a channel with circular or square cross section),the flow is confined in two directions, and the fluid motion generated by a point force in anydirection does decay exponentially at scales larger than W (Diamant 2009).

3.4. Screening of Hydrodynamic Interactions by Confinement

In bulk solution, hydrodynamic interactions between particles are long ranged. We turn now tothe important issue of how this feature of Stokes flow is altered by confinement. The simplestcase to consider is the quasi-1D situation of particles in a pore, d = 1, where the Stokeslet flowfield decays exponentially on the length scale W. As the interaction decays faster than any powerof r in this case, Equation 9 converges—hydrodynamic interactions in the quasi-1D system arenot long ranged and are said to be screened on length scales larger than W. The physical originof screening in this case is that the momentum imparted to the fluid by the point force is rapidlyabsorbed by the stationary walls.

The quasi-2D situation is more complex. For a point force in this geometry, p = 2, d = 2in Equation 9, leading to a logarithmic divergence of E. Nevertheless, as described below, bothexperiments and simulations of the diffusion of long flexible DNA molecule chains in microfluidicslits (Chen et al. 2004, Jendrejack et al. 2003a,b) are consistent with screening of hydrodynamicinteractions on the scale W. Resolving this seeming paradox requires moving beyond the naive

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Migration tensor:relationship betweenthe force dipolegenerated by an objectin fluid and thevelocity at which thatobject is convected bythe flow generated bythe dipole

scaling estimate E to examine more closely the structure of hydrodynamic interactions in a slit.Alvarez & Soto (2005) and Tlusty (2006) noted that at a given distance, the angular averageof the quasi-2D Stokeslet vanishes due to symmetry: Therefore, if a particle is suspended in ahomogeneous isotropic suspension of particles, the expected value of the far-field hydrodynamicforce on it due to the motions of the other particles vanishes. This screening due to cancellation inthe angular averaging does not arise in three dimensions. Tlusty (2006) calls this result “screeningby symmetry.”

3.5. Pair Interactions and Force Dipoles Near a Wall: The Migration Tensor

We consider a pair of suspended particles of radius a both a distance y � a above a plane walland a distance d � a from one another. Imagine that particle 2 exerts a central force f1 on particle1 and vice versa. (These particles could make up a bead-spring dumbbell.) The force balance onparticle 1 requires that it exert a force f1 on the fluid—this force will drive a fluid motion that willtend to move particle 2 both parallel to the wall and, more importantly, normal to it. Similarly, thefluid motion generated by particle 2 will also move particle 1 normal to the wall. If the particles areattracted to one another, each will generate a fluid motion that convects the other away from thewall, whereas if they repel one another, each will move toward the wall. This result was directlyobserved in suspensions of charged colloidal particles (Dufresne et al. 2000) and can be derived(Anekal & Bevan 2005, Dufresne et al. 2000) using the single-wall Green’s function of Blake(1971).

One can idealize the situation of a pair of interacting particles or indeed any single force- andtorque-free deformable particle (polymer chain, droplet, capsule, biological cell) as a symmetricpoint force dipole D. Using the point force solution of Blake, it is straightforward to determinethe velocity vmig at which the dipole is convected relative to the wall due to the fluid motion thatit generates. The result, first presented by Smart & Leighton (1991), is

vmig = M : D. (10)

The third-order tensor M is called the migration tensor; it can be found for any flow domaingiven the point force dipole solution in that domain. For a half-space domain with a no-slip wallat y = 0, it is given by

M = 1y2

M, (11)

where M is a constant tensor. The 1/y2 decay of this function arises from the dipolar nature ofthe particle-wall hydrodynamic interaction.

3.6. Equations of Motion for a Coarse-Grained FlexiblePolymer Molecule in Solution

Although it is not the purpose of the present review to describe bead-spring chain models in anydetail (e.g., see Bird et al. 1987, Ottinger 1996, Larson 2005), it will be useful to briefly present thebasic mathematical structure of these models. Following the notation introduced above for the pairmobility, we consider a system of N beads each with position ri and define R = (r1, r2, . . . , rN )T .Similarly, we let F = ( f 1, f 2, . . . , f N )T , where fi is the sum of the external (e.g., spring, excludedvolume, gravitational) forces exerted on bead i. In the limit of negligible fluid and particle inertia,the balance of hydrodynamic, Brownian, and external forces on each bead can be written as the


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Brownian dynamicssimulation:numerical timeintegration of astochastic differentialequation (e.g.,Equations 12 and 13)

following stochastic differential equation (Ottinger 1996):

d R =(

V∞ + M · F + kT∂

∂ R· M

)dt + B · d W, (12)

where

B · BT = 2kT M, (13)

and each element of dW is an increment of an independent Wiener process. Equation 13 is astatement of the fluctuation-dissipation theorem for this system. In most of the simulation resultsdiscussed below, these are the equations that are solved via so-called Brownian dynamics (BD)simulation. (In some contexts, the term BD-HI is used to denote Brownian dynamics simulationsthat incorporate hydrodynamic interactions (HI), i.e., the use of the full mobility tensor, whereasBD is used to denote simulations that neglect hydrodynamic interactions. In the present work,the term Brownian dynamics implies the use of the full mobility tensor.) Some computationalefficiency issues and alternate methods for simulating the evolution of coarse-grained polymermodels are discussed in Section 4.2.

4. DYNAMICS OF CONFINED FLOWING POLYMER SOLUTIONS

4.1. Results from Experiment, Theory, and Computation

In this section, we indicate how the recent literature weaves together the issues of polymer confor-mations described in Section 2.2 with those of hydrodynamics in confined geometries describedin Section 3.

