Cross-streamline migration of a semiflexible polymer in a pressure driven flow S. Reddig and H. Stark Citation: J. Chem. Phys. 135, 165101 (2011); doi: 10.1063/1.3656070 View online: http://dx.doi.org/10.1063/1.3656070 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i16 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 23 Mar 2012 to 130.149.114.120. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Transcript
Cross-streamline migration of a semiflexible polymer in a pressure drivenflowS. Reddig and H. Stark Citation: J. Chem. Phys. 135, 165101 (2011); doi: 10.1063/1.3656070 View online: http://dx.doi.org/10.1063/1.3656070 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v135/i16 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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Microfluidic devices have emerged as powerful tools formanipulating, controlling, and analyzing various processesin chemistry, physics, and biology such as DNA sequencing,polymerase chain reaction, cell sorting, and cell culturing.1, 2
On the other hand, microfluidic channels are ideal toolsfor basic research. They allow controlled studies on theinfluence of confinement or mimic biological systems whereconfinement is essential. Examples are the flow of red bloodcells through blood vessels or single actin filaments in theactin network of the cell cortex. Most importantly, using thepressure driven or Poiseuille flow through a microchannel,one can controllably drive suspended objects out of equi-librium and thereby induce novel and intriguing dynamicstructure formation in complex fluids. In this article weaddress the flow-induced migration of a semiflexible polymeracross streamlines by Brownian dynamics simulations. Thiseffect is commonly called cross-streamline migration. Weexplain it by analyzing the Smoluchowski equation for theprobability distribution of the polymer’s center of mass.
Already in 1836, Poiseuille observed in his studies ofblood flow that an absence of red blood cells near the confin-ing walls,3 which was caused by cells migrating perpendicu-lar to the flow direction. More than 100 years later, Segre andSilberberg investigated rigid spheres flowing through circulartubes with a diameter of ∼1 cm and found that the spheresaccumulate at a distance of 0.6 times the tube radius fromthe centerline.4 So, in addition to depletion at confining walls
a)Author to whom correspondence should be addressed. Electronic mail:[email protected].
reported in earlier studies,3, 5 migration of the particles awayfrom the channel’s centerline occurred. Further experimentsconfirmed this observation,6–8 also in microfluidic deviceswith flow velocities of the order of m/s.9 All these studieswere explained by inertial forces acting on the particles sincethe Reynolds number Re was always larger than one.10–12 Atlow Reynolds numbers close to zero, single spherical parti-cles just follow the streamlines in a laminar flow as a result ofthe kinematic reversibility of the Stokes equations and cross-streamline migration does not occur.13 This was explicitlydemonstrated in Refs. 9 and 14. At low Reynolds numbers,rigid particles distributed homogeneously across the channeldue to Brownian motion, whereas at finite Reynolds numbersthey migrated to a position between the channel wall and cen-terline. However, in dense colloid suspensions the lateral par-ticle profiles also become inhomogeneous.15
Flexible polymers also show cross-streamline migrationin the regime of low Reynolds numbers, which has been in-tensively studied. A depletion of polymers near walls in pla-nar shear flow as well as in Poiseuille flow was observedin computer simulations16–23 and experiments.24–26 Since apolymer interacts hydrodynamically with bounding walls, mi-gration towards the centerline occurs, where the thickness ofthe depletion layer increases with the strength of the flow.This explanation was confirmed by analytical arguments27, 28
and it describes well repulsion from bounding walls. In addi-tion, simulations reported migration away from the channel’scenterline,18–20, 22 especially under strong confinement.19 Asa result, the maximum concentration of the polymer occursat a finite distance from the centerline and the depletion inthe center increases with flow strength.19, 20, 22 The spatiallyvarying shear rate of a Poiseuille flow changes orientation
165101-2 S. Reddig and H. Stark J. Chem. Phys. 135, 165101 (2011)
and conformation of a polymer within the channel. Polymersare stretched close to the walls and they are coiled at thecenter.16–21 This gives rise to a position-dependent diffusivityand thereby a diffusion current away from the center,18–20, 22, 27
which generates the observed bimodal concentration profile.Cross-streamline migration of semiflexible polymers,
such as α-actin filaments, is much less studied. Since theirbending rigidity significantly determines flow-induced con-formations in the Poiseuille flow, we also expect an influenceon cross-streamline migration. Indeed, recent experimental29
and simulation30, 31 studies report a much more pronouncedbimodal concentration profile across the channel when com-pared to flexible polymers. This effect occurs even under lessstrong confinement. As for flexible polymers, the concen-tration profile is also explained by the competition betweenhydrodynamic polymer-wall interactions and enhanced diffu-sion away from the centerline.
