Hydrodynamic interaction near elastic interfaces

Hydrodynamic interaction between particles near elastic interfacesAbdallah Daddi-Moussa-Ider1, a) and Stephan Gekle1

Biofluid Simulation and Modeling, Fachbereich Physik, Universität Bayreuth,Universitätsstraße 30, Bayreuth 95440, Germany(Dated: August 24, 2016)

We present an analytical calculation of the hydrodynamic interaction between two spherical particles nearan elastic interface such as a cell membrane. The theory predicts the frequency dependent self- and pair-mobilities accounting for the finite particle size up to the 5th order in the ratio between particle diameter andwall distance as well as between diameter and interparticle distance. We find that particle motion towards amembrane with pure bending resistance always leads to mutual repulsion similar as in the well-known caseof a hard-wall. In the vicinity of a membrane with shearing resistance, however, we observe an attractiveinteraction in a certain parameter range which is in contrast to the behavior near a hard wall. This attractionmight facilitate surface chemical reactions. Furthermore, we show that there exists a frequency range inwhich the pair-mobility for perpendicular motion exceeds its bulk value, leading to short-lived superdiffusivebehavior. Using the analytical particle mobilities we compute collective and relative diffusion coefficients. Theappropriateness of the approximations in our analytical results is demonstrated by corresponding boundaryintegral simulations which are in excellent agreement with the theoretical predictions.

Published as: J. Chem. Phys. 145, 014905 (2016), doi:10.1063/1.4955099

I. INTRODUCTION

The hydrodynamic interaction between particles mov-ing through a liquid is essential to determine the behaviorof colloidal suspensions1, polymer solutions2,3, chemicalreaction kinetics4,5, bilayer assembly6 or cellular flows7,8.As an example, hydrodynamic interactions result in a no-table alteration of the collective motion behavior of cat-alytically powered self-propelled particles9 or bacterialsuspensions10–14. Many of the occurring phenomena canbe explained on the basis of two-particle interactions15which in bulk are well understood. Some of the most in-triguing observations, however, are made when particlesinteract hydrodynamically in the close vicinity of inter-faces – a prominent example being the attraction of like-charged colloid particles during their motion away froma hard wall16–18.In the low Reynolds number regime hydrodynamic in-

teractions between two particles are fully described bythe mobility tensor which provides a linear relation be-tween the force applied on one particle and the result-ing velocity of either the same or the neighboring par-ticle. In an unbounded flow, algebraic expressions forthe hydrodynamic interactions between two15,19–23 andseveral24–29 spherical particles are well established. Ex-perimentally, the predicted hydrodynamic coupling hasbeen confirmed using optical tweezers30–33 and atomicforce microscopy34.The presence of an interface is known to drastically

alter the hydrodynamic mobility. For a single particle,this wall-induced drag effect has been studied extensivelyover recent decades theoretically and numerically near arigid35–44, a fluid-fluid45–50 or an elastic interface51–56.

a)Electronic mail: abdallah.daddi-moussa-ider@uni-bayreuth.de

While rigid interfaces in general simply lead to a reduc-tion of particle mobility, the memory effect caused byelastic interfaces leads to a frequency dependence of theparticle mobility and can cause novel phenomena suchas transient subdiffusion55. On the experimental side,the single particle mobility has been investigated us-ing optical tweezers57–59, evanescent wave dynamic lightscattering60–69 or video microscopy70–72. The influence ofa nearby elastic cell membrane has recently been inves-tigated using magnetic particle actuation73 and opticaltraps52,74,75.Hydrodynamic interactions between two particles near

a planar rigid wall have been studied theoretically17,76and experimentally using optical tweezers77,78 and dig-ital video microscopy79. Narrow channels80,81, 2Dconfinement82 or liquid-liquid interfaces have also beeninvestigated83,84. Near elastic interfaces, however, nowork regarding hydrodynamic interactions has so farbeen reported. Given the complex behavior of a singleparticle near an elastic interface (caused by the above-mentioned memory effect) such hydrodynamic interac-tions can be expected to present a very rich phenomenol-ogy.In this paper, we calculate the motion of two spherical

particles positioned above an elastic membrane both an-alytically and numerically. We find that the shearing andbending related parts in the pair-mobility can in some sit-uations have opposite contributions to the total mobility.Most prominently, we find that two particles approachingan idealized membrane exhibiting only shear resistancewill be attracted to each other which is just opposite tothe well-known hydrodynamic repulsion for motion to-wards a hard wall17. Additionally, we show that the pair-mobility at intermediate frequencies may even exceed itsbulk value, a feature which is not observed in bulk ornear a rigid wall. This increase in pair-mobility resultsin a short-lived superdiffusion in the joint mean-square

