Dissipative particle dynamics simulation of depletion layer and polymer migration inmicro- and nanochannels for dilute polymer solutionsDmitry A. Fedosov, George Em Karniadakis, and Bruce Caswell

Citation: The Journal of Chemical Physics 128, 144903 (2008); doi: 10.1063/1.2897761View online: https://doi.org/10.1063/1.2897761View Table of Contents: http://aip.scitation.org/toc/jcp/128/14Published by the American Institute of Physics

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Dissipative particle dynamics simulation of depletion layer and polymermigration in micro- and nanochannels for dilute polymer solutions

Dmitry A. Fedosov,1 George Em Karniadakis,1,a� and Bruce Caswell21Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA2Division of Engineering, Brown University, Providence, Rhode Island 02912, USA

�Received 27 September 2007; accepted 22 February 2008; published online 14 April 2008�

The flows of dilute polymer solutions in micro- and nanoscale channels are of both fundamental andpractical importance in variety of applications in which the channel gap is of the same order as thesize of the suspended particles or macromolecules. In such systems depletion layers are observednear solid-fluid interfaces, even in equilibrium, and the imposition of flow results in furthercross-stream migration of the particles. In this work we employ dissipative particle dynamics tostudy depletion and migration in dilute polymer solutions in channels several times larger than theradius of gyration �Rg� of bead-spring chains. We compare depletion layers for different chainmodels and levels of chain representation, solvent quality, and relative wall-solvent-polymerinteractions. By suitable scaling the simulated depletion layers compare well with the asymptoticlattice theory solution of depletion near a repulsive wall. In Poiseuille flow, polymer migrationacross the streamlines increases with the Peclet and the Reynolds number until the center-of-massdistribution develops two symmetric off-center peaks which identify the preferred chain positionsacross the channel. These appear to be governed by the balance of wall-chain repulsive interactionsand an off-center driving force of the type known as the Segre–Silberberg effect. © 2008 AmericanInstitute of Physics. �DOI: 10.1063/1.2897761�

I. INTRODUCTION

Polymer depletion and cross-stream migration phenom-ena in micro- and nanochannels are important in microfluidicdevices and a variety of biological systems. These effectsmight be relevant in physical processes such as adsorption,lubrication, wall slip, and polymer transport. Depletion lay-ers arise from steric wall repulsion,1,2 when a polymer solu-tion is placed in confined geometries. Depletion was ob-served in the region next to the fluid-solid interface inseveral experiments,3,4 and was simulated using variousmethods including Monte Carlo �MC�,5–7 lattice Boltzmann�LBM�,8,9 Brownian dynamics �BD�,9,10 and dissipative par-ticle dynamics �DPD�.11 An asymptotic analytical solution12

of the depletion layer for an ideal chain in the presence ofpurely repulsive wall predicts depletion to be effective atabout of one radius of gyration from the confining surface.Therefore, in micro- and nanochannels the layer is often ofthe same order as the channel width, and greatly affects thepolymer distribution across the channel.

In presence of flow �e.g., Poiseuille, Couette� the poly-mer migration phenomena changes the polymer distributionacross the channel. Several experimental observations3,4

show polymer migration from the walls towards the channelcenterline. However, simulations8,10,11,13 showed that poly-mer migration might proceed towards the walls as well as tothe channel centerline. Two models of polymer migrationwhich emphasize the importance of polymer hydrodynamicinteractions were recently proposed by Graham et al.14 and

by Ladd et al.13 The former predicts polymer migration awayfrom the walls and attributes this effect to wall-polymer hy-drodynamic interactions and a gradient in chain mobility.The latter states that polymer migration can proceed bothaway from and to the wall, and is determined by the balanceof several effects: hydrodynamic lift, rotation, and drift ofthe polymer. Recent simulations8–11 of polymer migration inPoiseuille flow showed a development of two symmetric off-center peaks �local maxima� in the polymer distribution be-tween the wall and the centerline. These peaks become morepronounced with increasing Peclet or Reynolds number. Incontrast, the polymer distribution in Couette flow yielded asingle local maximum at the channel centerline. Thus, thepresence of two symmetric off-center peaks in pressure-driven Poiseuille flow appears to be related to variableshear rates. The above unresolved issues suggest furtherinvestigation.

Here we employ dissipative particle dynamics15,16 to in-vestigate the depletion layer and polymer migration. DPD isa mesoscopic method that can potentially bridge the gap be-tween the atomistic and the continuum descriptions of fluids.The DPD particles represent clusters of molecules movingtogether in a Lagrangian fashion subject to soft quadraticpotentials. In contrast to molecular dynamics method, DPDemploys much larger time steps and particle sizes due to softparticle interactions. In particular, the DPD method appearsto be successful in simulations of complex fluids, such assuspensions of polymers, DNA, and colloids in a Newtonianincompressible solvent, etc., see Refs. 17–19. In this paper,we systematically investigate the dependence of wall-polymer depletion on the polymer model, level of chain rep-

a�Author to whom correspondence should be addressed. Electronic mail:gk@dam.brown.edu.