Brochard & de Gennes (1977) described scaling results for the diffusivity D and relaxation timeλ of a flexible polymer confined to a slit or tube (i.e., quasi-2D or quasi-1D confinement) in thecase Lp Rg W . These results are based on an argument that hydrodynamic interactions arescreened at distances greater than W and that under these conditions the chain friction coefficientshould scale linearly with molecular weight (Rouse scaling). Using this argument and Equation 4,and defining D∗ = D/Dbulk and W ∗ = W /Rbulk, they predict that

D∗ ∼ W ∗−2/3. (14)

Although the assumption of screening is correct, as discussed below, the issue is more subtle thanindicated by this paper. As mentioned above, in the slit case, screening actually occurs because ofthe symmetry of the hydrodynamic interactions.

With regard to relaxation time, this review does not go into detail, for two reasons. First, theeffects of confinement on relaxation time are closely related to those on diffusion, for which moredetailed measurements exist. Second, although many measurements are available in the literature,these are based on many different, although related, definitions of relaxation time. The readeris referred to Bakajin et al. (1998), Reisner et al. (2005), Hsieh et al. (2007), and Bonthuis et al.(2008) for measurements and discussions of relaxation time.

Diffusion in the quasi-1D moderately confined case was further considered by Harden & Doi(1992). They performed a self-consistent mean field theory calculation to predict the distributionof polymer segments in a tube. Then they combined this prediction with the exact solution forthe Green’s function due to a point force in a cylindrical tube and used the so-called Kirkwoodapproximation (Bird et al. 1987) for the chain diffusivity to predict its dependence on the degreeof confinement. Over the range of their computations, the results were well approximated byD∗ ∼ W ∗−0.61, rather than the exponent of −2/3 found from the above scaling arguments. Theauthors argue that the discrepancy between their results and the simple scaling arises because of

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λ-phage DNA: thegenomic DNA of theλ-phage virus; whenstained withfluorescent dye forvisualization, it isapproximately 21 μmlong

the depletion of polymer segments in a region of size ∼Lp near the tube walls. This argument iscorroborated by simulation results ( Jendrejack et al. 2003b) and experimental results (Balducciet al. 2006, Hsieh et al. 2007), as further discussed below.

Brunn and Grisafi (Brunn 1984, 1985; Brunn & Grisafi 1985) developed a polymer kinetictheory for bead-spring chain models in inhomogeneous flows, including hydrodynamic interac-tions between beads, but not accounting for the change in hydrodynamic interactions due to thepresence of a wall. They note that in inhomogeneous flows, the polymer diffusivity can vary withposition, because its degree of stretching varies with position, and that this variation in diffusivitycan lead to cross-stream migration. Let us consider Poiseuille flow in the case of weak confinement.At the channel center there is no deformation of polymer chains, so they retain their equilibriumdiffusivity. With increasing distance from the center, the strain rate and thus the polymer stretch-ing increase. Polymer diffusivity decreases as stretch increases (de Gennes 1979), and Brunn’stheory predicts therefore that chains should migrate from the region of highest diffusivity towardregions of lower diffusivity, i.e., away from the channel center during flow, in seeming contrastwith experimental observations. We return to the issue of migration away from the centerline inpressure-driven flow in the discussion of more recent results below.

These kinetic-theory results for Poiseuille flow have been extended to more complex bead-spring chain models (Nitsche 1996) and to Brownian rigid rods (Nitsche & Hinch 1997, Schiek& Shaqfeh 1997). These studies again predict migration toward higher gradients (i.e., towardthe wall) in Poiseuille flow, for the same reason as Brunn’s. Woo et al. (2004a,b) modeled theeffect of confinement on the dynamics of chains by considering how the entropic spring force ina bead-spring chain model would be modified. Chain-wall hydrodynamic interactions were takeninto account by introducing a position-dependent mobility of each bead, an approximation thatis appropriate in the limit of moderate to strong confinement, where hydrodynamic interactionsare screened. Chain migration per se was not studied, but predictions of chain conformationaldynamics and rheology were made, many of which remain to be tested by direct simulation orexperiment.

In related work, Jhon & Freed (1985) presented a formalism for studying the migration ofa polymer chain in flow near a solid surface that does in principle incorporate hydrodynamicinteractions between the chain and a single wall. They predict that a chain migrates away froma wall in shear, but it should be noted that their equation 11a, for the Green’s function due toa point force above a plane wall, is incorrect. The migration tensor evaluated using this Green’sfunction would actually predict migration toward the wall.

In the context of direct solutions of polymer solution dynamics, wall hydrodynamic effects werefirst introduced in a series of papers by Jendrejack et al. (2003a,b, 2004). These papers studied thedynamics in the dilute limit of bead-spring chain models of genomic DNA ( Jendrejack et al. 2002)of contour lengths from 4 to 420 μm in square microchannels of various sizes, from W ∼ Rg toW � Rg . (For λ-phage DNA stained with fluorescent dye, Rbulk ≈ 0.7 μm and L ≈ 21 μm.) TheGreen’s function for a point force in this geometry can be calculated exactly as an infinite series, butfor performing computations, Jendrejack et al. took a different approach. They wrote the Green’sfunction as a free-space piece plus a correction such that the sum of the two pieces satisfied noslip on the walls and periodicity at the channel entrance and exit. The correction was determinednumerically using a standard finite-element method and stored as a look-up table. Regularizationof the Stokeslet at short distances was achieved with an ad hoc generalization of the RPY tensor.The Brownian displacement B · dW was determined using the polynomial approximation methodproposed by Fixman (1986) (see also Jendrejack et al. 2000).