In this article we reexamine the problem and performBrownian dynamics simulations for a bead-spring chain withbending elasticity. Hydrodynamic interactions between thepoint-like beads are taken into account by the two-wall Greentensor of the Stokes equations. We indeed observe the bi-modal distribution for the center of mass and present resultson the orientational order of the end-to-end vector. Generaliz-ing the approach of Ma and Graham,27 we formulate and in-terpret our findings with the help of a Smoluchowski equationfor the center-of-mass probability distribution by carefully an-alyzing all contributions to the probability current. In contrastto flexible polymers, we show that the deterministic drift cur-rent, where hydrodynamic interactions along the polymer areessential, mainly determines the bimodal distribution acrossthe channel, whereas diffusional currents become less impor-tant with increasing polymer stiffness.
In Sec. II, we describe the model and our simulationmethod. Our results are presented in Sec. III. Here, we an-alyze the steady-state probability distribution for the center ofmass and study both conformation and orientational order ofthe filament in the channel. In Sec. IV, we derive the Smolu-chowski equation for the center-of-mass distribution and in-vestigate the different contributions to the probability current.Finally, we summarize our results in Sec. V.
II. MODELING
In this section we describe how we model a semiflexiblepolymer in a pressure driven Poiseuille flow. We first intro-duce the semiflexible polymer as a bead-spring chain withbending elasticity, explain how it couples to the Poiseuilleflow field, and finally summarize details of our Browniandynamics simulations. The semiflexible polymer is confinedbetween two parallel planar walls at positions z = −W andz = W, which are infinitely extended in the x–y plane (seeFig. 1). For simplicity, we assume that all parts of the chainonly move in the y–z plane at x = 0. This corresponds to theexperiments reported in Ref. 29, where only filaments wererecorded and analyzed whose motion occurred in a narrowregion (±0.5 μm) around the focal plane of the microscope.Furthermore, this also reduces our computational efforts con-siderably. Between the walls a pressure driven Poiseuille flow
FIG. 1. A semiflexible polymer modeled by a bead-spring chain with bend-ing rigidity is confined between two parallel planar plates with distance 2W.The space vector r i points from the centerline at z = 0 to bead i.
is created, which we express as
v(r) = v0
[1 −
( z
W
)2]
ey, (1)
where v0 denotes the maximum velocity at the centerlinez = 0, ey is the unit vector along the y axis, and W is thedistance between the centerline and the walls.
A. Semiflexible polymer
The semiflexible polymer is modeled by a bead-springchain that resists bending.32 In this model, N sphericalmonomers of radius a and with Stokes friction coefficientγ = 6πηa are connected by N − 1 frictionless springs withspring constant H (see Fig. 2). Assuming a harmonic bond po-tential or Hooke’s law for the springs, the total bond energyreads as32
US = H
2
N−1∑i=1
(qi − l)2 , (2)
where qi = |r i+1 − r i | denotes the distance between beads i+ 1 and i, r i is the position vector of bead i, and l is the bondlength. To account for polymer stiffness, we apply the bendingenergy,33, 34
UB = κ
l
N−2∑i=1
(1 − qi+1
qi+1· qi
qi
). (3)
Here, qi = r i+1 − r i is the bond vector, κ is the bend-ing rigidity, which is related to the persistence lengthLp = κ/kBT,32 and · means scalar product. Finally, the forceacting on bead i due to bending and stretching the bead-springchain follows from
Fi = −∇i(US + UB), (4)
where ∇ i is the nabla operator with respect to the positionvector r i .
FIG. 2. Sketch of the bead-spring chain. N beads with position vectors r i areconnected by springs. The vector qi = r i+1 − r i connects bead i to i + 1 andR = rN − r1 denotes the end-to-end vector.
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165101-3 Migration of a semiflexible polymer J. Chem. Phys. 135, 165101 (2011)
FIG. 3. Field lines of the flow fields induced by a point force acting close to a wall either (a) along or (b) perpendicular to a wall.35
In addition, we let the beads interact by a hard-sphere(HS) potential
UH ={
∞, rij < 2a
0, else,(5)
which implements their excluded-volume interaction. In theBrownian dynamics simulation (see Sec. II C), we approxi-mate the hard-sphere potential by the steep repulsive potentialUHS = h
∑i,j (2a/rij )48 with h/kBT = 1 × 104. The beads are
also repelled from the walls through the hard-core potential,
UW ={
∞, W − |zi | < a
0, else,(6)
where zi is the z coordinate of r i and W is the distance fromthe centerline to the walls. When a bead overlaps with thewall during the Brownian dynamics simulation, we just moveit back to bead-wall contact.