THE JOURNAL OF CHEMICAL PHYSICS 128, 144903 �2008�

0021-9606/2008/128�14�/144903/14/$23.00 © 2008 American Institute of Physics128, 144903-1

resentation, solvent quality, and relative wall-polymer-solvent interactions. By means of suitable scaling ofsimulated depletion layers we compare our results to theasymptotic lattice theory solution12 of depletion near a repul-sive wall. Also, we investigate polymer migration in Poi-seuille flow, and we offer an argument which attributes poly-mer migration to wall-polymer hydrodynamic interactionsand to the well-known Segre–Silberberg effect.20,21

The paper is organized as follows. In the next section wepresent some details of the DPD governing equations, thepolymer models, and the boundary conditions. In Sec. III wepresent simulation results for a polymer solution in a channelin the static case �no net flow�. In Sec. IV we present migra-tion results for dilute-solution Poiseuille flow at different Pe-clet and Reynolds numbers. We conclude in Sec. V with abrief discussion.

II. DPD MODEL

A. DPD governing equations

DPD is a mesoscopic particle method, where each par-ticle represents a molecular cluster rather than an individualatom, and can be thought of as a soft lump of fluid. The DPDsystem consists of N point particles of mass mi, position ri,and velocity vi. DPD particles interact through simplepairwise-additive forces corresponding to a conservativeforce Fij

C, a dissipative force FijD, and a random force Fij

R. Thetotal force exerted on a particle i by particle j consists of thethree terms given by

FijC = Fij

C�rij�r̂ij , �1�

FijD = − ��D�rij��vij · r̂ij�r̂ij , �2�

FijR = ��R�rij��ijr̂ij , �3�

where rij =ri−r j, rij = �rij�, r̂ij =rij /rij, and vij =vi−v j. The co-efficients � and � determine the strength of dissipative andrandom forces, respectively. �D and �R are weight functions,

�ij is a normally distributed random variable with zero mean,unit variance, and �ij =� ji. All forces act within a sphere ofradius rc, the cutoff radius, which defines the length scale inthe DPD system. The conservative force Fij

C�rij� is typicallygiven by

FijC�rij� = �aij�1 − rij/rc� , for rij � rc

0, for rij � rc,� �4�

where aij =�aiaj and ai, aj are conservative force coefficientsfor particles i and j, respectively.

The random and dissipative forces must satisfy thefluctuation-dissipation theorem22 in order for the DPD sys-tem to maintain equilibrium temperature T.

�D�rij� = ��R�rij��2 �5�

�2 = 2�kBT , �6�

where kB is the Boltzmann constant. The usual choice for theweight functions is

�R�rij� = ��1 − rij/rc�p, for rij � rc

0, for rij � rc,� �7�

where p=1 for the original DPD method. However, otherchoices �e.g., p=0.25� for these envelopes have beenused23–25 in order to increase the viscosity of the DPD fluidand bring the Schmidt number in DPD to values representa-tive of real liquids. The p values used in simulations will bedesignated below.

The time evolution of velocities and positions of par-ticles is determined by the Newton’s second law of motion,

dri = vidt , �8�

dvi =1

mij�i

�FijCdt + Fij

Ddt + FijR�dt� . �9�

The above equations of motion were integrated using themodified velocity-Verlet algorithm.16

B. Polymer models

The polymer model in our simulations is based on thewell-known linear bead-spring polymer chain representation.Each bead in a polymer chain is subject to three DPD forcesmentioned in the previous section and intra-polymer forcesarising from neighboring bead-to-bead interactions. Here weconsider flexible chains, so two consecutive segments of achain have no preferred angle between them. Below we out-line two spring laws which define force contributions ofbead-to-bead interactions.

TABLE I. DPD simulation parameters.

n rc � � kBT

3 1 4.5 3 1

FIG. 1. �Color online� The equilibrium boundary condition with shear pro-cedure �EBC-S�.

144903-2 Fedosov, Em Karniadakis, and Caswell J. Chem. Phys. 128, 144903 �2008�

1. FENE spring

Each pair of particles connected by the finitely exten-sible nonlinear elastic �FENE� spring is subject to the non-linear potential.

UFENE = −k

2rmax

2 log1 −�ri − r j�2

rmax2 � , �10�

where rmax is the maximum spring extension and k is thespring constant. When the distance between two connectedbeads approaches rmax, the spring attractive force goes toinfinity, and therefore the length greater than rmax is notallowed.

2. Fraenkel spring

Two connected beads interact through the quadratic po-tential with a fixed equilibrium length req.

UFraenkel =k

2��ri − r j� − req�2, �11�

where k is the spring constant. In contrast to the FENEmodel, this type of spring virtually has no limit on the maxi-mum allowed distance between two beads. However, theminimum of potential energy corresponds to the length req

which is preferred equilibrium interbead distance. Some-times this model is also called the harmonic spring.

C. Wall boundary conditions

In order to enforce no-slip boundary condition �BC� atthe fluid-solid interface we employ the equilibrium BCmodel with adaptive shear correction �EBC-S�.25 At the pre-processing stage, the computational domain covers both fluidand solid wall regions and is assumed to be periodic in alldirections. The hydrostatic simulation is run until the equi-

librium state is reached. In the solid region the particles arethen frozen at some instant of time, and later are used tomodel solid walls in combination with bounce-back reflec-tion at the fluid-solid interface. In addition, we use an adap-tive shear procedure illustrated in the Fig. 1. Subregions ofthe computational domain of width L=rc adjacent to thefluid-solid interface in both fluid and wall regions are con-sidered. We divide the fluid and wall subregions into bins ofheight h, whose value determines the accuracy of the near-wall velocity profile. If the velocity profile changes in thedirection parallel to the wall these subregions can be dividedinto bins along the wall. During the simulation, in each bin inthe fluid subregion the time-averaged velocity vav is col-lected over a specified number of time steps. The velocitiesof the particles inside each bin in the wall subregion are setto be opposite, i.e., −vav, to the average velocity in the cor-responding fluid subregion bin, which is symmetric with re-spect to the fluid-solid interface. The wall particles �shown asopen circles� do not move in the simulations, and carry onlythe velocity information needed for the calculation of thedissipative force. A complete description of the BC modelcan be found in Ref. 25.