Hydrodynamic interaction near elastic interfaces 2

z0

a a

γ λx

h

Figure 1. Illustration of the problem setup. Two smallparticles labeled γ and λ of radius a are located a distanceh := xλ − xγ apart and a distance z0 above an elastic mem-brane. The dimensionless length scales of the problem areε := a/z0 and σ := a/h.

displacement.The remainder of the paper is organized as follows. In

Sec. II, we introduce the theoretical approach to com-puting the frequency-dependent self- and pair-mobilitiesfrom the multipole expansion and Faxén’s theorem, upto the 5th order in the ratio between particle radius andparticle-wall or particle-particle distance. In Sec. III,we present the boundary integral method which we haveused to numerically confirm our theoretical predictions.In Sec. IV, we provide analytical expressions of the par-ticle self- and pair-mobilities in terms of power series,finding excellent agreement with our numerical simula-tions. Expressions of the self- and pair-diffusion coef-ficients are derived in Sec. V. Concluding remarks aremade in Sec. VI.

II. THEORY

We consider a pair of particles of radius a suspendedin a Newtonian fluid of viscosity η above a planar elasticmembrane extending in the xy plane. The two particlesare placed at rγ = (xγ , 0, z0) and rλ = (xλ, 0, z0), i.e.the line connecting the two particles is parallel to theundisplaced membrane. We denote by h := xλ − xγ thecenter-to-center separation measured from the left (γ) tothe right (λ) particle (see Fig. 1 for an illustration).The particle mobility is a tensorial quantity that lin-

early couples the velocity Vγα of particle γ in directionα to an external force in the direction β applied on thesame (Fγβ) or the other (Fλβ) particle. Transforming tothe frequency domain we thus have85 (ch. 7)

Vγα(ω) = µγγαβ(rγ , rγ , ω)Fγβ(ω) + µγλαβ(rγ , rλ, ω)Fλβ(ω) ,

where Einstein’s convention for summation over repeatedindices is assumed. The particle mobility tensor in thepresent geometry can be written as an algebraic sum oftwo distinct contributions

µγλαβ(rγ , rλ, ω) = bγλαβ(rγ , rλ) + ∆µγλαβ(rγ , rλ, ω) , (1)

where bγλαβ is the pair-mobility in an unbounded geom-etry (bulk flow), and ∆µγλαβ is the frequency-dependentcorrection due to the presence of the elastic membrane.An analogous relation holds for µγγαβ .For the determination of the particle mobility, we con-

sider a force density f acting on the surface Sλ of theparticle λ, related to the total force by

Fλβ(ω) =∮Sλ

fβ(r′, ω)d2r′ ,

which induces the disturbance flow velocity at point r

vα(r, rλ, ω) =∮Sλ

Gαβ(r, r′, ω)fβ(r′, ω)d2r′ , (2)

where Gαβ denotes the velocity Green’s function(Stokeslet), i.e. the flow velocity field resulting from apoint-force acting on rλ. The disturbance velocity atany point r can be split up into two parts,

vα(r, rλ, ω) = v(0)α (r, rλ) + ∆vα(r, rλ, ω) , (3)

where v(0)α is the flow field induced by the particle λ in

an unbounded geometry, and ∆vα is the flow satisfyingthe no-slip boundary condition at the membrane. In thisway, the Green’s function can be written as

Gαβ(r, r′, ω) = G(0)αβ (r, r′) + ∆Gαβ(r, r′, ω) , (4)

where G(0)αβ is the infinite-space Green’s function (Oseen’s

tensor) given by

G(0)αβ (r, r′) = 1

8πη

(δαβs

+ sαsβs3

), (5)

with s := r− r′ and s := |s|. The term ∆Gαβ representsthe frequency-dependent correction due to the presenceof the membrane. Far away from the particle λ, the vec-tor r′ in Eq. (2) can be expanded around the particle cen-ter rλ following a multipole expansion approach. Up tothe second order, and assuming a constant force density,the disturbance velocity can be approximated by76,86–88

vα(r, rλ, ω) ≈(

1 + a2

6 ∇2rλ

)Gαβ(r, rλ, ω)Fλβ(ω) , (6)

where ∇rλ stands for the gradient operator taken withrespect to the singularity position rλ. Note that for a sin-gle sphere in bulk, the flow field given by Eq. (6) satisfiesexactly the no-slip boundary conditions at the surface ofthe sphere, i.e. in the frame moving with the particle,both the normal and tangential velocities vanish. UsingFaxén’s theorem, the velocity of the second particle γ inthis flow reads76,86–88

Vγα(ω) = µ0Fγα(ω) +(

1 + a2

6 ∇2rγ

)vα(rγ , rλ, ω) , (7)

where µ0 := 1/(6πηa) denotes the usual bulk mobil-ity, given by the Stokes’ law. The disturbance flow vα

Hydrodynamic interaction near elastic interfaces 3

incorporates both the disturbance from the particle λand the disturbance caused by the presence of the mem-brane. By plugging Eq. (6) into Faxén’s formula given byEq. (7), the αβ component of the frequency-dependentpair-mobilities can be obtained from

µγλαβ(ω) =(

1 + a2

6 ∇2rγ

)(1 + a2

6 ∇2rλ

)Gαβ(rγ , rλ, ω) .

