Wall attraction and repulsion of hydrodynamically interacting particles
Steffen Schreiber, Jochen Bammert, Philippe Peyla, Walter Zimmermann
aa r X i v : . [ c ond - m a t . s o f t ] S e p Wall attraction and repulsion of hydrodynamically interacting particles
Steffen Schreiber , Jochen Bammert , Philippe Peyla and Walter Zimmermann Theoretische Physik I, Universit¨at Bayreuth, 95440 Bayreuth, Germany and Laboratoire Interdisciplinaire de Physique, UMR,Universit´e Joseph-Fourier, 38402 Saint Martin d’Heres, France (Dated: October 25, 2018)We investigate hydrodynamic interaction effects between colloidal particles in the vicinity of awall in the low Reynolds-number limit. Hydrodynamically interacting pairs of beads being draggedby a force parallel to a wall, as for instance during sedimentation, are repelled by the boundary.If a pair of beads is trapped by harmonic potentials parallel to the external flow and at the samedistance to a wall, then the particle upstream is repelled from the boundary while its neighbordownstream is attracted. The free end of a semiflexible bead-spring polymer-model, which is fixedat one end in a flow near a wall, is bent towards the wall by the same reason. The results obtainedfor point-like particles are exemplarily confirmed by fluid particle dynamics simulations of beads offinite radii, where the shear induced particle rotations either weaken or enhance the effects obtainedfor point-like particles.
I. INTRODUCTION
The flow properties of suspensions depend very muchon the interaction between particles via the fluid, theso-called hydrodynamic interaction (HI) [1–4]. In mi-crofludics, where the distance between particles and wallsbecomes often small, the dynamics of particles can bestrongly influenced by the wall-induced hydrodynamic in-teraction effects between rigid as well as for soft particles,such as vesicles or polymers. Accordingly, particles mayexperience displacements across the unperturbed stream-lines of an external flow and therefore, may lead to par-ticle redistributions across the pipe diameter, if HI andinertia effects are taken into account. Thus, studies ofsuspensions are essential both from the fundamental andfrom the practical point of view.In the case of small but finite values of the Reynoldsnumber, Segr´e and Silberberg discovered the effect ofcross-stream migration of particles to specific positionsaway from the centerline of a tube flow [5–10]. This parti-cle focusing is understood to arise from the force balancebetween a wall effect that drives the particles to the cen-ter of a channel and a shear-gradient-induced migrationpushing the particles towards the boundary.In the over-damped Stokes limit in fluid dynamics theinterplay between the HI and the deformability of softparticles like tank-treading vesicles in shear flows leadsto a lift force close to boundaries [11–13]. For single softparticles such as oil drops or vesicles, their deformabilitycombined with the shear gradient in Poiseuille flow causescross-stream migration [7, 14], even in the absence ofwall effects. In these cases the interesting question arises,whether and to which extend particle-wall HI affects thecross-streamline migration too [15]. Wall-induced cross-stream migration may occur also for polymers in suspen-sion in a pipe flow, which is a long studied and impor-tant problem with a number of recent insights [16–20], orduring sedimentation [21] with particle depletions closeto a boundary [22] and during active motions of micro-swimmers in confinement [23]. Moreover, it has been found experimentally and theoretically that swimmingmicroorganisms may be attracted by solid walls [24].Wall-grafted polymer-brushes, which are also usedfor tuning surface properties in microchannels, are an-other example [25–27], where the particle-wall interac-tions play an important role for the dynamics of poly-mers and which differs significantly from that in thebulk. For instance, the cyclic motion of polymers teth-ered at a surface depends crucially on the interplay be-tween the polymer-wall interaction and the shear flow[28–34]. Here, the HI plays a major role similar as forthe related oscillatory motion of three trapped and hy-drodynamically interacting particles in shear flow [35].A related problem is the dynamics of cellulose fiber sus-pensions close to a wall, which is important for paper-manufacturing and therefore intensively investigated inorder to better control the fiber orientation [36].The examples mentioned so far focus mainly on thebehavior of single rigid or soft particles (with more dy-namical degrees of freedom) in fluid flow. For severaldisconnected particles there are a number of other inter-esting hydrodynamically induced interaction effects evenin the bulk and in the absence of fluctuations. Besidesthe oscillatory dynamics of three sedimenting free par-ticles [37] and of three trapped particles in shear flow[35], one finds also HI induced attraction or repulsion be-tween asymmetric rotors [38], and an attraction betweentethered polymers in plug flow [39]. For a diluted sus-pension of Brownian particles in shear flow an enhancedself-diffusion in shear flows is reported [40], which is ex-plained by a wall-induced migration of free particles [41].In this work we focus on boundary induced hydrody-namic interaction effects between particles fixed closedto a wall in Stokes flow or dragged particles in a qui-escent fluid as described by the well known techniqueintroduced by Blake for point particles [42] (whereof ex-tensions may be found for instance in Refs. [17, 43, 44]).The reminder of the article is organized as follows: Therelated basic equations of motion for point-like particlesare presented in Sec. II, including a summary of Blake’sresults. The results of our numerical and analytical inves-tigations on the particle-wall HI are presented in Sec. III,where as a basis of our analysis the flow lines arounda single particle close to a wall are given in Sec. III A.Two or more sedimenting particles near a boundary areconsidered in Sec. III B, where a cross-streamline migra-tion away from the wall can be qualitatively explainedin terms of the flow lines around a single particle. Thecase of two trapped point like particles exposed to a flowis described in Sec. III C. The results of the two trappedparticles are qualitatively confirmed by fluid particle dy-namics simulations [45, 46] for particles with finite radiiand they illustrate the major effects that are relevantfor the applications to semiflexible bead-spring modelsin Sec. IV. In Sec. IV A a semiflexible bead-spring chaintethered close to a wall and exposed to a flow is treatedand in Sec. IV B semiflexible polymers perpendicularlyanchored at a wall in shear flow. The article closes inSec. V with a discussion of the results and suggestions ofpossible experiments.