III. HYDROSTATICS OF CONFINED DILUTE POLYMERSOLUTIONS

In this section we present results of DPD simulations forthe static case of dilute polymer solutions confined betweenparallel plates separated by gap H. Dilute solutions are com-posed as a single polymer chain immersed in a Newtonian-type fluid solvent. Numerical measurements are taken of thepolymer center-of-mass distribution, the bead distribution,and the stresses across the channel.

TABLE II. Simulation parameter sets for different bead-spring models.

spring k rmax req ass app aps awp Rg

FENE 10 2 N/A 25 25 17.5 17.5 1.6264FENE 20 2.5 N/A 25 25 17.5 17.5 1.4437Fraenkel 10 N/A 0.7 25 25 17.5 17.5 1.9207

FIG. 2. Center-of-mass �left� and bead�right� distributions for three N=16bead chains.

144903-3 Polymer migration in channels J. Chem. Phys. 128, 144903 �2008�

A. Simulation parameters

Table I presents the parameters used in DPD simulations.The number density n of a solution includes solvent andpolymer particles. The conservative force coefficients will bespecified in text corresponding to a particular simulation. Allsimulations employed the modified velocity-Verlet integra-tion scheme with �=0.516, which corresponds to the standardvelocity-Verlet scheme widely used in molecular dynamicssimulations. The time step was set to 0.01. Solid walls wereplaced at y=0 and y=H, and modeled by freezing DPD par-ticles at equilibrium in combination with bounce-back reflec-tion at the fluid-solid interface. The number density of thewalls was that of fluid. In addition, adaptive shear procedure�EBC-S type25� was used in order to enhance dissipative in-teractions and ensure no-slip condition at the fluid-solid in-terface. For EBC-S we used a 1�5�1 bin grid for all wallsin order to compute the near-wall velocity profile and mimica counter flow in the wall region.

B. Simulations with several bead-spring models

Employing two different spring models we performedseveral DPD simulations of a chain in a static solvent in thechannel described above. The polymer chain consists of 16beads. Three sets of simulation parameters are shown inTable II, where ass is the repulsive force coefficient betweensolvent-solvent particles, app—polymer-polymer �beads�,aps—polymer-solvent, and awp—wall-polymer particles, re-spectively. Rg is the polymer radius of gyration obtainedfrom equilibrium simulations of dilute solution in largeenough periodic domain. The choice of repulsive interactionsdefines a polymer solution with good quality solvent, whichwas pointed out in Ref. 26. The exponent p in Eq. �7� was setto 0.25 that corresponds to the kinematic viscosity of a fluid0.54, which was obtained using the periodic Poiseuille flow

method of Ref. 27. The height of the channel H was set to3Rg. In the other two dimensions, system is periodic and hasa length more than 2H in order to ensure no dependence ofthe results from the domain size. The simulation times wereset to allow the chain diffusion distance to be at least 40H.Furthermore, 32 statistically independent copies �trajecto-ries� of each simulation were run simultaneously on a BlueGene supercomputer. The combination of trajectories andlong enough run times provided us with smooth chain center-of-mass results even though only one polymer chain ispresent in the simulation domain.

The results revealed that the polymer center-of-mass andthe bead distributions across the channel collapse onto onecurve for all three simulations. Figure 2 shows the center-of-mass distribution �left� and polymer bead distribution �right�for three polymer chains with the distance from the wallnormalized by Rg. These and all subsequent distributions areshown over the half channel because of symmetry with re-spect to the centerline, and hence they have been normalizedwith area of one half. Thus, it appears that polymer distribu-tion in dilute solution is independent of polymer model used,and is correlated only by a mesoscopic polymer characteris-tic length such as the radius of gyration or the end-to-enddistance.

C. Effect of the solvent quality

Polymers and macromolecules can have dissimilar prop-erties and behavior in different solvents. In fact, polymercharacteristics can drastically change depending on the sol-vent quality. It is well known that the quality of the solventexhibits different scaling laws for static and dynamic poly-mer properties �e.g., Rg, D�. Here, we performed DPD simu-lations to identify the effect of solvent quality on the polymerdistribution in the channel. We compare ideal chains to

TABLE III. DPD parameter sets for solvent quality calculations.

spring solvent k rmax req ass app aps awp Rg

FENE good 20 2 N/A 25 25 17.5 17.5 4.7373FENE poor 20 2 N/A 25 25 25 25 3.9235Fraenkel ideal 10 N/A 0.7 0 0 0 0 3.5691

FIG. 3. Effect of the solvent qualityon the center-of-mass �left� and bead�right� distributions for N=100 chains.

144903-4 Fedosov, Em Karniadakis, and Caswell J. Chem. Phys. 128, 144903 �2008�

FENE chains in both poor and good solvents respectively.Each chain consists of N=100 beads and is placed in a chan-nel of gap H=3Rg. The simulation parameters are summa-rized in Table III. The different spring model for the idealchain is required due to the absence of DPD repulsive inter-actions, which for a FENE spring would yield zero equilib-rium distance between connected beads because it exertsonly attractive force. However, Fraenkel spring exhibits anonzero equilibrium length explicitly. Furthermore, the re-sults shown in Fig. 2 exhibited no dependence on the springmodel. The above parameters are chosen to match the sol-vent quality as in Ref. 26. The exponent p �Eq. �7�� is set to1.0, so that kinematic viscosity of the fluid is 0.2854.