Several important sets of results emerged from these simulations. The first, described inJendrejack et al. (2003b), is an explicit prediction of the diffusivity of a flexible chain as a


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–5 –5 –5 –5

5 5 55z (µm) z (µm)y (µm) y (µm)

0 0 0 0

a b

Figure 3Predicted steady-state center-of-mass probability (concentration) distribution versus cross-sectional chain position for dimers ofλ-DNA in a square channel with W = 10.6 μm (W ∗ ≈ 10). (a) Equilibrium distribution. (b) Distribution at γ = 308 s−1 (Wi = 92).Figure reprinted with permission from Jendrejack et al. (2003a). Copyright 2003 by the American Physical Society.

function of the degree of confinement. The crossover from bulk to confined behavior was foundto begin at W ∗ ≈ 10, and in the region 1/W ∗ � 1 a power-law behavior D∗ ∼ W ∗−0.5 was found.This exponent is different from the value of −2/3 predicted by Brochard & de Gennes (1977).Consistent with the work of Harden & Doi (1992), the authors attribute the deviation to the factthat there is not a wide scale separation between the channel size and the equilibrium distancebetween beads in the model (approximately 4 persistence lengths). On the other hand, results forthe equilibrium stretch closely followed the W ∗−2/3 prediction of Daoud & de Gennes (1977) onceW ∗−1 � 5.

Turning from dynamics at equilibrium to dynamics in flow, Jendrejack et al. (2003a, 2004)examined pressure-driven flow of DNA solutions in square microchannels over roughly the samerange used for the diffusion simulations. One of the primary results of this study is the predic-tion in the weakly confined regime of a depletion layer whose size at high Wi is much largerthan the chain radius of gyration (Figure 3). The degree of migration increases with increasingWeissenberg number, consistent with experimental observations that DNA (and other polymers)can be separated by pressure-driven flow through a capillary (Seo et al. 1996, Sugarman 1988). Adistinctive feature of the predicted concentration profiles in the square channel is their volcano-like shape, as seen in Figure 3: The concentration is essentially zero near the walls, rises rapidlytoward the center, but then displays a slight dip in the region very near the centerline. The mi-gration away from the wall is driven by the wall contribution to the hydrodynamic interactions,and the slight migration away from the centerline results from the phenomenon predicted byBrunn, in which chains migrate to regions of lower mobility. The volcano shape is also found bySaintillan et al. (2006) in their simulations of chains of freely jointed Brownian rods in a slit. Toaccount for rod-wall hydrodynamic interactions, these authors used a regularized form of Liron& Mochon’s (1976) Green’s function for a slit that was introduced by Staben et al. (2003). Thiswork also revisited the limiting case of a polymer consisting of a single rigid Brownian rod (seeNitsche & Hinch 1997 and Schiek & Shaqfeh 1997, discussed above), demonstrating that the rodsactually migrate away from the walls, albeit weakly. Park et al. (2007) performed a similar analysis,arriving at the same result.

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In the moderately confined regime, substantial migration does not occur, although other inter-esting phenomena are found. Section 2.2 above discusses the concept of blobs of chain segmentsthat are of size W, independent of molecular weight. In their simulations, Jendrejack et al. (2004)found that in this regime, the wall shear rate at which the polymer chains start to stretch is alsoindependent of molecular weight: Polymer stretching begins when the Weissenberg number fora blob of size W becomes of order unity. Presently there is not an explicit experimental test ofthis prediction, although, as noted by Hsieh et al. (2007), there are data that are consistent withit. Stein et al. (2006) presented images of λ-phage DNA during pressure-driven flow in a slit withheight 250 nm (approximately Rbulk/3) at a wall shear rate of approximately 90 s−1. In unboundedflow at this shear rate, chains would be stretched, but no stretching was observed in their exper-iment. Similarly, Larson et al. (2006) reported stretch versus mean velocity for BAC12m9 DNA(which is about four times longer than λ-phage) during pressure-driven flow in a 1-μm-high slitmicrochannel. From this data one can estimate that there is not substantial chain stretching until astrain rate of approximately 1,000 s−1, again much larger than would be expected for this moleculein unbounded shear flow.

Complementary to the simulations just described, Jendrejack et al. (2004) and Ma & Graham(2005) revisited the dilute solution kinetic theory for bead-spring dumbbell models of polymersin solution, incorporating both intrachain and chain-wall hydrodynamic interactions. For thecase of a dumbbell (i.e., a two-bead chain) in solution in a semi-infinite domain bounded by ano-slip wall, substantial analytical progress can be made. In particular, in the limit where thedumbbell size is much smaller than its distance from the wall, a closed form expression for theflux j of polymer chains can be found. This expression depends on the polymer number densityc, bulk fluid velocity field v∞, polymer conformation tensor α = 〈qq〉 (where q = r2 − r1 is thedumbbell end-to-end vector and 〈·〉 denotes ensemble averaging over q), polymer stress tensorτ , and (conformation-dependent) Kirkwood diffusity evaluated in an unbounded domain DK ,bulk.The expression is

j = c v∞ + c8〈qq〉 : ∇∇v∞ + M : τ p

− c ∇ · DK , bulk − DK , bulk · ∇c . (15)

The first term in this expression is convection, and the last term is normal Fickian diffusion; theother terms lead to migration. Let us consider first the term containing the migration tensor andthe stress tensor. Each dumbbell induces a force dipole flow in the surrounding solvent—the stresstensor is the ensemble average of this dipole. In the presence of a wall, the force dipole induces afluid velocity M : τ/c at the position of the dumbbell (Smart & Leighton 1991). This term wasmissing in previous theories of polymer migration. The term containing the divergence of DK ,bulk isthe effect noted by Brunn (1984). In nonhomogeneous flow, the term containing ∇∇v∞ predictsthe lag of a macromolecule behind the solvent along the streamline (Aubert & Tirrell 1980)but no cross-streamline migration in rectilinear flows, and unless the velocity nonhomogeneityis so large that it cannot be ignored even on the length scale of the polymer molecule, thisterm is small. [Nevertheless, it may be important in curvilinear flows that are maintained forvery long times (see, e.g., Macdonald & Muller 1996).] It is important to note that substantialwall-induced hydrodynamic migration also arises in suspensions of deformable particles such asdroplets or capsules (Leal 1980, Olla 1999). Hudson (2003) has written an expression similar toEquation 15 for the transport of emulsion drops that undergo migration and shear-induced ratherthan Brownian diffusion.