II. MODEL EQUATIONS
We consider the dynamics of colloidal particles in thelimit of a vanishing Reynolds number, where the laminarflow is described by the Stokes equation for an incom-pressible Newtonian fluid. If not stated otherwise we as-sume point-like particles with an effective hydrodynamicradius a .The dynamics of the N beads at the positions r i ( i =1 , . . . , N ) is governed by N coupled equations˙ r i = H ij F j + u ( r i ) , (1)where u ( r i ) is an externally applied flow as for examplea linear shear flow u ( r ) = u ( z )ˆ x = ˙ γz ˆ x , (2)with the shear rate ˙ γ . The force F j is the sum over allpotential forces acting on the j -th bead. Depending onthe specific system, these may include stretch or bendingforces as well as forces due to trap potentials, which canbe derived from a potential V according to F j = −∇ j V, (3)where ∇ j denotes the gradient with respect to r j . Therelevant expressions for V are specified in Sec. III andSec. IV. H ij refers to the mobility matrix describing the HI be-tween the beads i and j . In the presence of a plane wallwith a no-slip boundary condition for the fluid at thewall, H ij is given by the Blake tensor [42], H ij ( r i , r j ) = S H ij ( r i , r j ) − S H ij ( r i , r ′ j )+ D H ij ( r i , r ′ j ) − SD H ij ( r i , r ′ j ) , (4)where r ′ j = ( x j , y j , − z j ) is the position of the mirrorimage of bead j at the opposite side of the boundary. The first contribution to H ij accounts for the HI in theunbounded domain described by the Oseen tensor [3], S H αβij ( r i , r j ) = πηr ij (cid:18) δ αβ + r αij r βij r ij (cid:19) for i = j , πηa δ αβ for i = j , (5)and the second one for the HI between the beads andtheir mirror images, S H αβij ( r i , r ′ j ) = 18 πη ˜ r ij δ αβ + ˜ r αij ˜ r βij ˜ r ij ! , (6)where η is the viscosity of the fluid. We furthermore usethe abbreviations r ij = r i − r j = r ij ˆ r ij , (7a)˜ r ij = r i − r ′ j = ˜ r ij ˆ˜ r ij (7b)and the components of the vector r ij (˜ r ij ) are denoted by r αij (˜ r αij ), where α = x, y, z . In Eq. (4) the contribution D H αβij ( r i , r ′ j ) = 14 πη ˜ r ij z j (1 − δ βz ) δ αβ − r αij ˜ r βij ˜ r ij ! (8)is the Stokes doublet ( D ) and SD H αβij ( r i , r ′ j ) = 14 πη ˜ r ij z j (1 − δ βz ) δ αβ ˜ r zij − δ αz ˜ r βij + δ βz ˜ r αij − r αij ˜ r βij ˜ r zij ˜ r ij ! (9)is the source doublet ( SD ). In our numerical calculationshigher order corrections to H ij due to the finite size ofthe spheres are included up to the order a (Rotne-Pragerapproximation). The final equations of motion are givenby ˙ r i = ˜ H ij F j + u ( r i ) , (10)where the mobility matrices˜ H ij = (cid:18) a ∇ i + a ∇ j (cid:19) H ij (11)fulfill the relation H ij = H Tji , which is important for theoverall symmetry of the problem - as pointed out inRef. [44].