Figure 3 shows the center-of-mass distribution �left� andthe polymer bead distribution �right� for the ideal �Fraenkel�polymer and FENE chains in poor and good solvents. Incontrast to the insensitivity of the distributions to differentpolymer models �Fig. 2�, solvent quality has an effect on thepolymer distribution. The ideal chain exhibits larger walldepletion �more confined distribution around the center�

compared to chains in good and poor solvents. A good sol-vent yields the smallest wall depletion and poor solventcurve falls in-between good solvent and ideal chain. The walldepletion force on a chain is expected to be repulsive due tothe cost in free energy for the loss of the available polymerconfigurations near the wall. Furthermore, the depletion po-tential and resulting wall force are weaker for chains withexcluded volume �EV� interactions �good solvent� than forideal chains as theoretically predicted by Schlesener et al.,28

where they concluded that EV interactions effectively reducethe depletion effect. As a result we observe larger wall deple-tion for ideal chains. This is illustrated by the pressure dis-tribution across the slit. Figure 4 shows the excess pressuredistribution in the channel with the solvent virial contribu-tion and without the interbead-spring forces. Hence, onlywall-polymer and solvent-polymer interactions contribute tothe wall depletion. The excess pressure is obtained by sub-traction of the equilibrium pressure in a large box. The pres-sure gradient for the ideal chain as we approach the wallfrom the centerline is larger than that of poor and good sol-vent. The larger pressure difference drives the polymer fur-ther away from the wall and contributes to larger wall deple-tion. The least pressure gradient and thus the smallest walldepletion was found for the good solvent case.

For complete analysis of the results we compute averagepolymer lengths in all three directions �x, y, and z� across thechannel which characterize relative polymer shapes in theslit. We define local radius of gyration depending on thedistance from the wall y as

Rg�y� = ��Rgx�y��2 + �Rg

y�y��2 + �Rgz�y��2�1/2, �12�

where Rgx�y�= ��1 /N�i=1

N �xi−xcm�y��2 , �· denotes time aver-aging, xi are bead coordinates in the x-direction, and xcm�y� isthe center of mass at y. Rg

y�y� and Rgz�y� are defined analo-

gously. Figure 5 presents the local radius of gyration normal-ized by the unconfined Rg �left� and the ratio of Rg

x�y� andRg

y�y� �right� which identifies the relative polymer shapeacross the channel. The left-hand plot in Fig. 5 shows thatwhen confined in a slit the polymer occupies a volumeslightly smaller than it would in the unconfined state, andthat the chain takes on a more compact volume as it ap-proaches the wall. The right-hand plot of Fig. 5 shows that

FIG. 4. Excess pressure across the channel for N=100 bead chain in sol-vents of different quality.

FIG. 5. Local radius of gyration �left�and relative shape of the polymer�right� for N=100 bead chain in sol-vents of different quality.

144903-5 Polymer migration in channels J. Chem. Phys. 128, 144903 �2008�

during the approach to the wall the chain elongates in the xand z directions relative to the y direction, and that its shapeis nearly independent of solvent quality. As expected by sym-metry Rg

z�y� was found to be statistically identical to Rgx�y�.

D. Wall-polymer-solvent interactions

In the simulations presented above wall-polymer andsolvent-polymer interactions were identical. These are neu-tral walls for which no explicit adsorption or repulsion isexpected. However, real walls are known to induce adsorp-tion or repulsion �electrostatic� of polymers. A manipulationof relative wall-polymer-solvent interactions enables us toexplicitly introduce net-attractive or repulsive wall forces onthe polymer. This might correspond to adsorption and repul-sion, respectively, but it requires further investigation. To thisend, a polymer of N=25 beads in a good solvent was placedin a slit of gap H=3Rg. The simulation parameters are sum-marized in Table IV. Figure 6 presents the center-of-mass�left� and the bead �right� distributions for both neutral andrepulsive walls. These results are compared to the center-of-mass distribution obtained by Ladd et al. in Ref. 8 using theLBM method for an N=16 bead chain in a channel also ofgap H=3Rg. The repulsive wall exhibits larger depletioncompared to neutral wall. Since the results of Ladd et al. fallbetween the neutral and the repulsive curves they correspondto a slightly repulsive wall. To verify that the boundary con-dition model �EBC-S� does not introduce extraneous effects,we have run analogous simulations using periodic BCs.Here, the computational domain is doubled in y direction togive a gap of 2H=6Rg. The left half contains polymer solu-tion confined between two parallel reflective walls at y=0and y=H, and the right half is filled with the solvent. Thesystem is set to be periodic in the y direction. Such a setup

corresponds to a perfectly neutral wall with no extraneouseffects. The results were found to differ by no more than thestatistical error.

E. Influence of the channel width

To this point we have presented results for channels offixed gap H=3Rg. In this section the effect of channel widthon the wall depletion layer is investigated with simulationsof a N=16 bead chain confined in slits having gaps of H=3Rg, 4Rg, 5Rg, and 8Rg, and with the parameters shown inTable V. Figure 7 presents the center-of-mass distribution ofthe simulated polymer confined in channels with gaps listedabove. It also includes center-of-mass distributions normal-ized by their maximum magnitude cmax �correspondence isshown by arrows�. For all channel widths the normalizeddistributions collapse onto a single curve which clearly dem-onstrates that the wall-polymer depletion region reaches nofurther than about 2.5Rg. Consistency of this conclusion istested by plotting the local radius of gyration normalized bythe unconfined Rg and the ratio of components Rg

x�y� andRg

y�y�. Figure 8 shows that beyond a distance of about 2.5Rg

both the local size �left� and the shape �right� of the polymerare unaffected by the wall.