B. Hydrodynamic interactions
In the regime of low Reynolds numbers (Re � 1), thecreeping flow induced by the volume force density f (r)follows from the Stokes equations and the condition forincompressibility:13, 36
η∇2v(r) − ∇p(r) + f (r) = 0, (7)
div v(r) = 0, (8)
where v(r) is the fluid velocity, p(r) is the pressure field, andη is the viscosity of the fluid. For a point force acting at po-sition r ′ on the fluid, f (r) = F0δ(r − r ′), the solution of thelinear Eqs. (7) and (8) is called Stokeslet,13
v(r) = T (r, r ′)F0, (9)
where T (r, r ′) is the Green tensor of the Stokes equations.Equation (9) gives the flow field at position r induced by apoint force at position r ′. For an unbounded fluid, the Green
tensor is called Oseen tensor,13
TO(r, r ′) = 1
8πη|r − r ′|(
1 + (r − r ′) ⊗ (r − r ′)|r − r ′|2
). (10)
Green’s tensor for a fluid confined between two planarwalls was first derived by Liron and Mochon in 1975.37 Inanalogy to electrostatics, they set up an infinite series of im-ages in order to obey the no-slip boundary condition at thetwo bounding walls . In our work we will use an integral rep-resentation of the two-wall Green tensor derived by Jones.38 Itis more symmetric and easier to handle than the original form.It splits up naturally into the Oseen tensor and a reflectionalpart T 2W (r, r ′) = TO(r − r ′) + TRef l(r, r ′). Figure 3 showsthe flow fields induced by a point force situated close to a walland acting either parallel or perpendicular to the wall. In oursimulations, we store the values of the two-wall Green tensoron a grid and interpolate linearly between the grid points forintermediate locations.
An approximate Green tensor to satisfy the no-slipboundary conditions at both walls can be constructed fromthe Blake tensor,39 which is the Green tensor in the presenceof a single bounding planar wall. For any location of the pointforce within the two-wall geometry, one takes the Blake ten-sor of the lower wall to describe the flow field below the cen-terline and the Blake tensor of the upper wall to describe thefield above the centerline. For point forces close to a wall,this approximation agrees well with the exact Green tensorin the region of the point force up to the centerline but de-viates strongly when one crosses the centerline. For a pointforce on the centerline, we compare in Fig. 4 the two-walltensor (on the left) to the approximate tensor (on the right).Clearly, there are quantitative differences between both flowfields. Since we wanted to have quantitatively correct results,we decided to work with the full two-wall Green tensor.
C. Brownian dynamics simulation
The beads in the model polymer also experience stochas-tic forces due to collisions with fluid molecules from the sur-rounding solvent. As usual, we take them into account as
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165101-4 S. Reddig and H. Stark J. Chem. Phys. 135, 165101 (2011)
FIG. 4. Field lines of the flow fields induced by a point force at the centerline; calculated by the two-wall Green tensor (a) and by the approximate Green tensor(b) based on the Blake tensor.35
Gaussian white noise that obeys the fluctuation-dissipationtheorem.40 Using a compact notation, the resulting Langevinequation in differential form reads as41, 42
dX =[
V + 1
kBTDF + div D
]dt +
√dt Hn , (11)
with D = kBT μ = 1
2H HT . (12)
Here, we have introduced 3N-dimensional vectors to collectthe positions of the beads, X = [r1, . . . , rN ], the forcesacting on them, F = [F1, . . . , FN ], and the velocitiesfrom the external flow field at the positions of the beads,V = [v(r1), . . . , v(rN )]. The mobilities μij form the mo-bility tensor μ and D stands for the generalized diffusiontensor, where kB is Boltzmann’s constant and T is the absolutetemperature. The tensor H is determined such that it fulfillsEq. (12). Finally, n denotes a random vector with expectationvalues for the mean and the second moment,
〈n〉 = 0 and 〈n ⊗ n〉 = 1. (13)
The first line in Eq. (11) describes the drift motion of thebeads, including the noise-induced spurious drift (divD),and the second line their Brownian motion.41 To numericallyintegrate Eqs. (11), we use a predictor-corrector schemesummarized in Ref. 42 to avoid the explicit calculation ofdivD. An Euler step without the spurious drift generates anintermediate configuration X∗(t + �t) = X(t) + �X∗ with
�X∗ =[
V + 1
kBTDF
]�t +
√�t Hn. (14)
Then the corrector step calculates the final configurationX(t + �t) = X(t) + �X with
�X = 1
2
[V + V ∗ + 1
kBT
(DF + D∗ F∗)] �t
+1
2[1 + D∗ D−1]
√�t Hn. (15)
Here the symbol ∗ means that these quantities are calculatedfor the intermediate configuration X∗.
D. Parameters
In the experiments in Ref. 29, α-actin filaments with acontour length of about L = 8 μm were analyzed in a mi-crochannel with a square cross section of width 2W = 10 μm.The actin filament is a thin semiflexible biopolymer with adiameter of 8nm and a persistence length of Lp = 13 μm.This gives the experimental value Lp/L = 1.6. The velocityof the imposed Poiseuille flow at the centerline varied fromv0 = 0 mm/s to v0 = 2.4 mm/s.