III. RESULTS FOR BASIC MODELS
In this section we determine the influence of wall-induced hydrodynamic interactions on the dynamics oftwo and three beads which are either dragged in a fluidparallel to a boundary or hold in potentials near a walland exposed to flows. To this end Eq. (10) with Eq. (11)is solved numerically for the viscosity η = 1 and approx-imate analytical solutions are given in some cases. A. A single trapped bead in shear flow close to awall
The influence of a wall on the HI between two beadscan be illustrated by the streamlines around one point-like particle with an effective hydrodynamic radius a ata distance d w from a solid boundary. The particle isexposed to the flow given by Eq. (2) and is trapped by aharmonic potential V = k pot r b − r pot ) , (12)where r b is the position vector of the particle, r pot theposition of the potential minimum, with a spring constantof k pot = 1. Equivalently, one could consider a point-likeparticle dragged by a constant external force F = f ˆ x parallel to the wall in a quiescent fluid. The streamlinesaround the point-like particle are the same in both cases ifa comoving frame is chosen, where the particle’s positionis held fixed by the potential at r pot = (0 , , d w ).The trap force F s = − k pot ( r b − r pot ) as a functionof the unperturbed flow velocity u ( r ) is obtained bysolving Eq. (10) for a vanishing bead velocity ˙ r b = 0:0 = ˜ H ( r b , r b ) F s + u ( r b ) . (13)We find for the Oseen approximation F sx = f x d w d w − a and F sy = F sz = 0 , (14)with f x = − πηau ( d w ) the force required to keep theparticle fixed in the absence of a wall, if it is exposed to aflow with the velocity u ( d w ). When u ( d w ) is constantfor varying d w , e.g. , by adjusting the shear rate appro-priately, the force F sx exerted on the bead increases withdecreasing d w . This effect will be investigated further insubsection III C.The vanishing force F sz = 0 perpendicular to the wallin Eq. (14) reflects the time reversibility of the Stokesequation [1]. If a single bead in shear flow would mi-grate perpendicularly to the wall, i.e. F sz = 0, then forsymmetry reasons the drift would point in the same di-rection after reversing the flow. But then the motionwould not be reciprocal and the time-reversibility of theStokes equation would be violated. Therefore, F sz = 0is forbidden and there is no migration of a single beadperpendicular to the wall.The flow field u ( r ) around a bead fixed at r b =(0 , , d w ) shown in Fig. 1 is given by u ( r ) = (cid:18) a (cid:19) H ( r , r b ) F s + u ( r ) . (15)In the absence of a boundary, where only the first con-tribution in Eq. (4) has to be taken into account, thestreamlines are up/down and left/right symmetric withrespect to the center of the bead. In the presence of a FIG. 1. In the xz plane the streamlines around the bead fixedat r b = (0 , , d w = 3) are in the presence of a plane no-slipboundary at z = 0 asymmetric with respect to the axis at z = d w . One obtains the same asymmetry of the flow linesin the co-moving frame of a point particle dragged parallel tothe boundary along the dotted line at z = d w . The flow lineswere calculated via Eq. (15), whereby the red line marks thetrajectory of a test particle starting and terminating belowthe center of the bead but passing the bead on top. wall this up/down symmetry is broken and the stream-lines are deformed as indicated in Fig. 1. This effect iscaused by the HI between the fixed bead and the bound-ary, which is described by the second, third, and fourthcontribution to the mobility matrix in Eq. (4).If one introduces a small tracer particle, starting at x < z < d w , it follows one of the dis-played streamlines as indicated by the red streamline inFig. 1 and may pass the bead at the side opposite fromthe wall, i.e. z > d w . According to the ± x symmetrythe streamline reaches the range z < d w again for large x . This is a consequence of the time-reversibility of theStokes equation, which is also valid in the presence ofa solid boundary. The wall-induced deformation of thestreamlines has interesting consequences as discussed inthe following sections. B. Particles dragged parallel to a boundary - amodel for sedimentation
Here we investigate the motion of several particleswhich are dragged by a constant external force F par-allel to a wall, as for example by the gravitational forceduring the sedimentation of particles.We first consider the motion of two beads dragged in anunbounded fluid. If the force F acts parallel or perpen-dicular to the connection vector between both particles, r = r − r , cf. Fig. 2(a) and (b), they move in eithercase parallel to the force, as indicated in Fig. 2 by parta) and b). If r encloses an angle Θ = 0 and Θ = π/ F , the particles move obliquely with re- −0.5 −0.25 0 0.25 0.5−0.0100.010.02 Θ / πφ / π FIG. 2. Two particles are dragged through a fluid by a force F anti-parallel to the vertical z axis. The beads move parallelto F , if the connection vector r encloses with F an angleΘ = π/
2, as in part (a), or Θ = 0, as in part (b). Forother values of Θ the HI effects cause an angle φ between thedirection of the particle velocity v and F , as indicated in (c)[48]. An approximation of the dependence of φ on Θ is givenby Eq. (16) and plotted in (d). spect to the drag force due to the HI between the beadsas indicated in Fig. 2(c) (see also [48]). The deflectionangle φ between the direction of the particle motion andthe force has its maximum value at Θ = ± π/ φ = arctan sin Θ cos Θ1 + d a + cos Θ ! , (16)where d = | r | is the distance between the particles.In the presence of a boundary the particle-particlehydrodynamic interaction via the boundary comes intoplay. As already indicated by the flow lines around a sin-gle particle in Fig. 1, a wall breaks the symmetry aroundthe particle with respect to the z direction. For two parti-cles, which are initially located at the same distance froma wall and pulled parallel to it, as depicted in Fig. 3(a),we show in