F. The bead number N effect

Many theoretical results in polymer physics areasymptotic in the limit of large bead number N. Examplesare the scaling laws for the radius of gyration and for thediffusion coefficient. To test the level of chain representationon the depletion layer we performed simulations for a set ofchains having bead numbers N=16, 25, 100, and 500 in achannel of gap H=3Rg, and with the parameters shown inTable VI.

TABLE IV. DPD parameter sets used in wall-polymer-solvent interactioncalculations.

spring wall k rmax ass app aps awp Rg

FENE neutral 10 2 25 25 17.5 17.5 2.1903FENE repulsive 10 2 25 25 17.5 25 2.1903

TABLE V. Simulation parameters for the channels of various gaps.

spring k req ass app aps awp Rg

Fraenkel 10 0.7 25 25 17.5 17.5 1.9207

FIG. 6. Effect of wall-polymer-solventinteractions on the center-of-mass�left� and bead �right� distributions fora N=25 bead chain.

144903-6 Fedosov, Em Karniadakis, and Caswell J. Chem. Phys. 128, 144903 �2008�

Figure 9 shows the calculated center-of-mass distribu-tions for this set of chains, and also the LBM results for N=16 in a channel of the same gap by Ladd et al.8

The gap dimensions for the simulations discussed aboveassumed the appropriate characteristic length to be the un-confined Rg. An alternative length scale, independent of mo-lecular concepts, is derivable from concentration distribu-tions such as those of Fig. 9. This depletion layer thickness is defined as,

Rg= �

0

� �1 −c�z�cmax

�dz , �13�

where z=y /Rg, c�z� is the center-of-mass distribution, andcmax=maxz�0�c�z��. From the distributions of Fig. 6, Eq. �13�yields /Rg=0.837 for the neutral wall, /Rg=0.954 for therepulsive wall, and for Ladd et al. /Rg=0.868.

Longer chains are subject to a stronger depletion effectmeasured as a larger depletion layer thickness. Thus, it ap-pears that, when the allowable configuration space is re-

stricted to be a half-space, the longer chains suffer a largerloss in free energy and are subject to larger steric depletionforces. However, the analytical solution of Ref. 12 for idealchains assumes the center-of-mass distribution will convergeto an asymptotic curve as N becomes very large. In Fig. 10the center-of-mass distributions, normalized by their maxi-mum values cmax, for different bead numbers are comparedwith the asymptotic analytical solution. In the left-hand plotthe distance from the wall is normalized by the equilibriumvalue of the radius of gyration Rg, and in the right-hand plotby the depletion layer thickness . The LBM results8 are foran N=16 bead polymer in a channel of width H=5Rg. Thecurves in the figure show that, with lateral shifting, the nu-merical models closely capture the functional forms of theanalytic solution, and that the depletion layer thickness pro-vides the means for shifting the numerical distributions ontothe analytic solution to produce an almost common curve.The agreement is remarkable since the numerical model fea-tures solvent explicitly represented by DPD particleswhereas in the lattice model the solvent is implicit. Figure 10�left� shows that, as expected, the shorter chains have distri-butions closer to the wall, and that, as N goes from 100 to500, the discrepancy between the numerical and the analyticdistribution becomes vanishingly small. For the latter thesmall discrepancy may be due to the solution having beencarried out with a poor solvent condition, which as previ-ously noted slightly weakens the depletion. The LBM curvehas the largest discrepancy probably due to the relative wall-polymer-solvent interactions mentioned in Sec. III D and itssmall bead number N. In addition, our results agree well withthe depletion layers calculated from Monte Carlo simulationsby Berkenbos et al.7 for tubes. They also investigated thelimit of very small tubes, where the confinement greatly re-stricts the chain configurations. These distributions no longerresemble those in Figs. 7 and 10.

For the same simulations, Fig. 11 presents the local

TABLE VI. DPD parameters used in bead number effect simulations.

spring k rmax ass app aps awp

FENE 10 2 25 25 25 25

FIG. 7. Influence of gap size on the center-of-mass distribution for a chainof N=16 beads.

FIG. 8. Local radius of gyration �left�and relative shape �right� for a chainof N=16 beads for various gaps.

144903-7 Polymer migration in channels J. Chem. Phys. 128, 144903 �2008�

Rg�y� normalized by its unconfined value �left� and the ratioof Rg

x�y� and Rgy�y� �right�. The fact that the longest chains

have the smallest normalized local Rg�y� suggests that in theslit they adopt a more close-packed form than their smallercounterparts even though their relative shapes are about thesame.