With Equation 15, it is possible to make some explicit predictions in various special cases. Thesimplest of these is an explicit prediction of the steady-state concentration profile c( y) in simple


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shear flow above a plane wall:

c (y) = c b exp (−Ld /y), (16)

where cb is the bulk concentration and Ld is the depletion-layer thickness. For a general dumbbellmodel, and neglecting the conformation dependency of diffusivity,

Ld = 9√

π

128N 1 − N 2

c kTh∗ Rbulk, (17)

where h∗ ≈ 0.25 is the hydrodynamic interaction parameter for the dumbbell, and N1 and N2

are the first and second normal stress differences (Bird et al. 1987); these are both linear in c soit cancels out of the expression. For the FENE (finitely extensible, nonlinearly elastic) dumbbellmodel, N 2 = 0 and N 1 ∼ c W i2/3, yielding a distinct scaling prediction from this model of

Ld ∼ W i2/3 Rg . (18)

When W i � 1, depletion layers can be much larger than the equilibrium molecular size.With this expression, Ma & Graham (2005) also made predictions of the spatial development of

the depletion layer (the migration analog of the Graetz-Leveque problem) as well as the temporaldevelopment upon start-up of steady shear. These predictions are both in the form of similaritysolutions and share two important features. The first is a transient pile-up of concentration inthe boundary layer—the concentration profile is nonmonotonic because rapidly migrating chainsnear the wall tend to catch up with more slowly migrating chains initially farther from the wall.The second is that the time (meaning residence time in the spatially developing case) for thedevelopment of the steady-state depletion layer scales as the time L2

d /Dbulk required for chains todiffuse over a length scale Ld . For the spatially developing case, and using the scaling estimatefrom the FENE dumbbell model, this implies an entrance length that scales as Wi3Rg.

Many extensions of the framework established in Ma & Graham (2005) have been made.Hernandez-Ortiz et al. (2006a,b) showed that in the single-wall case, Brownian dynamics simu-lations with dumbbell and chain models were in good agreement with the analytical theory. In aslit geometry, the analytical theory can be extended by using a single-reflection approximation, inwhich the migration effects due to the two walls of the slit are simply summed, ignoring the factthat this summation leads to violation of the no-slip boundary condition. For moderately confinedchains at very high Wi, weak migration toward the wall can occur because in this case the hydro-dynamic migration effects of the two walls essentially cancel, and the chain, which is stretched inthe flow direction, is actually slightly compacted in the wall-normal direction, allowing its centerof mass to come closer to the wall than it can at equilibrium (see de Pablo et al. 1992).

Butler et al. (2007) generalized Ma & Graham’s (2005) theory to allow for the presence of anexternal force on the dumbbells. A particularly interesting prediction is that the combination ofshear flow and an external field can lead to migration toward the wall, an effect that competeswith the wall-induced hydrodynamic migration. This effect arises from the following mechanism:The shear flow causes the polymer to be tilted on average with respect to the flow direction. Ifthe external force on the polymer drives it in the direction opposite to the flow, the anisotropyof the hydrodynamic drag on the stretched chain tends to drive transverse migration of the chaintoward the region of lower velocity, i.e., toward the wall. The anisotropy of drag results from theintrachain hydrodynamic interactions. This result is consistent with the experimental observationsof Zheng & Yeung (2002, 2003) under combined electric and flow fields. A closely related work isthat of Hoda & Kumar (2007), who study the interaction between hydrodynamic migration andelectrostatic attraction for a polyelectrolyte flowing near an oppositely charged surface. This studyalso removed the far-field approximation for the chain-wall hydrodynamic interactions used byMa & Graham (2005), an important feature when considering adsorption. The authors predicted

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that the competition between electrostatic attraction and hydrodynamic repulsion could lead to asteady-state concentration profile with a maximum, although it is unclear whether the parameterregime in which this phenomenon occurs is experimentally accessible. (In most experiments withgenomic DNA, for example, the Debye layer thickness is ∼1 nm, which is much smaller thanthe bulk equilibrium size of the chain and thus much smaller than the scale at which a dumbbellmodel is expected to be valid.) Sendner & Netz (2008) consider the dynamics of dumbbell modelswith nonzero equilibrium length as models of semiflexible polymers. In particular, they extend thescaling result given in Equation 18 to the semiflexible case and corroborate the scaling predictionwith Brownian dynamics simulations.

The entrance length prediction of Ma & Graham (2005) implies that a long channel may berequired for full development of the hydrodynamic depletion layer in steady flow. Chen et al. (2005)and Jo et al. (2009) circumvented this issue by considering a zero-mean oscillatory pressure-drivenflow. In the first half-cycle of the oscillation, a constant flow rate is imposed, with the directionof flow changing sign in the second half-cycle. In this case there is no net bulk flow, and thedevelopment of the depletion layer can be observed at a single position in the microchannel.This system is characterized by both a Weissenberg number W i = λ|γ | and the strain in eachhalf-cycle γ = γ / f , where f is the inverse period of the oscillations. Substantial migration onlyoccurs at high Wi and high γ . For genomic DNA with radii of gyration ∼1 μm, depletion-layerthicknesses of 10 μm or more were predicted with Brownian dynamics simulations at high Wi andγ . As shown in Figure 4, the predictions were in reasonable agreement with experimental results,although the simulations slightly overpredicted the depletion-layer thicknesses and the time (2–3 min) required to reach steady state. Both the simulation and experimental studies showed theexistence of off-center peaks in the concentration profile. It is unclear whether this feature is moreclosely related to the pile-up prediction of the boundary layer theory treatment of Ma & Graham(2005) or the steady-state volcano-peak result of Jendrejack et al. (2003a, 2004).