Fig. 4 the trajectories of point particle’s de-termined numerically from Eq. (10). The open and thefilled circles in Fig. 4 represent the particle positions atequal times.The bead in front is repelled by the wall immediatelyfrom the start, while the particle behind is at first at-tracted to the wall, as indicated in Fig. 3a). Later onthe rear bead moves away from the wall as well. In orderto understand this effect it is useful to consider the earlyand the later stage of the motion shown in Fig. 4 for eachparticle. For the early regime the flow-lines around a sin-gle bead, as shown in Fig. 1, allow an estimate about the FIG. 3. Each of the two beads in part a) is dragged by a force F parallel to a wall. The particle in front is repelled from thewall, while the bead behind is attracted, as expected by theform of the streamlines around a single bead in Fig. 1. Inpart b) the time has been reversed and thus the direction ofthe applied force and the particle motion are reversed too. za x/a FIG. 4. The trajectories of two beads in the xz plane areshown, which are dragged through the fluid by an externalforce F k ˆ x parallel to the boundary at z = 0. The initialpositions of both beads are located at r = (0 , , d w ) and r = ( d b , , d w ) with d b = 3 a and d w = 4 . a . The particlepositions are indicated by open and filled circles along thetrajectories at equal times. The bead behind (dashed line) isat first attracted towards the wall due to the wall-mediatedHI, whereas the particle in front (solid line) is always repelled.Later on the bulk HI dominates and both particles move awayfrom the wall, as expected according to Eq. (16). motion of the two particles along the z direction. In thefront of a pulled particle the velocity of the fluid has acomponent in the positive z direction and in the negative z direction behind it. Accordingly, in the case of a pairof dragged beads the particle in front is repelled fromthe wall whereas the one behind is attracted towards theboundary as sketched in Fig. 3(a). During this processof wall-particle repulsion and attraction the connectionvector r becomes skewly oriented to the drag force, i.e. Θ = 0 as indicated in Fig. 2c), with the particle closerto the wall behind the other.If the connection vector is skewly oriented, the bulkeffect, as described above, comes into play in the secondstage of the motion. It causes in the case, when the par-ticle closer to the wall moves behind the other one, a driftof both particles perpendicular to the external force andaway from the wall. The bead closer to the wall movesslower than the other one because its effective friction isenhanced closer to the wall (cf. Eq. (14)) and thereforethe bead distance d increases and Θ decreases. Conse-quently | φ | decreases in agreement with Eq.(16). In thelong-time limit the particles’ trajectories align with theapplied forces, in which case the wall distance of bothparticles practically saturate. Therefore, the essentialeffect of the hydrodynamic particle-wall-particle interac-tion on the motion of the beads shown in Fig. 4 is thereorientation of their connection vector r . When r is oblique to the drag force, the bulk effect provides themajor contribution to the migration of the particles awayfrom the wall.Near a point-like particle at r = (0 , , d w ), which isdragged parallel to a boundary, the induced velocityof the fluid around it can be determined analyticallyby taking the wall effects into account in the Oseen-approximation. In this case the non-zero z -component ofthe velocity of the fluid at the positions r ± = ( ± d b , , d w )is as follows: v z ± = ± fπη d b d w (4 d w + d b ) / . (17)If a free test particle is placed at r ± near the draggedparticle, it moves with the fluid and its induced verticalvelocity component is given by the fluid velocity v z ± .These velocity components have the same magnitude inthe front and in the rear of the dragged particle, whereasthey point in opposite directions as indicated also by thestreamlines shown in Fig. 1. According to this qualitativereasoning one expects in the case of a pair of beads, whichare dragged parallel to a wall, that the particle in front isrepelled from the wall whereas the rear one is attractedtowards the wall.Eq. (17) displays also the reciprocity of trajectoriesin Stokes flow. If the time and therefore the directionof the applied forces are reversed, then in the discussedsetup the bead in front is again repelled from the walland the rear one is driven to the wall, as indicated inFig. 3(b). After the time-reversal the role of the beads isinterchanged, with v z + = − v z − and the motion is recipro-cal to the one before, thus obeying the time reversibilityof the Stokes-equation.Fig. 5 shows the trajectories of two particles as ob-tained by integrating Eq. (10) for two particles draggedby a force F k ˆ x along a wall at z = 0 and interactingvia the Rotne-Prager approximation. The bead connec-tion vector r at the initial position encloses an angleΘ < π/ F . In this case, the bead closer to the wall, i.e. z = 0, is slightly in front of the other one, and due tothe bulk effect described above (cf. Fig. 2), both beadsfirst approach the wall. As the HI with the boundaryand thus the friction becomes stronger the bead closer to the wall becomes slower than the more distant particle(cf. Eq. (14)). Consequently, the particle further fromthe boundary finally overtakes the other one and in thiscase r becomes again oblique with respect to F withΘ > π/ r and F (Θ = 0, Θ = π/