IV. POISEUILLE FLOW OF DILUTE POLYMERSOLUTIONS

This section results are presented for the Poiseuille flowof dilute polymer solutions confined between parallel platesseparated by gap H. The Poiseuille flow is driven by a uni-form pressure gradient as a force applied equally to bothpolymer and solvent DPD particles. Following8 dynamic ef-fects of the flow will be interpreted primarily with the Pecletnumber �Pe� defined as

Pe = �̇Rg

2

D= 4Sc Re

Rg2

H2 , �14�

where �̇=2Vc /H is the mean shear rate, Vc is the centerlinevelocity, and D is the polymer center-of-mass diffusion co-efficient measured in equilibrium. The Schmidt number �Sc�and the channel Reynolds number �Re� are defined, respec-tively, by

Sc =

D, Re =

VcH

2 =

�̇H2

4 . �15�

The discussion below invokes other Reynolds numbersbased on the radius of gyration �Reg� and Stokes–Einsteinradius rb �Ref. 24� of a polymer bead �Reb� which can bedefined, respectively, as

Reg =�̇Rg

2

, Reb =

�̇rb2

. �16�

In some works,9,10 Poiseuille flow results are interpretedin terms of the Weissenberg number Wi=��̇, where � is de-rived from the long-time relaxation of the stretched polymerchain in a stagnant solvent. Since both D and � are derivedfrom equilibrium data, they differ by at most a constant for agiven chain. Furthermore, Wi can be expressed as a productsimilar to Eq. �14� with Sc replaced by � /Rg

2, which is alsoa purely material property.

A. Velocity profiles

The Poiseuille velocity profile between walls placed aty=0, H for a non-Newtonian fluid with a power-law shearviscosity is given by

V�y� = Vc1 − � y − H/2H/2 �1+1/n� ,

�17�

Vc =n

1 + n��g

��1/n�H

2�1+1/n

,

where n is the power-law index, g the uniform driving forceper unit mass, � the power-law shear-stress coefficient, and �the mass density.

FIG. 9. Effect of bead number N on the center-of-mass distribution in a slitof H=3Rg.

FIG. 10. Normalized center-of-massdistributions for different bead num-bers compared with the analytical so-lution. The wall distance y is normal-ized by the unconfined Rg �left� anddepletion layer thickness �right�.

144903-8 Fedosov, Em Karniadakis, and Caswell J. Chem. Phys. 128, 144903 �2008�

Figure 12 shows velocity profiles for an N=100 bead-chain solution in a gap of 3Rg at Pe=50, 100, and 200. Thecalculated velocity profiles of the suspension are well de-scribed by the power law �Eq. �17�� with n=0.88. Thedashed curves are the corresponding Newtonian profiles forn=1. The average velocity of the particles within a slice �yof the gap is assigned to be the local continuum velocity atthe slice center. Hence, continuum wall velocities cannot becalculated directly. However, as expected from the particleboundary conditions of Sec. II C, there appears to be no slipat the wall since nearby velocities extrapolate to zero withinstatistical error. The Re, Reg, and Reb numbers scale linearlywith Pe, and for the flow of Pe=200 they are Re=39.49,Reg=17.55, and Reb=0.055, respectively, where Rg=3.9235,rb=0.22, and =0.2854.

B. Polymer migration

In addition to the hydrostatic depletion already pre-sented, Poiseuille flow gives rise to further cross-flow migra-tion of the polymer. This dynamic migration was investi-

gated with a chain of N=16 beads with the parametersshown in Table VII. The solvent viscosity for all cases is =0.54.

The simulations were performed at Pe numbers of 50,100, and 200 for several channels. Figures 13–15 correspondto the gap widths H=3Rg, 5Rg, and 8Rg, respectively, anddisplay the effect of the Pe and the Re numbers on the resultsfor �i� the center-of-mass distributions, and �ii� chain confor-mation distributions imaged by computation of the threecomponents of the local radius of gyration Rg

x�y�, Rgy�y�,

Rgz�y� normalized by their equilibrium components Rg /�3. In

the figures the mean of the distribution is displayed as ahorizontal line.

The center-of-mass distributions of Figs. 13–15, respec-tively, show the dynamic depletion layer to be steadily re-duced relative to the hydrostatic case as Pe is increased, andthat the effect is strongest for the smallest gap. In contrast,migration of polymer from the centerline towards the wallsbecomes more pronounced as the gap size increases, andbetween Pe=100 and 200 the significant development is adistribution with two off-center peaks, which is consistentwith previous work,8 and similar results obtained by variousother methods were reported in Refs. 9–11 and 13. Factorscontributing to cross-stream migration are listed by Laddet al.13 as lift, rotation, stretching, and drift of the polymerfrom the wall. Graham et al.10 attribute migration to twoeffects: chain-wall hydrodynamics and chain mobility gradi-ents due to different conformations such as a stretched chain.The Rg distributions show the chain to be strongly stretchedin flow direction x when subjected to high shear rates. Inview of the hydrostatic preference for the centerline region, itseems counterintuitive that a chain prefers to occupy the re-gion of higher shear rates away from the centerline with itsvanishing shear rate. It is plausible to attribute the forceswhich drive the chain away from the center to be similar tothose that control the effect of Segre–Silberberg in which20,21

TABLE VII. Simulation parameters used in polymer migration calculations.

spring k rmax ass app aps awp Rg

FENE 10 2 25 25 25 25 1.36205

FIG. 11. Local radius of gyration �left�and relative shape �right� of the poly-mer for various N.

FIG. 12. Poiseuille velocity profiles for several Pe’s, N=100.