To be close to the experimental values, we choose the fol-lowing parameters for our simulations. We set the width of thechannel to the experimental value, 2W = 10 μm, and refer allour lengths to W. The polymer consists of N = 16 monomersand the equilibrium bond length to the nearest neighbor isl/W = 0.1 so that the total contour length amounts to L/(2W)= 0.75. The bead radius is set to a/W = 0.005 meaning l/a= 20 which allows to treat the beads as point particles whenhydrodynamic interactions are calculated. The spring constantis set to H = 1.2 × 104(kBT/W2) and we will explore how themodel polymer behaves when the persistence length variesfrom Lp/L = 1 to Lp/L = 16. We use the viscosity of water,ηH2O = 0.01 P, and perform the Brownian dynamics simula-tions at a temperature T = 300 K. The imposed flow velocityat the centerline varies from v0 = 0 mm/s to v0 = 2.5 mm/s.The configurational update of the polymer is calculated af-ter a time step of �t = 10−5 s. Since the evaluation of thetwo-wall Green tensor at each time step is very time con-suming, we stored its values on a grid with 200 × 200 gridpoints and with a mesh size of 0.01W. Tensor values at inter-mediate locations are determined from the grid points by lin-ear interpolation. The probability distribution and all averagespresented in Sec. III A–III B are calculated after the poly-mer reached its steady state for various independent initialconditions.
III. RESULTS
The imposed parabolic Poiseuille flow of Eq. (1) drivesthe filament out of equilibrium and determines its steady state,
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165101-5 Migration of a semiflexible polymer J. Chem. Phys. 135, 165101 (2011)
FIG. 5. Snapshots of the bead-spring chain for different persistence lengthsat the centerline (top) and near the wall (bottom) at v0 = 2.5 mm/s. (a) Lp/L= 1, (b) Lp/L = 4, and (c) Lp/L = 16.
which we quantify, e.g., by the center-of-mass probability dis-tribution n(zC). In this section we will investigate how flowstrength, hydrodynamic interactions, polymer stiffness, andthe ratio between bond length and bead radius, l/a, influ-ence the polymer conformation, the center-of-mass distribu-tion n(zC), and the orientational order of the filament’s end-to-end vector.
Figure 5 shows different snapshots of the filament nearthe centerline and near a bounding wall. For Lp/L = 16, avideo is provided in the supplemental material.43 The filamentconstantly moves up and down along the lateral direction.Most of the time, it is aligned parallel to the flow directionalthough it is not perfectly straight. At the centerline, wealso observe that the filament bends to a typical U shapethat traces the Poiseuille flow profile (see Fig. 5, top). Thisconfiguration is not stable when the filament moves awayfrom the centerline. Outside the center, the filament alsotumbles when the front end moves closer to the walls whereit experiences a smaller flow velocity (see Fig. 5, bottom).After one tumbling event, the filament aligns along the flowfield until thermal motion moves the front end again closer tothe wall. Increasing v0 also increases the local shear rates andthereby the tumbling frequency.
A. Center-of-mass probability distribution
One quantity to characterize the filament within the mi-crochannel is the steady-state probability distribution n(zC)for its center of mass across the channel. We always normalizeit to one,
∫ W
−Wn(z)dz = 1. The experiments of Ref. 29 clearly
revealed that the distribution depends on how strongly the fil-ament is driven out of equilibrium. In Fig. 6, we plot the dis-tribution for different flow strengths v0 from the centerline tothe bounding wall. Without flow the center of mass location zC
is equally distributed within the interval |zC|/W ≤ 1 − L/2W= 0.25. In thermal equilibrium, such a behavior is expectedfor the freely moving filament with arbitrary orientation. Forlocations zC closer to the walls, the orientation of the stiff fil-ament is more and more restricted due to the steric interac-tion with the wall and the distribution decreases to zero at the
FIG. 6. Center-of-mass probability distribution plotted from the centerline(zC = 0) to the wall (zC = W) for different flow velocities v0 and a persistencelength of LP/L = 1. The inset compares the simulation results for zero flowvelocity v0 to the hard-needle distribution of Eq. (16).
walls. For a rigid rod, one readily derives the distribution
n(zC) =
⎧⎪⎪⎨⎪⎪⎩
π
2(π − L/W ), |zC |/W ≤ 1 − L/2W
arcsin u
π − L/W, 1 − L/2W < |zC |/W ≤ 1
(16)with
u = 1 − |zC |/W
L/2W, (17)
which is compared to the simulated profile in the inset ofFig. 6. Deviations result from thermal fluctuations of the fil-ament. When we turn on the Poiseuille flow (see Fig. 6,v0 = 0.5 mm/s), the probability distribution close to thewall first increases. We understand this behavior. The tum-bling events illustrated in Fig. 5 (bottom) and initiated by thePoiseuille flow allow the filament to move closer to the wall.Increasing the flow velocity further, the filament is depletedmore and more at both walls. This is initiated by the hydrody-namic repulsion of the filament from the wall, which pushes itaway from the wall.18, 27, 28 The repulsion becomes clear withthe help of Fig. 3 on the left. Due to the strong shear flowclose to the wall, the filament is under tension which initi-ates a flow field that drives the model polymer away fromthe wall.18, 27 In particular, at both ends of the rod tensionalforces point into the rod. Each of these forces initiates a flowfield similar to the one of Fig. 3(a) which drives the other endof the rod away from the wall. This phenomenon is differentfrom the orientational lift forces reported in Ref. 28. The mostpronounced behavior of the system is the depletion of the dis-tribution at the centerline. It leads to a bimodal probabilitydensity with a maximum at a finite distance from the center.The maximum increases with the flow strength. At the max-imum, the migration towards the centerline induced by thehydrodynamic filament-wall interaction has to be cancelledby a probability current away from the centerline. Based onliterature,18, 23, 27 Refs. 29–31 attribute such a current to thespatially varying diffusivity of the filament across the chan-nel: the U-shaped conformation at the centerline has a largerdiffusivity than the straight filament outside the center. InSec. IV, we will demonstrate that the current away from the
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165101-6 S. Reddig and H. Stark J. Chem. Phys. 135, 165101 (2011)
FIG. 7. Center-of-mass probability distributions for different persistencelengths LP/L at a fixed center flow velocity v0 = 2.5 mm/s.