2, Θ ≷ π/ za x/a FIG. 5. The trajectories of two beads in the xz plane, whichare dragged by an external force F k ˆ x parallel to the bound-ary at the xy plane, i.e. z = 0. The initial bead connectionvector r is slightly tilted with respect to the z axis so thatthe bead which is closer to the wall is slightly in front of theother one. The open and the filled circles visualize the po-sitions of the two particles at equal times. First the beadsapproach the wall until the boundary effect causes a reorien-tation of r , so that the particles are finally repelled fromthe wall. The trajectories of three particles which are initiallyaligned perpendicularly to a wall are shown in Fig. 6.The initial distances between the beads and between thelowest particle and the wall are 3 a . As explained be-fore, the bead which is furthest from the boundary isoften slightly faster than the other two. Due to the bulkHI between the particles, which then have different x positions, the lowest and the highest bead perturb theflow in such a way that the middle particle experiencesfriction just as small as for the upper one. For the ex-ample in Fig. 6 the upper two particles finally build apair and move away from the wall in a similar manneras in the case of two beads described above in Fig. 5,whereas the lower (third) particle moves similar to a sin-gle particle nearly parallel to the wall. For other initialconditions similar formations of pairs of beads are found,which move finally away from the wall. za x/a FIG. 6. The trajectories of three beads in the xz plane, whichare dragged by an external force F k ˆ x away from their initialpositions: (0 , , a ), (0 , , a ) and (0 , , a ). The positions ofthe particles at equal times are indicated by open, grey andblack filled circles along the trajectories. Due to the interplayof bulk and wall HI effects the upper beads form a pair afteran intermediate regime and finally drift away from the walldue to bulk HI effects. The third bead remains nearly at itsinitial distance from the wall. The motion of an assembly of many particles is gov-erned by the same principles as described above. By acomplex interplay of bulk and wall effects pairs of beadsform and dissolve, but the overall tendency is a migra-tion of the particles away from the wall, similar to theresults reported in Ref. [22], where the situation of smallbut finite values of the Reynolds number is investigated.Another case, where the hydrodynamic particle-wall-particle interaction has to be considered, is when twobeads are dragged perpendicularly to a wall with identi-cal wall distances. This is equivalent to the situation oftwo sedimenting particles, when they approach the bot-tom boundary. In this case, the two beads repel eachother slightly and the repulsion becomes significant atwall distances smaller than 10 a . Possible experimentalsetups for investigations of the effects predicted in thissubsection are discussed in Sec. V. C. Two trapped beads in shear flow close to a wall
In this section wall-induced HI effects are investigatedfor a system composed of two trapped particles in a linearshear flow near a wall. The harmonic trap potential isgiven by V = k pot (cid:2) ( r − r pot , ) + ( r − r pot , ) (cid:3) , (18) where k pot = 1 is the spring constant and r and r are the positions of the two beads. The locations ofthe potential minima are r pot , = (0 , , d w ) and r pot , =( d b , , d w ) with the connection vector r pot , − r pot , par-allel to the wall. The results discussed are obtained forpoint-like particles and compared to the case of finite-sized beads which undergo in addition shear-induced ro-tation. FIG. 7. Two point-like particles are trapped in a shear flow u ( z ) by harmonic potentials with their minima at a distance d w from the wall at z = 0 and a mutual separation d b . Withthe coordinate z of the particle downstream ∆ b = d w − z measures its displacement from the potential minimum. The two point-like particles are displaced by the flowin x direction away from the minima of the trapping po-tential. Simultaneously, the wall-induced HI causes dis-placements along the z direction until the influence ofthe boundary is balanced by the potential forces. Sim-ilar to the previous section, the bead upstream is effec-tively repelled from the wall and the bead downstream isattracted. Note, that the deflection of the particle down-stream is different from the displacement of the bead up-stream, which is due to the bulk effect described in theprevious subsection III B and can be explained as fol-lows: Because of the finite angle Θ between the beadconnection vector and the trap forces, cf. Fig. (2), bothparticles are shifted away from the wall. The boundaryeffect dominates, because the trap forces prevent the an-gle Θ from becoming too large, but due to the describedshift the bead downstream is attracted towards the wallby a smaller distance than the one upstream is repelledfrom the wall.The wall-induced particle displacement along the z di-rection is characterized by the shift∆ b = d w − z , (19)where z is the steady-state position of the bead down-stream.In the following we determine the displacement ∆ b atthe steady-state as a function of the flow velocity as wellas of the distances d w and d b . As a first approach, ∆ b is approximated analytically for two point-like particlesas shown in Fig. 7. The forces needed to hold the beadsin place in an external flow can be estimated in a similarmanner as described in Sec. III A, cf. Eq. (14). In theOseen approximation their x components are given by F sx = f x d b d w d w (2 d b + 3 a ) − ad b (20)with f x = − πηau ( d w ). For the particle downstreamone obtains the initial velocity v z − in z direction viaEq. (17) with f = F sx . The resulting Stokes force F sz = 6 πηav z − can be used to estimate the elonga-tion ∆ b in the steady state via the counteracting force F pot z = k pot ∆ b , which must fulfill F sz + F pot z = 0. Thiscalculation yields∆ b = 864 πηa d b d w u ( d w ) k pot (4 d w + d b ) / [8 d w (2 d b + 3 a ) − ad b ] . (21)∆ b depends linearly on the unperturbed flow velocity u ( d w ), which itself increases linearly as a function of d w according to Eq. (2). Since we are mainly inter-ested in the d w -dependence of the wall-induced HI ef-fects and not in its dependence on the absolute value of u ( d w ), we adjust the shear rate according to the rela-tion ˙ γ = u ( d w ) /d w , such that the flow velocity u ( d w )becomes independent of d w : u = u ( d w ) zd w ˆ x . (22) ∆ b a d w / a x10 -3 d b = 10a d b = 10a (ana.) d b = 20a d b = 20a (ana.) FIG. 8. The bead deflection ∆ b is plotted as a function ofthe wall distance d w for two different distances d b = 10 a, a between the potential minima, as indicated in Fig. 7, andfor the flow field given by Eq. (22) with u ( d w ) = 0 . The analytical expression for ∆ b in Eq. (21) exhibitscharacteristic maxima as a function of d w and as a func-tion of d b . The dependence of ∆ b on d w is shown in Fig. 8for the two different values d b = 10 a, a . In this figurethe analytical results are compared with the numericallydetermined values, which were obtained via Eq. (10). De-spite the assumptions included in Eq. (21), both curvesagree with each other surprisingly well qualitatively, es-pecially at larger distances d w and d b . The maximum of ∆ b ( d w ) in Fig. 8 can be traced backto Eq. (17), which represents the z component of theperturbed fluid velocity due to a dragged particle neara wall. This expression already exhibits a maximumas a function of d w . However, for very large wall dis-tances d w ≫ d b the influence of the boundary must van-ish and ∆ b must approach 0. The dependence of ∆ b on ∆ b a d b / a p o i n t - li k e p a r t i c l e s r o t a t i n g p a r t i c l e s a n a l y t i c a l x10 -2 FIG. 9. ∆ b is plotted as a function of the bead distance d b forthe setup shown in Fig. 7 and Fig. 10. The distance to thewall is d w = 2 . a and the flow velocity is u ( d w ) = 0 . the particle-particle distance d b is shown in Fig. 9 for d w = 2 . a and for the flow velocity u ( d w ) = 0 . b ( d b ) in the numerical solution (solid line),is less pronounced than in the corresponding analyticalexpression (dash-dotted line).Up to this point we considered finite size effects of theparticles up to second order in a (bead radius). Parti-cles with a finite diameter also rotate in shear flows andtherefore cause an additional contribution of third order,which is neglected. What is more, the rotation is alsoinfluenced by the vicinity of a wall [49, 50]. In order toquantify the influence of the rotation on the wall-inducedHI and ∆ b , we performed computer simulations for finite-sized particles using the method of fluid particle dynam-ics (FPD) [45, 46]. In these simulations the particles aretrapped in harmonic potentials at a distance d w = 2 . a to the wall and, as indicated in Fig. 10, they can freelyrotate when exposed to a linear shear flow. In the FPDsimulations for rotating particles the flow velocity at theparticle positions u ( d w ) = 0 .
004 is the same as in sim-ulations of the point-like partiles. The squares in Fig. 9represent the results from FPD simulations.As displayed in Fig. 10 the shear flow profile in FPDsimulations was realized by moving the upper and thelower boundary into opposite directions by a constantvelocity v . For such a configuration the shear inducedparticle rotation counteracts the effects of a wall (cf.Fig. 10). In spite of this counteraction the particle down-stream is again effectively attracted to the wall and theparticle upstream repelled, similar to the case of point-like particles. Accordingly, ∆ b is smaller for rotating par-ticles than for point like particles of the same effective ra-dius. In Fig. 7 and Fig. 10 the shear rates at the positionof the particles have an opposite sign. This has for pointlike particles no influence on the elongation ∆ b . However,in FPD simulations of particles of a finite bead-diameterthe bead rotations change their sign with the shear rate.In contrast to the case sketched in Fig. 10, shear inducedbead rotations as indicated in Fig. 7 support the elonga-tion ∆ b and ∆ b becomes in this case larger than for pointlike particles. FIG. 10. The shear-induced particle rotation influences thewall-induced particle attraction and repulsion. For the givensetup the deflection of both beads from their initial position z = d w is reduced due to the rotation effects. The influence ofthe rotational interaction is investigated exemplarily by fluidparticle dynamics simulations of two trapped particles. The shear rate in the FPD simulations was chosen suchthat the ratio between the difference of the flow velocityat the lower and at the upper side of the sphere, ∆ u s , andthe mean velocity u ( d w ) was ∆ u s u ( d w ) = a ˙ γu ( d w ) = 0 . b iscaused by the wall-induced HI since ∆ b > b becomes large for small values of d b and d w . Inorder to estimate ∆ b for possible experiments, we choose d b = 3 a and d w = 2 a . Additionally, a typical potentialstrength is k pot ≃ − N/m , the viscosity of water is η ≃ − N s/m and typical flow velocities in microflu-idic environments are of the order of u ( d w ) ≃ − m/s .Using these values one obtains∆ b a ≃ ηu ( d w ) k pot ≃ − , (23)which might be below the currently possible experimen-tal resolution. However, ∆ b can be enhanced by using a liquid with a higher viscosity than water as for exampleglycerol with η ≃ N s/m , but then the maximum at-tainable flow velocities may be smaller. Furthermore, theeffect can be amplified by placing several beads in a rowand measuring the deflection of the final bead. An am-plification of ∆ b by about 10% can be reached by usingfive beads in a row. Compared to this estimate, sedimen-tation experiments close to a wall, as described in sectionV, seem to be more appropriate for the detection of thewall-induced HI effects.The results discussed until now apply to linear shearflow, but the most important property, that the stream-lines are parallel to the wall, is also shared by other flowprofiles like plane Poiseuille flow. The spatially varyingshear rate in Poiseuille flow causes higher order effects,but the major results presented here remain qualitativelyvalid to Poiseuille flow, as well. IV. APPLICATIONS TO SEMIFLEXIBLEBEAD-SPRING CHAINS
In this section we explore applications of the wall-mediated HI effects of tethered semiflexible bead-springmodels in a flow and fixed near a wall in Sec. IV A orgrafted to the boundary in Sec. IV B.