144903-9 Polymer migration in channels J. Chem. Phys. 128, 144903 �2008�

a neutrally buoyant rigid sphere or ellipsoid29 in a channelwill migrate to an equilibrium position between wall andcenterline due to the combined hydrodynamic effects ofwall-particle interactions, velocity profile curvature andshear forces. The components of Rg show that, on average,the confined chain resembles an ellipsoid when viewed as anentity. However, flexibility and elasticity are not includedexplicitly in the Segre–Silberberg analysis although thesefactors are implicit in the definition �14� of the Peclet num-ber. An alternative view is the chain as an ensemble of con-nected point-particles where each behaves as a sphere with aStokes–Einstein radius,24 and therefore is subject to theSegre–Silberberg forces. As reported in Refs. 30 and 31 theSegre–Silberberg effect for a suspension of particles stronglydepends on the ratio of particle size to the channel width andReynolds number of the flow based on the particle size. Forthe case of Pe=200, we have Re number in the range of28.4–201.2 for the gaps 3Rg−8Rg, and Reg=12.6, Reb

=0.33, respectively. In our simulations the Re numbers are inthe range of those given in Ref. 31 where results for Poi-seuille flow predict particle migration to a stable positionin-between the wall and the centerline. This appears to sup-port our interpretation of the above results as the Segre–Silberberg effect. The results are qualitatively similar for thethree channels. However, wall-molecule interactions appearto be a determining factor for the polymer distribution insmall channels �H�3Rg�. Wall-polymer repulsive forces are

dominant within a layer of about 2.5Rg, and therefore insmall slits the development of the off-center peaks in thedistributions is observed only for relatively high Pe numberwhen Segre–Silberberg forces are able to overwhelm wallsteric repulsion. In contrast, for larger channels �H�5Rg� theoff-center peaks in the polymer distributions appear at lowerPe, because wall-polymer forces act only within a near-walllayer and may not be present in the centerline region whereshear forces have a migration effect. In addition, the distri-bution peaks coincide approximately with the maximumstretch �max.Rg

x, min.Rgz� of the chain in the streamwise di-

rection. The scatter in the distribution data clearly increaseswith channel size. This statistical defect for wider channels isa consequence of more expensive calculations for larger par-ticle systems and longer times required for a chain to migratea number of times across the slit.

The above argument suggests that polymer migration ischaracterized by at least three numbers: Pe, Re, and Rg /H. Inthe presence of inertia �Re�0� the Segre–Silberberg effect isto be expected, and this is in accord with the relatively highRe number cases presented above. Graham et al.9 used theBrownian dynamics �BD� method to show ever increasingpolymer migration towards the channel centerline with in-creasing Pe or Wi, where hydrodynamic inertia effects areexcluded by use of only Stokes interactions for which theSegre–Silberberg effect vanishes. In particular, Stokes hydro-dynamics accounts for very strong wall-polymer interactions.

FIG. 13. Polymer center-of-mass �up-per left� and conformation distribu-tions of Rg

x �upper right�, Rgy �lower

left�, and Rgz �lower right� in Poiseuille

flow, N=16, H=3Rg.

144903-10 Fedosov, Em Karniadakis, and Caswell J. Chem. Phys. 128, 144903 �2008�

However, as Wi increases, the channel Reynolds numbermust also increase. Analysis with linearized inertia �timedependent32 �p. 354�, Oseen33� suggests that as inertia in-creases from zero the typical leading Stokes interaction inthe reciprocal distance is continuously replaced by the first-order inertial term typical of the unbounded case. This tran-sition would appear to mark the inception of the Segre–Silberberg effect. The complexity of bead-spring suspensionsdoes not permit “a priori” estimates of this Re range, but theRe values actually realized are germane to an assessment ofthe effect in Ref. 9. To this end, we have performed a numberof DPD simulations with greatly reduced Re number. A 16-bead polymer solution having viscosity =14.019 confinedin the channel of H=3Rg is subject to Poiseuille flow at Pe=50, 100, and 200. For Pe=200 we have Re=0.041, Reg

=0.018, and Reb=4.66�10−4, respectively. Figure 16 showsthe center-of-mass distribution and chain conformations. Thecenter-of-mass distributions of Fig. 16 differ only slightlyfrom the hydrostatic case, in accord with the often-used de-scription of low Re flow as “quasistatic.” In contrast, thechain conformations resemble those at high Re shown in Fig.13 for the same H /Rg rather than the hydrostatic conforma-tions of Fig. 8. Unfortunately, the BD results of Graham etal.9 do not include the hydrostatic case which would providea critical comparison with the results of this paper. For Pe�100 polymer migration proceeds towards the channel cen-terline which suggests the dominance of wall-polymer inter-

actions. However, as Pe→200, the center-of-mass distribu-tion indicates a slight polymer migration away from thecenterline which suggests that the Segre–Silberberg forceshave become comparable with wall-polymer interactions.Moreover, evidence for the Segre–Silberberg effect at lowRe number is provided by the perturbation theory of Ref. 30,the simulations of Ref. 31 and the experiments of Ref. 34. Inconclusion, the results of this work indicate that polymermigration is governed by the Pe, Re, and Rg /H numberswhich characterize wall-polymer interactions and the Segre–Silberberg effect. However, when Re→0 �approximatelyO�10−3�� the Segre–Silberberg effect is negligible and poly-mer migration is then governed only by Pe and Rg /H, inaccord with Refs. 9 and 10.

Other results, not included in this paper, indicate thatpolymer migration in Poiseuille flow is independent of thesegment spring model �FENE and Fraenkel�, and is governedalmost entirely by the three numbers mentioned above whichincorporate the characteristic polymer length and time scales�Eq. �14� and �15��. However, we expect that increasing thenumber of beads in a chain will alter chain-wall interactionswhich this work has shown to affect migration. The hydro-static results �Fig. 10� suggest that for large enough N themigration effect should be independent of the chain represen-tation. However, even in the static case, 500-bead chains

FIG. 14. Polymer center-of-mass �up-per left� and conformation distribu-tions of Rg

x �upper right�, Rgy �lower

left�, and Rgz �lower right� in Poiseuille

flow, N=16, H=5Rg.