center is mainly due to deterministic drift motion which be-comes dominant over the diffusional current when the rigidityof the filament increases.
Figure 7 shows the center-of-mass distributions at a fixedflow velocity v0 = 2.5 mm/s for different persistence lengthsLP. Here, the depletion layer at the walls decreases slightlywith increasing bending rigidity, whereas the bimodal distri-bution becomes more pronounced. For larger rigidity, the mi-gration away from the centerline increases. As a result, thedepletion at zC = 0 becomes stronger and the position ofthe maximum is shifted towards the walls which causes thedecreasing depletion layer. We will explain this behavior inSec. IV. For LP/L = 16, the local minimum and the maximumof the distribution differ by a factor of two, which is compa-rable to the results of Ref. 29.
Most of our simulations are done for a bead distance toradius of l/a = 20. This might mimic the thin actin filament.However, it also underestimates the overall friction betweenthe filament and the solvent. In Fig. 8, the center-of-mass dis-tributions for different bead sizes a at a fixed flow velocity v0
= 2.5 mm/s are shown. Increasing the overall friction of themodel polymer with a enhances tensional forces within thepolymer and thereby the drift currents, discussed in Sec. IV.
FIG. 8. Center-of-mass probability density for different ratios l/a at a fixedpersistence length LP/L = 2 and a fixed flow velocity v0 = 2.5 mm/s. Inorder to change the ratio l/a, we keep the length L and the number of beadsN constant and vary the radius a of the beads. This corresponds to increasingthe thickness of our model polymer.
FIG. 9. Center-of-mass probability distribution simulated without hydro-dynamic interactions for different flow velocities v0 and persistence lengthLP/L = 1.
As a result, the bimodal distribution and the depletion at thewalls become more pronounced.
Hydrodynamic interactions between different parts of themodel polymer and with the wall are crucial for the observedbimodal distribution and the depletion at the walls. In Fig. 9,we show the resulting center-of-mass distributions at differentflow strengths when hydrodynamic interactions are switchedoff during the simulations. The equilibrium profile at zeroflow field is the same as in Fig. 6 since it should not depend ondynamic properties such as hydrodynamic interactions. Fornonzero flow velocity, the minimum in the center vanishescompared to Fig. 6 whereas the depletion close to the wall isless pronounced. It even decreases with increasing v0 sincethe hydrodynamic repulsion from the wall is missing and thetumbling polymer is stronger confined due to larger viscousshear stresses.29, 44
B. Polymer orientation within the channel
To study how the Poiseuille flow influences the filamentshape and orientation within the channel, we calculate theend-to-end vector R and determine its orientational orderaround the channel axis by the order parameter S,
S(zC) = 〈cos(2θ )〉. (18)
Here θ is the angle between R and the y axis and S describesthe orientational order in a two-dimensional system.45 A poly-mer with randomly oriented end-to-end vector gives S = 0,whereas S = 1 means that the polymer is perfectly alignedalong the flow direction and S = −1 indicates perfect orienta-tion perpendicular to the flow direction.
Figures 10(a) and 10(b) show, respectively, the orienta-tional order parameter and the average end-to-end distance〈R〉 for different flow strengths. Without flow, S is nearly zerowithin the region |zC|/W ≤ 0.25, where the filament can freelyrotate and where it assumes all orientations with equal prob-ability. Outside this region, the filament’s orientation is moreand more restricted due to the steric polymer-wall interactionand S grows monotonically until it reaches S = 1 at the walls.For a rigid rod, one readily derives the order parameter as a
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165101-7 Migration of a semiflexible polymer J. Chem. Phys. 135, 165101 (2011)
FIG. 10. Order parameter S and average end-to-end distance 〈R〉 plotted versus the lateral center-of-mass position zC/W: (a) and (b) for a fixed stiffness Lp/L =1 and different flow velocities; (c) and (d) for a fixed flow velocity v0 = 2.5 mm/s and different stiffnesses Lp/L. Inset in (a): For v0 = 0 mm/s, the simulatedorder parameter S is compared to the analytic result of Eq. (19).