A. Polymer model fixed near a wall
Polymers tethered with one end in a uniform flow is amodel system for exploring the importance of HI alongpolymers at various stages of its flow-induced conforma-tions [51–56]. Here we consider a semiflexible bead-springpolymer model with its first bead tethered at a distance d w from a wall as shown in Fig. 11 and we explore theimportance of wall-induced hydrodynamic interaction ef-fects.The stationary chain conformation in shear flow isagain determined via Eq. (10) by using the potential en-ergy for the elastic forces along the polymer V = k trap r − r trap ) + N − X i =1 k str | r i − r i +1 | − d n ) + N − X i =2 k bend χ i ) , (24)with r trap = (0 , , d w ) the location of the minimum ofthe trap potential of strength k trap = 1. k str = 500 is thestretching stiffness of the springs and d n = 5 a is the equi-librium distance between neighboring beads. The bend-ing stiffness is k bend = 100 and the bending angle at the i -th bead is χ i = arccos [( r i − r i − ) · ( r i +1 − r i )]. The dis-tances between the beads along the chain are practicallyfixed on the time scale of the bending dynamics, whichcan be seen by the ratio of the relaxation times of stretch-ing and bending: τ str /τ bend = 2 k bend / ( d k str ) ≃ / N = 10beads as an example. FIG. 11. A semiflexible chain, fixed with one end at a distance d w from a wall and exposed to a linear shear flow, is attractedtowards the wall with its free end by a distance ∆ c . Thisshift depends on d w as shown in Fig. 12. For the displayedconfiguration the parameters are d w = 5 . a and u ( d w ) =0 . In an unbounded fluid the tethered polymer would as-sume a straight conformation, parallel to the streamlinesof the external flow. However, in the presence of a wallthe free end of the chain is attracted via the wall-inducedHI. This is similar to the bead attraction downstreaminvestigated in the previous section and to the hydrody-namically induced attraction between two tethered poly-mers in plug flow and Brownian motion as described inRef. [39]. A measure of the effective wall attraction isthe deflection ∆ c = d w − z N of the N -th bead towardsthe wall, which is shown in Fig. 11 for d w = 5 . a and theflow profile (22) with u ( d w ) = 0 . ∆ c a d w / a (b) 0.8 0.9 1 ∆ c a (a) FIG. 12. The deflection ∆ c of the free end of a tetheredchain consisting of N = 10 beads (cf. Fig. 11) is shown as afunction of the distance d w to the wall for the flow field givenby Eq. (22) with u ( d w ) = 0 . ∆ c is plotted as a function of the wall distance inFig. 12 for the constant flow velocity u ( d w ) = 0 . z = d w . The solid line in part (a) describes the case wherein the wall-mediated HI between all beads is takeninto account. In contrast to the curve for two beadsin Fig. 8, it displays two maxima and decreases mono-tonically afterwards. The first maximum of ∆ c ( d w ) at d w ≃ a is mainly caused by the wall-induced HI betweennearest-neighbor beads along the chain. In order to sub-stantiate this interpretation, we plot ∆ c ( d w ) in Fig. 12(b)for the model situation where the wall-mediated HI isonly taken into account between nearest neighbor beads.In this case, the magnitude of ∆ c ( d w ) is much smaller andhas indeed only one maximum at d w ≃ . a . The smallshift between this maximum and the first maximum ofthe curve in Fig. 12(a) is caused by the boundary-inducedHI between beads which are further apart. The secondmaximum at d w = 7 . a in Fig. 12(a) is also a result ofhydrodynamic bead-bead interactions via the wall overlarger distances than nearest neighbors.Besides the deflection of the free end, the chain exhibitsalso a small curvature. This can be explained as follows:The trapping potential prevents the fixed end of the chainfrom moving away from the wall. On the other hand eachbead is driven to the wall due to the flow perturbationsfrom all its neighbors upstream. Therefore, the beadscloser to the free end are increasingly attracted towardsthe wall, which leads to a slight bending of the chain. B. Perpendicularly anchored semiflexible chains inflow
As wall-grafted polymers are relevant in several appli-cations [27] we investigate the influence of the boundaryon the behavior of semiflexible chains, which are perpen-dicularly grafted to a wall and exposed to a shear flow asshown in Fig. 13.For our model calculations we use a potential energydescribing the bending and stretching of the N c chainsas given by V = N c X j =1 V j with (25) V j = N − X i =1 k str | r ji − r ji +1 | − d n ) + N − X i =0 k bend χ ji ) . (26)Here r ji and χ ji are the position vector and the bendingangle of the i -th bead in the j -th chain. The constants k str , d n and k bend have the same values as in the previoussubsection, but the chains are now composed of N = 9beads. However, the second sum in Eq. 26 starts with i =0 and therefore includes additional bending contributionsat the boundary in order to ensure that the chain relaxesback to an orientation perpendicular to the wall afterswitching off the flow.0A single chain is bent towards the flow direction un-til the forces exerted on the beads due to the externaldrag are balanced by the stretching and bending forcesaccording to Eq. (26). If the flow velocity is very large,the chain even goes beyond the alignment with the flowlines and bends towards the wall similar to the resultspresented in Sec. IV A. FIG. 13. Three bead-spring chains, which are perpendicularlyanchored at a boundary and exposed to a linear shear flow.For the grey conformation the wall-mediated HI is neglectedand only the bulk HI effects are taken into account.