Fang et al. (2005) reported fluorescence microscopy measurements of the configuration andconcentration of DNA molecules in simple shear flow near a surface in a microchannel withW � Rbulk. They reported depletion layers as thick as approximately 10 μm for λ-phage DNA(which has Rbulk ≈ 700 nm) at large Wi. This study was extended by Fang et al. (2007), whopresented more extensive experimental measurements of depletion-layer thicknesses for dilutesolutions of λ-phage DNA, as well as Brownian dynamics simulations of chains near a singlewall [the latter are similar to the single-wall simulations of Hernandez-Ortiz et al. (2006b)]. Theexperimental results displayed qualitative but not quantitative agreement with the dumbbell theorypredictions of Ma & Graham (2005); the experimentally observed depletion-layer thickness wassubstantially thinner than the prediction. Bead-spring chain simulations were in better agreementwith the experimental results, only overpredicting the depletion layer thickness by about a factor oftwo, but overpredicting the timescale for steady state to be reached by an order of magnitude. Theseauthors concluded that more refined polymer models were necessary to obtain more quantitativeagreement with data. Finally, Fang & Larson (2007) examined the concentration dependency ofthe depletion layer in DNA solutions, finding that as the concentration exceeds 0.1 of the overlapconcentration c∗ for the chains, the hydrodynamic migration effect begins to diminish, althoughit does not vanish until the concentration exceeds approximately 3c∗. This result is qualitativelyconsistent with the computational results of Hernandez-Ortiz et al. (2007, 2008) (further describedbelow) as well as with the observation that, in bulk solution, hydrodynamic interactions begin tobe screened at concentrations above approximately 0.1 c∗.

As described above, both scaling and computational predictions have been made regarding thediffusion of chains in the moderately confined regime. Chen et al. (2004) reported measurementsand comparisons to simulations for λ-phage DNA and two- and threefold concatemers thereof.


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–20

–10

20

3

2

1

0

2

1

0

t (s)0

0

a

b

0

10

y (µ

m)

100 200 300

100 200 300

–20

–10

20

t (s)

0

10

y (µ

m)

100 200 300

Concentration/average concentration

Concentration/average concentration

Figure 4(a) Simulation and (b) experimental results for the time evolution of the concentration distribution ofT2-DNA molecules in a 40-μm square microchannel and subjected to oscillatory pressure-driven flow withf = 0.25 Hz. Experimental results are at W i ≈ 60 and computational results at W i = 50. Brightness isproportional to concentration. Experimental results at each time instant are averaged over the axial positionin the channel. Computational results are averaged over 100 simulations with random initial positions for themolecules. Figure adapted with permission from Chen et al. (2005). Copyright 2005 American ChemicalSociety.

The experiments in this work were performed in a slit geometry and were in good agreement withthe simulation results and reasonable agreement with the scaling theory. The simulations examinedboth slit and square cross sections, yielding good agreement with the static scaling predictions ofDaoud & de Gennes (1977) for chain extension in both geometries. The simulation results forthe slit geometry were closer to the prediction of a −2/3 power-law exponent than were thosefor the square channel (as discussed above), but some deviation still remained. Rough agreementwith the −2/3 exponent for confined diffusion was also experimentally found with DNA by Steinet al. (2006).

The experimental aspect of Chen et al.’s (2004) work was extended in Balducci et al. (2006)and Hsieh et al. (2007). The former paper reports experimental observations of diffusion of DNAin slit channels in the moderate and strong confinement regimes. Figure 5a shows experimentalresults for diffusion as a function of degree of confinement, as well as some of the simulationresults presented in Chen et al. (2004)—the good agreement between experiment and simulationis apparent. Furthermore, they demonstrated, as shown in Figure 5b, that once the slit height iseven slightly smaller than the bulk radius of gyration, diffusion follows Rouse scaling (D ∼ N −1

p )

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8

0.1

2

3

4

5

678

1

D/D

bulk

80.1

2 4 6 81

2 4 6 810

2

4

68

0.1

2

4

6

81

2

2 3 4 5 6 7 8 91

2

Rg,bulk/h

D (μ

m2 s

–1)

MW/MW,λ-DNA

a

b

Blob scaling (–2/3)

Zimm scaling (–0.6)

Rouse scaling (–1.0)

M13mp18 DNA½ λ-DNAλ-DNA2 λ-DNASimulation (Chen et al. 2004)Experiment (Chen et al. 2004)

W = ∞545 nm280 nm190 nm100 nm50 nm

Figure 5(a) Experimental and simulation results for the diffusion of DNA in a slit microchannel. (b) Diffusivity as afunction of molecular weight, relative to λ-DNA for DNA in slit microchannels of various heights. For all ofthe confined cases, the diffusivity closely follows Rouse scaling. Figure reprinted with permission fromBalducci et al. (2006). Copyright 2006 American Chemical Society.

closely, indicating that hydrodynamic interactions are screened over the coil size of the polymer.On the other hand, as noted above, the experimental results for D/Dbulk versus Rbulk/W deviatefrom the −2/3 exponent. These results demonstrate that for the DNA molecular weights underconsideration here, there is not sufficient scale separation between the chain persistence lengthand degree of confinement to justify the assumption in the scaling theory that the polymer chainsform well-developed blobs—the double condition Lp W Rbulk is not satisfied. This work


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also generalizes the observation by Alvarez & Soto (2005) and Tlusty (2006) of screening bysymmetry. Hsieh et al. (2007) also reported measurements of rotational relaxation time for DNAin a slit channel and corroborate the observation that, in the experimental regime studied, scaleseparation is insufficient for predictions based on a blob model to strictly hold.