function of |zC|/W,
S(zC) =
⎧⎪⎪⎨⎪⎪⎩
1, |zC |/W ≤ 1 − L/2W
u√
1 − u2
arcsinu, 1 − L/2W < |zC |/W ≤ 1,
(19)where u is given in Eq. (17). The inset of Fig. 10(a) showsvery good agreement with our simulations. Deviations resultfrom thermal fluctuations of the filament. They also explainthat the end-to-end distance in Fig. 10(b) deviates slightlyfrom one. In the presence of external flow both the orderparameter and the end-to-end distance have a minimum inthe center of the channel which we attribute to the U-shapedconformations illustrated at the bottom of Fig. 5. S and 〈R〉then increase monotonically until a nearly constant value orplateau is reached around |z|/W = 0.2. Here the filament isaligned along the flow (S > 0.7) and tumbling events causethe end-to-end distance to be smaller than for zero flow field.The end-to-end distance shows a weak minimum near thewall which we attribute to an increasing number of tumblingevents since the shear stress on the filament within the channelbecomes largest. Finally, very close to the wall, when tum-bling events can no longer occur, the filament is perfectlyaligned with the flow. The order parameter S reaches its max-imum value one and the end-to-end distance even shows val-ues above L for large flow strengths. Interestingly, the plateauvalue of S increases with the flow strength while 〈R〉 de-
creases. Higher shear rates align the filament better alongthe flow direction but they also initiate more tumbling eventswhich reduce 〈R〉 but not S. We repeated our simulations with-out hydrodynamic interactions acting across the filament andfrom the wall. The results agree well with the behavior dis-cussed in Figs. 10(a) and 10(b). The same behavior occurswhen we reduce the ratio l/a of bond length to bead radiusfrom the value l/a = 20 used in most of our simulations tol/a = 5, where hydrodynamic interactions between the beadsare stronger. This indicates that overall shape and orientationof the filament is determined by its overall friction with thesolvent and, in particular, by the applied Poiseuille flow.
For stiffer filaments, the end-to-end distance 〈R〉 in-creases as indicated in Fig. 10(d). The plateau value of theorder parameter in Fig. 10(c) does not change strongly whenthe stiffness increases. Interestingly, the minimum value forS at the centerline first decreases to even negative values andthen increases again. The negative values are caused by theU-shaped conformation where the end-to-end vector is per-pendicular to the centerline. A persistence length LP/L be-tween 4 and 8 seems to be the best choice for realizing theU-shaped conformation.
IV. KINETIC THEORY FOR A SEMIFLEXIBLEPOLYMER
In this section we derive and analyze a Smoluchowskiequation for the center-of-mass probability distribution of the
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165101-8 S. Reddig and H. Stark J. Chem. Phys. 135, 165101 (2011)
bead-spring chain with bending elasticity. We thereby gen-eralize the approach of Ma and Graham who formulated akinetic theory for a bead-spring dumbbell.27 We derive theSmoluchowski equation in Sec. IV A and study and discussin detail the different contributions to the lateral center-of-mass current in Sec. IV B in order to identify the cause forcross-streamline migration.
A. Smoluchowski equation for center-of-mass current
The probability distribution ψ(r1, . . . , rN ; t) for findingthe bead-spring chain in a state determined by the bead coor-dinates r1, . . . , rN at time t is governed by the Smoluchowskiequation,
∂
∂tψ = −∇i · j i , (20)
where we have introduced the probability density current ofbead i
j i = r iψ − Dij∇jψ. (21)
Here,
r i = v0(r i) + μij Fj (22)
denotes the deterministic drift velocity and Dij = kBT μij isthe diffusion tensor connected to the mobility tensor by theEinstein relation. Note that we use the convention where wesum over bead indices that occur twice in an expression. Sincewe are interested in the center-of-mass probability distribu-tion, we introduce the respective center-of-mass position andbond vectors,
rC = 1
N
N∑i=1
r i , (23)
qn = rn+1 − rn, n = 1, . . . , N − 1. (24)
Using the new coordinates and,
∇i = 1
N∇rC
+ ∇qi−1 − ∇qi, (25)
we rewrite the Smoluchowski equation (20) as
∂ψ
∂t= −∇rC
· (rCψ − DrC
∇rCψ − DT
qi∇qi
ψ)
−∇qi· (qiψ − Dqi
∇rCψ − Dij∇qj
ψ). (26)
We have introduced the respective deterministic velocities forcenter of mass and bond i,
The average over all diffusion tensors Dij is the Kirkwooddiffusivity DrC
, whereas Dqiand Dij are diffusivities related
to the bond vectors. The superscript T in Eq. (26) denotes thetransposed tensor. We now write the full probability distribu-tion ψ in the new coordinates rC, q1, . . . , qN−1 and introducethe center-of-mass probability distribution
n(rC, t) =∫
. . .
∫ψ(rC, q1, . . . , qN−1; t)dq1 . . . dqN−1.