For three chains, which are perpendicularly grafted toa wall, Fig. 13 shows the steady state chain conforma-tions for the case in which the wall-mediated contribu-tions to the HI are neglected (grey) and for the case inwhich the HI via the boundary is completely included(black). In the free draining limit, where the HI betweenthe beads are disregarded, the three perpendicularly an-chored polymers would be bent identically by the flow.If only the effects of bulk HI are taken into account, theouter polymers perturb the flow in such a way, that themiddle polymer experiences a smaller drag force. Hence,the first and the last polymer, i.e. the left and the rightone in Fig. 13, are bent almost identically, while the onein between is bent less strongly, which is indicated by thegrey configuration in Fig. 13.Similar to the results from the previous subsection,the wall-mediated HI leads to a stronger bending of thechains towards the wall, which is displayed by the de-viation of the black chain configurations from the greyones in Fig. 13. This attractive effect increases forchains which lie further downstream, because the wall-attraction of each chain bead is enhanced by its neighborsupstream as described in Sec. IV A. The screening effectfor the middle chain due to the interactions in the bulkis therefore superimposed by the wall-induced effect.Consequently, the wall-mediated HI along this semi-flexible brush leads to a stronger bending of the chainsand thus to a reduction of the brush height. So the effec-tive diameter of a polymer-grafted tube is increased andthe flow resistance of the grafted chains is reduced, whichmight be interesting for studies on wall-grafted brushesof semiflexible polymers or for a wall decorated by thinflexible pillars.
V. CONCLUSIONS
In our investigations of wall-induced effects on the stat-ics and the dynamics of hydrodynamically interactingparticles we first calculated the streamline deformationaround a single trapped point-like particle close to awall in order to develop a qualitative picture about theparticle-particle interaction near a wall. It was shownhow the deformations of the flow lines around a fixedparticle allow an estimation of the direction of the forceacting on a nearby second particle, which was confirmedby numerical calculations.For two or three beads being dragged parallel to a wall,a scenario with resemblance to sedimenting particles un-der gravity, it was shown that the beads always migrateaway from the wall. The origin of this behavior is dueto the interplay between effects from the bulk and theboundary, and it is similar to the lift force discussed forvesicles and polymers [11, 12, 17, 34].The analysis of the configurations of two beads, whichare trapped by harmonic potentials close to a wall and ex-posed to an external shear flow, provided further insightabout the hydrodynamically mediated particle wall inter-action. We found a repulsion from the wall for the parti-cle upstream and an attraction towards the boundary forthe one downstream. Varying the particle-wall distanceand the particle-particle distance a characteristic max-imum in the deflection of the downstream particle wasfound, which could be described also analytically givingfurther insight on the parameter dependence of this phe-nomenon. The behavior obtained for two point-like par-ticles was confirmed by using the complementary methodof Fluid Particle Dynamics [45, 46], which accounts forthe finite particle radii and the effects of particle rotationon the hydrodynamic particle-particle interaction.As an application of the basic effects found for the two-bead configuration, we investigated a semiflexible bead-spring chain, where one end is held in a linear shear flowat a distance d w from the boundary. The chain is at-tracted towards the boundary as a function of d w and weidentified the wall-induced contributions to the HI as thesource for this behavior. The phenomenon is related toa recent study on the flow induced polymer-polymer at-traction [39] mediated through inter-chain HI, where thesecond polymer causes very similar effects as the bound-ary in this work. Both are examples of hydrodynamicallyinduced particle-particle attraction, which has recentlybeen found for rotors as well [38].What is more, three perpendicularly anchored semi-flexible bead-spring chains were investigated as a simplemodel for a polymer brush and their response to an ex-ternal linear shear flow was obtained. It was found , thatwall-effects cause a stronger effective attraction towardsthe wall for the polymers downstream than for the onesupstream. Whether three semiflexible polymers, perpen-dicularly anchored at a wall, but not along a line, andof different length, have also the propensity to oscilla-tory motion as reported for three beads in shear flow in1Ref. [35], is an interesting further question.While our analysis in this work is exclusively used for alinear shear flow as external stimulus, we nonetheless ex-pect a similar qualitative behavior for other flow profileswith parallel streamlines. The reason is, that the defor-mations of the streamlines near walls as shown in Fig. 1are in general the same for other laminar flow profileswith parallel streamlines.Our results may be tested by different experiments. Afirst one would be to measure the wall-induced displace-ment of an array of beads trapped close to wall by lasertweezers while imposing an external flow similar to thesetups shown in Fig. 7 and Fig. 11. A variation of theabove setup may be to measure the deflection of a can-tilever in proximity of a wall and exposed to a flow asillustrated for a model polymer in Sec. IV A.Another alternative to probe our findings is to line uptwo particles in a row close to the wall of a container,extended in its vertical direction, and to track the tra- jectories of the two sedimenting particles. According toour predictions, the particle moving in front should be re-pelled from the wall whereas the particle behind shouldat first be attracted. As soon as the connection vectorbetween the particles becomes sufficiently oblique withrespect to the boundary, the bulk effect is expected tobecome dominant, so that both beads are carried bothaway from the wall. ACKNOWLEDGMENTS
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