4.2. Complex Geometries, Finite Concentrations, and Computational Issues

Returning to the issue of simulation approaches for polymer solutions in confined geometries, it isimportant to consider the computational expense of the Stokes flow-based approaches describedabove. Direct construction and matrix multiplication of M both cost O(N 2) operations per timestep. Exact determination of B costs O(N 3) operations, although the Chebyshev polynomial ap-proximation introduced by Fixman (1986) reduces this cost so that it scales linearly with the costof a matrix-vector multiplication. In periodic domains, however, these scalings can be dramaticallyimproved by use of Ewald-sum-based and P3M (particle-particle-particle-mesh) methods, whichuse fast Fourier transforms (Deserno & Holm 1998a,b, Hasimoto 1959, Hockney & Eastwood1988, Smith et al. 1987, Toukmaji & Board 1996), and one can compute these interactions inperiodic domains in O(N ln N) operations. The Brownian term can then also be approximatedin O(N ln N) operations using the Fixman method. (Strictly speaking, these scalings refer to thenumber of mesh points used for spatial discretization, but for many situations of interest, thisis proportional to N.) In the context of suspensions of rigid Brownian particles, this approach isembodied in the accelerated Stokesian dynamics method of Banchio & Brady (2003) and Sierou& Brady (2001).

For studying confinement effects, these Fourier-transform-based methods are not directly ap-plicable [although boundaries can be built in artificially, as done, for example, by Freund (2007)in a study of wall effects on blood flow]. Nevertheless, the idea used in those methods of splittinga solution to the Stokes equation into singular, short-ranged parts and smooth long-ranged partsremains useful even for nonperiodic domains. This observation is used in the general geometryEwald-like method of Hernandez-Ortiz et al. (2007), which has O(N) or O(N ln N) scaling. Withthis approach, Hernandez-Ortiz et al. (2008) studied the effects of concentration on migrationduring Poiseuille or Couette flow in a slit geometry with W ≈ 10Rbulk, finding that as the con-centration increases toward the overlap concentration, the depletion-layer thickness decreases,with migration vanishing for c � 0.2c ∗. This agrees qualitatively but not quantitatively with theexperimental observations of Fang & Larson (2007), although the specifics of both the polymerand confinement were substantially different in the two cases.

This study also described a simulation of a solution at finite concentration in a more complexgeometry. Flow over a channel with grooves oriented perpendicular to the flow direction wasconsidered. As shown in Figure 6a, at low concentration and high Wi, the groove was almostcompletely depleted of chains, and this observation was explained by arguing that once chainsdiffuse out of the groove into the main flow, they can only move back into the groove by diffusingacross the hydrodynamic depletion layer in a time that is less than the time it takes for the chainto cross the mouth of the groove, an unlikely event. Interestingly, at concentrations approachingoverlap, the concentration difference between bulk and groove is substantially reduced to onlyabout a factor of two, but only if hydrodynamic interactions are included in the simulation. Anargument is made that at higher concentration, for the main flow to drag one chain out of thegroove it has to drag many, because of the hydrodynamic coupling between the chains in thegroove. Because of this mechanism, and because the mobility of chains in the groove is reduced bythe higher degree of confinement there, chains are less susceptible to being dragged into the bulkflow. These predictions and the mechanisms used to explain them are experimentally untested,

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Figure 6Predicted concentration profiles, normalized with maximum concentration (white), for flow of a polymersolution over a grooved wall. The top wall moves to the right while the bottom (grooved) wall is stationary.(a) Prediction for dilute solution. (b) Prediction for c /c ∗ = 0.12. Figure adapted with permission fromfigures 6 and 8 of Hernandez-Ortiz et al. (2008). Copyright 2008 by the Korea-Australia Rheology Journal.

although they are not inconsistent with experimental observations that above c ∗, concentrationsin side channels can actually be higher than in the bulk (Agarwal et al. 1994).

All the computational and theoretical studies described above have been based on a Stokes flowtreatment of the solvent motion. In recent years a number of other approaches to the treatmentof the solvent dynamics have gained substantial attention, either as potentially more efficientalternatives to Stokes flow simulation methods (but see below) or as treatments that capture moreof the small-scale physics of the solvent than a continuum approach can. The lattice Boltzmannmethod (Dunweg & Ladd 2009) discretizes the Boltzmann equation rather than the Stokes orNavier-Stokes equation, nominally resulting in a highly efficient, parallelizable, method. Usta et al.(2005, 2006) used this method to study polymer solutions in the weakly and moderately confinedregimes, finding a roughly −2/3 scaling exponent for diffusion as a function of confinement, aswell as hydrodynamic migration and the characteristic volcano-peak structure described abovefor Poiseuille flow in a slit. The stochastic rotation dynamics method (Malevanets & Kapral1999) introduces particles that have artificial collision dynamics at short times but whose behaviorat long times reproduces fluctuating hydrodynamics. Watari et al. (2007) used this method tostudy bulk and confined polymer solution dynamics, qualitatively reproducing the hydrodynamic