(32)Integrating Eq. (26) over all bond vectors qi , one is able toderive a Smoluchowski equation for the center of mass,
∂n
∂t= −∇rC
· jC, (33)
with the center-of-mass probability current,
jC = [〈rC〉q − ∇rC〈DrC
〉q + 〈∇rCDrC
〉q+ ⟨∇qi
DTqi
⟩q
]n − 〈DrC
〉q∇rCn. (34)
The brackets 〈. . . 〉q denote an ensemble average over all bondvectors qi or polymer conformations with fixed rC ,
Note that the second and third terms on the right-hand side ofEq. (34) are not necessarily the same. In integrating Eq. (26)over all bond vectors qi , we use the reasonable assumptionthat all surface terms vanish. We, therefore, can immediatelyskip the terms in the second line of Eq. (26). In the remain-ing terms, ψ is replaced by nψ from Eq. (36) and partial in-tegrations are performed so that no gradient acts on ψ andEq. (35) can be applied. This procedure finally gives thecenter-of-mass probability current, Eq. (34).
Equations (33) and (34) determine the center-of-massprobability distribution. Whereas the last term on the right-hand side of Eq. (34) describes conventional diffusion,27 theremaining terms proportional to the distribution n formally aredrift terms. However, only the first term is due to determinis-tic motion of the center of mass. The second to fourth termsresult from the diffusional currents in Eq. (21). As we demon-strate explicitely in the following section, all of these terms,in principle, can lead to cross streamline migration. Here weadd some general remarks.
The first term on the right-hand side of Eq. (34) de-scribes migration due to hydrodynamic interactions andthe applied external flow. Without hydrodynamic interac-tions the cross mobilities μij vanish and Eq. (27) reducesto rC = ∑
i[v0(r i)]/N . The frictional term∑
i μii Fi
= (∑
i Fi)/(6πηa) = 0 does not contribute to the center-of-mass motion since the total force acting on the filament hasto be zero. As a result, without hydrodynamic interactions
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165101-9 Migration of a semiflexible polymer J. Chem. Phys. 135, 165101 (2011)
cross-streamline migration cannot occur. The second termon the right-hand side of Eq. (34) is the divergence of theKirkwood diffusivity and a natural candidate for migrationaway from the centerline as demonstrated in Refs. 18 and27. Finally, the third and fourth term vanishes whenever thedivergence of the diffusion tensors or mobilities is zero. Inparticular, this is the case in an unbounded fluid when onetreats hydrodynamic interactions on the level of the Oseentensor or in the next higher order via the Rotne-Prager tensor.In our case, the third and fourth term will be non-zero due towall-induced hydrodynamic interactions.
B. Analysis of the lateral center-of-mass current
Due to translational symmetry along the channel direc-tion, the center-of-mass distribution does not vary along the yaxis and the z-component of the center-of-mass current, i.e.,the current across the channel is
jC,z = [〈zC〉q − ∂zC〈DrC,zz〉q + 〈∂zC
DrC,zz〉q+ ⟨
∂qi ,yDTqi ,zy
+ ∂qi ,zDTqi ,zz
⟩q
]n
−〈DrC,zz〉q∂zCn. (37)
In steady state the center-of-mass current jC is constant.Since the current at the walls has to vanish, jC, z is zero ev-erywhere across the channel.
In Fig. 11, we plot all contributions of the center-of-masscurrent jC, z proportional to n for different bending rigiditiesof the filament. The sum of these currents balances thediffusional current −〈DrC,zz〉q∂zC
n. Concentrating on LP/L= 1, we recognize that close to the wall the deterministicdrift current 〈zC〉qn is directed away from the wall, e.g., it ispositive at zc = −W, and it dominates all the other currents.Hence, the hydrodynamic repulsion from the walls is re-sponsible for the depletion at the walls. On the other hand,close to the centerline, the current is directed towards thewall and, therefore, causes depletion at the centerline.For LP/L = 1, the diffusional current −∂zC
〈DrC,zz〉qndue to the gradient of the conformation-averagedKirkwood diffusivity also points away from the center-line and contributes to the observed depletion. We understandthis since the U-shaped conformation close to the center-line has a larger diffusivity than the straight conformationoccurring outside the centerline.18, 27 However, alreadyat a stiffness of LP/L = 2, the deterministic drift cur-rent is clearly the dominant part for causing centerlinedepletion and at LP/L = 16 all the diffusional currents arenegligible. The experiments of Ref. 29 were performed forLp/L = 1.6. So we conclude that the observed centerlinedepletion is mainly due to the deterministic drift current. Thethird term 〈∂zC
DrC,zz〉qn is approximately zero for each Lp/Land the fourth term 〈∂qi ,yD
Tqi ,zy
+ ∂qi ,zDTqi ,zz
〉qn leads to acurrent towards the centerline due to hydrodynamic repulsionfrom the wall.