144903-11 Polymer migration in channels J. Chem. Phys. 128, 144903 �2008�

were subject to the statistical difficulties mentioned above,and reliable results will be even more expensive to attain inthe dynamic case.

The effect of solvent quality on migration is investigatedwith the 16-bead chain employed above in the channels withgaps of H=3Rg and 4Rg for Pe=50 and 100. Figure 17shows the center-of-mass distribution for solvents of differ-ent quality. Results predict stronger migration away from thecenterline for good solvents than poor solvents. We attributethis to weaker wall depletion of polymers in good solvents aswas pointed out in Sec. III C for the static case, and poten-tially stronger Segre–Silberberg effect due to the larger vol-ume taken up by chains in good solvents compared to themore compact, less swollen, shape in poor solvents. As in theresults of Figs. 13–16, the migration effect in smaller chan-nels is attenuated by the stronger chain-wall interactions.

V. SUMMARY

For dilute polymer solutions in narrow channels we haveinvestigated the hydrostatic depletion near walls for severalbead-spring models, channel widths, number of beads, sol-vent quality, and wall-polymer-solvent interactions. Thechannel width was varied from 3 to 8 times Rg, the uncon-fined polymer radius of gyration. The center-of-mass andbead distributions and polymer shape measured by the com-ponents of the local radius of gyration were found to be

independent of the bead-spring model used in the simula-tions. Wall depletion and polymer shapes were found to besimilar for channel gaps H�3Rg, and indeed distributionsfor all channel gaps collapse onto a single curve by normal-ization with the maximum concentration cmax, when othervariables are fixed. The chain length specified with beadnumber N in the chain representation affects the depletionlayer with short chains having narrower depletion layers thanlonger ones. However, as N becomes large �N�500� thedepletion layer is independent of N, and the polymer distri-bution across the channel converges to the lattice theory so-lution for ideal chains near a purely repulsive wall. By scal-ing the distance from the wall with , an integral measure ofdepletion layer thickness �Eq. �13��, the center-of-mass dis-tributions for all N can be collapsed onto the lattice theoryasymptotic solution. Thus, with appropriate scaling, distribu-tions for N�O�10� adopt the shape of the very long, flexiblechains of the lattice theory.

However, when a chain is immersed in a solvent of dif-ferent quality its depletion layer properties change. A goodsolvent yields a thinner depletion layer than does a poorsolvent, and by comparison the ideal chain has the strongestwall depletion. Finally, relative wall-polymer-solvent inter-actions or simply polymer boundary conditions also have a

FIG. 15. Polymer center-of-mass �up-per left� and conformation distribu-tions of Rg

x �upper right�, Rgy �lower

left�, and Rgz �lower right� in Poiseuille

flow, N=16, H=8Rg.

144903-12 Fedosov, Em Karniadakis, and Caswell J. Chem. Phys. 128, 144903 �2008�

strong effect on the depletion layer. Potentially, these inter-actions could be combined to control the depletion layerthickness and to model a wall adsorption.

In the dynamic case of Poiseuille flow we found thatdilute polymer solutions exhibit slightly non-Newtonian be-havior with velocity profiles corresponding to a power-lawindex of n=0.88 �Eq. �17��. The hydrostatic depletion layeris affected by the flow, but beyond about a distance of �O�1� the center-of-mass distributions exhibit with increas-ing Pe and Re numbers ever stronger indications of migra-

tion toward an intermediate position between wall and cen-terline. At high enough Pe number Poiseuille flow induces inthe distributions the new feature of two off-center peakswhich correspond to the most probable channel positions.Simultaneously, the chains tend to stretch out in the directionof the shear planes. Thus, chains prefer an intermediate po-sition between wall and centerline and not the middle of thechannel where the shear rate vanishes. We attribute this tothe hydrodynamic Segre–Silberberg effect which forcesspheres and ellipsoids away from the centerline due to the

FIG. 16. Polymer center-of-mass �up-per left� and conformation distribu-tions of Rg

x �upper right�, Rgy �lower

left�, and Rgz �lower right� in Poiseuille

flow, N=16, H=3Rg. Low Re number.

FIG. 17. Influence of solvent qualityon center-of-mass distributions for Pe=50,100, H=3Rg �left�, and H=4Rg

�right�.

144903-13 Polymer migration in channels J. Chem. Phys. 128, 144903 �2008�

shear gradient in Poiseuille flow. At high enough Re whenhydrodynamic forces prevail over Brownian fluctuations, thequasistable polymer position in the channel appears to bedetermined by the balance of the wall-polymer hydrody-namic interactions and forces arising from Segre–Silberbergeffect. In case of low enough Re number when the Segre–Silberberg effect becomes negligible polymer migration isgoverned by Pe number and proceeds mostly to the channelcenterline due to wall-polymer interactions. We have shownthat polymer migration is independent of the bead-springmodel. However, the quality of the solvent, polymer bound-ary conditions, and the number of beads in the chain allaffect polymer migration in the slit. We expect no depen-dence on number of monomers for large enough N. Finally,two off-center peaks in the distribution develop at lower Pein the larger channels which is consistent with the knowninteraction length of several Rg for wall-polymer depletion.Hence the migration effect is more pronounced in the chan-nels of larger widths.

ACKNOWLEDGMENTS

We would like to thank A. J. C. Ladd for providing LBMresults for comparison. This work was supported by NSFgrants IMAG and CI-TEAM. Simulations were performedon the NSF/SDSC Blue Gene system.

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