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migration phenomenon. The dissipative particle dynamics (DPD) method (Espanol & Warren1995, Groot & Warren 1997, Ripoll et al. 2001) explicitly solves the equations of motion forcoarse-grained solvent particles that have both conservative and dissipative interactions, to modelthe dynamics of clusters of solvent molecules rather than individual ones. Early work treatingconfined polymers with this approach (Fan et al. 2003) was unable to predict the existence ofhydrodynamic migration, perhaps because of the finite Reynolds number of those simulations(see Hernandez-Ortiz et al. 2006b for a brief discussion of finite–Reynolds number effects onmigration). More recent work (Fedosov et al. 2008) has examined the issue of migration in DPDsimulations in some detail. At Reynolds numbers larger than 1, this study predicts depletion atthe centerline but no hydrodynamic migration from the wall. At Reynolds numbers smaller than1, there is some migration away from the wall but no migration away from the centerline (thissimulation was in a very small channel, W = 3Rbulk). The migration toward the centerline in thehigher–Reynolds number case was attributed to the Segre-Silberberg effect (Segre & Silberberg1962a,b), in which even rigid particles in pressure-driven flow at finite Reynolds number move toan equilibrium position that is off the channel centerline. The mobility gradient effect describedabove may also be active, but it is unclear which is dominant in this particular system. Recentwork (Millan & Laradji 2009) was able to qualitatively capture both the wall and mobility gradienteffects, producing the expected volcano-peak structure. Finally, Khare et al. (2006) and Kohale &Khare (2009) have performed direct molecular dynamics simulations of polymer solutions with anatomistic solvent in a slit. Their results demonstrate the persistence of the hydrodynamic migrationand chain mobility gradient contributions to cross-stream transport down to very small scalesand also highlight the potential importance of thermal diffusion (the Soret effect) in molecularsimulations and in experimental nanoscale systems.

Finally, we make a further comment on computation time: For lattice Boltzmann, the compu-tation time scales linearly with the number of lattice points—this is the same scaling [to within O(lnN)] as the general geometry Ewald-like method and the other accelerated Stokes flow approachesdescribed above. Stochastic rotation dynamics and DPD scale linearly with the number of “sol-vent” particles, a situation analogous to a continuum method that scales linearly with the number ofmesh points. Thus at this stage in the development of computational methods for confined polymersolutions or suspensions, all current methods lead to linear scaling with problem size. The morerelevant consideration, therefore, is the extent to which a given method can efficiently capturethe time and length scales of the important phenomena. This is a nontrivial issue even for latticeBoltzmann (Pham et al. 2009), but especially for stochastic rotation dynamics, DPD, and molecu-lar dynamics, which are trying to capture Navier-Stokes dynamics with a method that is inherentlylimited to time steps characteristic of the relaxation processes of the “solvent” molecules, whichare much shorter than the relaxation or diffusion time of a suspended particle or polymer chain.

5. CONCLUSIONS

The past decade has seen dramatic advances in the understanding of the single-molecule dynam-ics of confined dilute polymer solutions during flow—at least in simple unidirectional flow, theprimary phenomena at work seem to have been elucidated. Single-molecule and dilute solutionexperiments with fluorescently stained DNA have enabled the direct observation of confined poly-mer chains, and theoretical and computational methods are now able to make detailed predictionsof polymer behavior in confined flow. There are some cases in which quantitative agreement be-tween experiments and models can be obtained—for example, in the case of diffusion in a slit—butin other cases, such as cross-stream migration, agreement between experiment and simulations issemiquantitative at best.

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Many challenges remain in gaining a predictive understanding of transport in confined poly-mer solutions. Experiments (Del Bonis-O’Donnell et al. 2009) and simulations (Hernandez-Ortizet al. 2008) indicate intriguing and potentially technologically important phenomena during flowof polymer solutions in complex geometries. Chains tethered to surfaces display complex dynamicsin flow (Beck & Shaqfeh 2006), and the combination of tethering and confinement introduces newdynamics that would be absent otherwise (Zhang et al. 2009). All these cases admit only limitedunderstanding at this time. We also do not understand in any detail how the migration and dif-fusion phenomena discussed above depend on concentration, particularly in complex geometries,nor how they are altered in mixtures. Finally, an important transport topic, especially for DNAnanotechnology ( Jo et al. 2007), is that of electrokinetic effects; DNA is a charged molecule thatresides in an ionic solution. The tools that led in the past decade to the understanding of single-molecule confined dynamics during flow provide starting points for the coming decade’s work inaddressing these more complex issues.

SUMMARY POINTS

1. The transport of dissolved polymer molecules during flow in confined geometries arisesin many applications, from oil recovery to DNA nanotechnology, and displays manycomplexities including hindered diffusion and cross-stream migration. Direct visualiza-tion of fluorescently stained genomic DNA has allowed the direct observation of thesephenomena.

2. For polymers in dilute solution, the primary focus of this review, three general regimesof confinement can be identified, based on the relative length scales of the moleculeand the confining geometry. For a polymer chain with persistence length Lp and bulkradius of gyration Rbulk in a domain with characteristic scale W, these regimes are weakconfinement, where Rbulk W ; moderate confinement, where Lp W Rbulk; andstrong confinement, where Lp ∼ W Rbulk.

3. Confinement changes the fluid dynamics of suspended particles or macromoleculesin three important ways: (a) Hydrodynamic interactions are long ranged in bulksolutions and are screened in confined geometries, as walls absorb the momentumimparted to fluid by a moving object in the flow; (b) walls modify the flow driven bymoving objects and generically lead to cross-stream migration of deformable particlesor macromolecules; and (c) cross-stream migration can also occur due to Browniandiffusion in inhomogeneous flows in which the conformations and thus diffusivity ofpolymer molecules vary with position. In general this effect competes with the migrationdriven by the presence of walls.

4. Moderate or strong confinement changes the equilibrium conformations of flexible poly-mer molecules; in the moderately confined regime, relatively simple physical argumentslead to predictions of the scalings of equilibrium polymer shape and size that agree wellwith experiments and simulations.