We summarize our results here. Whereas the filament atthe centerline displays more compact conformations like theU shape, close to the walls it is mainly aligned along the flowlines due to the large shear rate. Therefore, the friction coeffi-
cient for motion across the channel increases from the center-line to the wall and its inverse, the Kirkwood diffusivity, de-creases monotonically from the centerline towards both walls.This initiates migration away from the centerline,18, 27, 28
which can cause centerline depletion,18, 20, 22, 27 and was there-fore used to interpret the recent experiments of Steinhauseret al.29, 30 Here we demonstrate that for increasing bend-ing rigidity, centerline depletion is mainly caused by the de-terministic drift current. In the U-shaped conformation ofFig. 12(a) the filament experiences bending forces. When itrelaxes, different parts of the filament interact hydrodynami-cally via flow fields initiated along the filament. These hydro-dynamic interactions cause the center of mass to move awayfrom the centerline as illustrated in the second video of thesupplemental material.43 Increasing the rigidity of the fila-ment also increases the bending forces which explains whythe deterministic drift currents in Fig. 11 close to the center-line increase with LP/L.
The shear-induced hydrodynamic repulsion of a dumbellfrom a wall has been treated in Refs. 18, 27, and 28. Simi-larly, close to the wall the shear flow stretches the filament andcreates tensional forces which initiate flow fields as sketchedin Fig. 3 on the left. They drive the filament away from thewall [see Fig. 12(b) and the third video of the supplementalmaterial43]. These tensional forces do not depend on the fila-ment’s bending rigidity. Therefore, in Fig. 11 the determinis-tic drift currents close to the wall do not change with LP/L.
V. SUMMARY AND CONCLUSION
In this article we treated a paradigmatic model for study-ing how the properties of a system change when it is drivenout of equilibrium. Motivated by the recent experiments,29 weanalyzed a single semiflexible polymer confined between twoplanar walls and under the influence of an imposed Poiseuilleflow with special emphasis on the observed cross-streamlinemigration. We performed Brownian dynamics simulations fora bead-spring chain with bending rigidity and used the two-wall Green tensor of the Stokes equations to take into ac-count the hydrodynamic interactions along the polymer andwith the wall. We carefully analyzed how polymer conforma-tions, center-of-mass distribution, and orientational order ofthe end-to-end vector within the channel depend on parame-ters such as the flow strength and the stiffness of the modelpolymer. Analytic expressions for hard needles at zero flowreproduce the simulation results and demonstrate how the be-havior of the model polymer changes when the Poiseuille flowis turned on. Our results are in agreement with experiments29
and simulations30, 31 that employ a different, particle basedmethod to simulate the viscous environment called multi-particle collision dynamics.
In particular, we observed the characteristic bimodalprobability distribution for the polymer’s center of massand showed that hydrodynamic interactions along the modelpolymer and with the wall are essential for this distribu-tion to occur, whereas shape and orientation of the fila-ment are mainly determined by the applied Poiseuille flow.Based on a Smoluchowski equation for the center-of-massprobability distribution, we investigated cross-streamline
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165101-10 S. Reddig and H. Stark J. Chem. Phys. 135, 165101 (2011)
FIG. 11. Plots of the different contributions of the lateral center-of-mass current across the channel at flow velocity v0 = 2.5 mm/s for different bendingrigidities: (a) LP/L = 1, (b) LP/L = 2, (c) LP/L = 4, and (d) LP/L = 16.
migration and the origin of the bimodal distribution in detail.Whereas the migration away from the wall is due to hydrody-namic interactions with the bounding walls, in agreement withRefs. 29–31 and work on flexible polymers,16–22, 24, 25 weclearly identified a deterministic drift current as the majorcause for migration away from the centerline, especially whenthe bending stiffness of the model polymer increases. The cur-
rent is set up when bent conformations of the polymer relaxtowards the straight filament. Diffusional currents due to aposition-dependent diffusivity become completely irrelevantwith increasing rigidity of the polymers. This demonstratesthat bending rigidity leads to a clear difference in the behaviorof flexible and semiflexible polymers in Poiseuille flow and inthe explanation of the observed cross-streamline migration.
FIG. 12. (a) At the centerline, the relaxing U-shaped filament initiates flow fields relative to the applied Poiseuille flow. The resulting hydrodynamic interactionsdrive the filament away from the centerline (see also the second video of the supplemental material43). The strength of the flow is given by the color code inarbitrary units. (b) Close to the wall the filament is under tension. This initiates flow fields illustrated in Fig. 3 on the left that drive the filament away from thewall (see also the third video of the supplemental material43).
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165101-11 Migration of a semiflexible polymer J. Chem. Phys. 135, 165101 (2011)
ACKNOWLEDGMENTS
We would like to thank T. Pfohl, R. Winkler, and G.Gompper for helpful discussions. This work was supportedby the Deutsche Forschungsgemeinschaft (DFG) through theresearch training group GRK 1558.
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video shows the lateral motion of the filament within the microchannel ina frame moving with the center of mass with flow velocity in the center isv0 = 2.5 mm/s and Lp/L = 16; the video shows the deterministic drift awayfrom the centerline by performing a deterministic simulation starting fromthe U-shaped conformation; and the video shows the deterministic drift to-wards the centerline by performing a deterministic simulation starting fromthe straight filament under tension.
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