5. Combining the scaling arguments for static polymer conformation under confinementwith the argument that hydrodynamic interactions are screened at scales larger than Wleads to predictions for the scaling of polymer diffusivity with molecular weight that agreewell with experiments on DNA. Predictions of scaling with the degree of confinement atfixed molecular weight agree somewhat less well because the separation of scales argumentLp W invoked in the scaling predictions is not satisfied in the experimental systems.


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6. Direct simulations of bead-spring chain models of flexible polymers in dilute solutioncan now efficiently incorporate confinement effects on flow, predicting the existence ofhydrodynamic depletion layers that can be much larger than the equilibrium molecularsize. These predictions are in qualitative and, to some extent, quantitative agreement withexperiment, although discrepancies still exist. Predictions from simulation of diffusionin confined geometries agree well with experimental results for DNA in the moderatelyconfined regimes.

7. For dilute polymer solutions in the weakly confined regime, where migration is mostimportant, kinetic-theory expressions for polymer flux have been derived for a numberof important cases, and in some situations analytical solutions for concentration distri-butions are available.

8. The transport of polymers in the nondilute regime and/or in complex confined geome-tries remains poorly understood, although new computational methods are beginning toaddress these regimes.

DISCLOSURE STATEMENT

The author is not aware of any affiliations, memberships, funding, or financial holdings that mightbe perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

The author gratefully acknowledges his long-time collaborators on the dynamics of confined DNA,Juan J. de Pablo and David C. Schwartz, as well as the graduate students and postdocs, RichardJendrejack, Hongbo Ma, Yeng-Long Chen, Raj Khare, Juan Hernandez-Ortiz, Eileen Dimalanta,Kyubong Jo, and Yu Zhang, who have worked with him on it. The author has also benefitted frominteractions with many other researchers, including Eric Shaqfeh, Ron Larson, and Pat Doyle.The author’s research in this area has been supported by the National Science Foundation, grantsECS/BES/CTS-0085560 and DMR-0425880 (Nanoscale Science and Engineering Center).

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FL43-FrontMater ARI 15 November 2010 11:55

Annual Review ofFluid Mechanics

Volume 43, 2011Contents

Experimental Studies of Transition to Turbulence in a PipeT. Mullin � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Fish Swimming and Bird/Insect FlightTheodore Yaotsu Wu � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �25

Wave TurbulenceAlan C. Newell and Benno Rumpf � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �59

Transition and Stability of High-Speed Boundary LayersAlexander Fedorov � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �79

Fluctuations and Instability in SedimentationElisabeth Guazzelli and John Hinch � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �97

Shock-Bubble InteractionsDevesh Ranjan, Jason Oakley, and Riccardo Bonazza � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 117

Fluid-Structure Interaction in Internal Physiological FlowsMatthias Heil and Andrew L. Hazel � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 141

Numerical Methods for High-Speed FlowsSergio Pirozzoli � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 163

Fluid Mechanics of PapermakingFredrik Lundell, L. Daniel Soderberg, and P. Henrik Alfredsson � � � � � � � � � � � � � � � � � � � � � � � 195

Lagrangian Dynamics and Models of the Velocity Gradient Tensorin Turbulent FlowsCharles Meneveau � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 219

Actuators for Active Flow ControlLouis N. Cattafesta III and Mark Sheplak � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 247

Fluid Dynamics of Dissolved Polymer Moleculesin Confined GeometriesMichael D. Graham � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 273

Discrete Conservation Properties of Unstructured Mesh SchemesJ. Blair Perot � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 299

Global Linear InstabilityVassilios Theofilis � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 319

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FL43-FrontMater ARI 15 November 2010 11:55

High–Reynolds Number Wall TurbulenceAlexander J. Smits, Beverley J. McKeon, and Ivan Marusic � � � � � � � � � � � � � � � � � � � � � � � � � � � � 353

Scale Interactions in Magnetohydrodynamic TurbulencePablo D. Mininni � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 377

Optical Particle Characterization in FlowsCameron Tropea � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 399

Aerodynamic Aspects of Wind Energy ConversionJens Nørkær Sørensen � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 427

Flapping and Bending Bodies Interacting with Fluid FlowsMichael J. Shelley and Jun Zhang � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 449

Pulse Wave Propagation in the Arterial TreeFrans N. van de Vosse and Nikos Stergiopulos � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 467

Mammalian Sperm Motility: Observation and TheoryE.A. Gaffney, H. Gadelha, D.J. Smith, J.R. Blake, and J.C. Kirkman-Brown � � � � � � � 501

Shear-Layer Instabilities: Particle Image Velocimetry Measurementsand Implications for AcousticsScott C. Morris � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 529

Rip CurrentsRobert A. Dalrymple, Jamie H. MacMahan, Ad J.H.M. Reniers,

and Varjola Nelko � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 551

Planetary Magnetic Fields and Fluid DynamosChris A. Jones � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 583

Surfactant Effects on Bubble Motion and Bubbly FlowsShu Takagi and Yoichiro Matsumoto � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 615

Collective Hydrodynamics of Swimming Microorganisms: Living FluidsDonald L. Koch and Ganesh Subramanian � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 637

Aerobreakup of Newtonian and Viscoelastic LiquidsT.G. Theofanous � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 661

Indexes

Cumulative Index of Contributing Authors, Volumes 1–43 � � � � � � � � � � � � � � � � � � � � � � � � � � � � 691

Cumulative Index of Chapter Titles, Volumes 1–43 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 699

Errata

An online log of corrections to Annual Review of Fluid Mechanics articles may be foundat http://fluid.annualreviews.org/errata.shtml

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fluid dynamics of dissolved polymer molecules in confined geometries

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