(8)For the self-mobilities, only the correction in the flow

field ∆vα due to the presence of the membrane in Eq. (3)is considered in Faxén’s formula (the influence of thesecond particle on the self-mobility is neglected here forsimplicity76,87). Therefore, the frequency-dependent self-mobilities read

µγγαβ(ω) = µ0 + limr→rγ

(1 + a2

6 ∇2r

)×(

1 + a2

6 ∇2rγ

)∆Gαβ(r, rγ , ω)

(9)

and analogously for µλλαβ .In order to use the particle pair- and self-mobilities

from Eqs. (8) and (9), the velocity Green’s functions inthe presence of the membrane are required. These havebeen calculated in our earlier work55 and their deriva-tion is only briefly sketched here with more details inAppendix A.

We proceed by solving the steady Stokes equationswith an arbitrary time-dependent point-force F actingat r0 = (0, 0, z0),

η∇2v −∇p+ F δ(r − r0) = 0 , (10)∇ · v = 0 , (11)

where p is the pressure field. The determination of theGreen’s functions at rλ is straightforward thanks to thesystem translational symmetry along the xy plane. Aftersolving the above equations and appropriately applyingthe boundary conditions at the membrane, we find thatthe Green’s functions are conveniently expressed by

Gzz(r, rλ, ω) = 12π

∫ ∞0Gzz(q, z, z0, ω)J0(ρq)qdq , (12a)

Gxx(r, rλ, ω) = 14π

∫ ∞0

(G+(q, z, z0, ω)J0(ρq)

+ G−(q, z, z0, ω)J2(ρq) cos 2θ)qdq , (12b)

Gyy(r, rλ, ω) = 14π

∫ ∞0

(G+(q, z, z0, ω)J0(ρq)

− G−(q, z, z0, ω)J2(ρq) cos 2θ)qdq , (12c)

Gxz(r, rλ, ω) = i cos θ2π

∫ ∞0Glz(q, z, z0, ω)J1(ρq)dq ,

(12d)

where ρ :=√

(x− xλ)2 + y2, θ := arctan(y/(x − xλ))with r = (x, y, z). Here Jn denotes the Bessel function

of the first kind of order n. The functions G±, Glz andGzz are provided in Appendix A. It is worth to men-tion here that the unsteady term in the Stokes equationsleads to negligible contribution in the correction to theGreen’s functions55, and it is therefore not considered inthe present work.The membrane elasticity is described by the well-

established Skalak model89, commonly used to de-scribe deformation properties of red blood cell (RBC)membranes90–92. The elastic model has as parametersthe shearing modulus κS and the area-expansion modu-lus κA. The two moduli are related via the dimensionlessnumber C := κA/κS. Moreover, the membrane resists to-wards bending according to Helfrich’s model93, with thecorresponding bending rigidity κB.

III. BOUNDARY INTEGRAL METHODS

In this section, we introduce the numerical methodused to compute the particle self- and pair- mobilities.The numerical results will subsequently be comparedwith the analytical predictions presented in Sec. II.For solving the fluid motion equations in the inertia-

free Stokes regime, we use a boundary integral method(BIM). The method is well suited for problems with de-forming boundaries such as RBC membranes94,95. Inorder to solve for the particle velocity given an ex-erted force, a completed double layer boundary integralmethod (CDLBIM)96,97 has been combined with the clas-sical BIM98. The integral equations for the two-particlemembrane systems read

vβ(x) = Hβ(x) , x ∈ Sm ,

12φβ(x) +

6∑α=1

ϕ(α)β (x) 〈ϕ(α),φ〉 = Hβ(x) , x ∈ Sp .

(13)

where Sm is the surface of the elastic membrane andSp := Spγ ∪Spλ is the surface of the two spheres. Here vdenotes the velocity on the membrane whereas φ repre-sents the double layer density function on Sp, related tothe velocity of the particle γ via

Vγβ(x) =6∑

α=1ϕ

(α)β (x) 〈ϕ(α),φ〉 , x ∈ Spγ . (14)

where ϕ(α) are known functions97. The brackets standfor the inner product in the space of real functions whosedomain is Spγ , and the function Hβ is defined by

Hβ(x) := −(Nm∆f)β(x)− (Kpφ)β(x)+G(0)βµ (x,xλc)Fµ ,

with xλc being the centroid of the sphere labeled λ uponwhich the force is applied. The single layer integral isdefined as

(Nm∆f)β(x) :=∫Sm

∆fα(y)G(0)αβ (y,x) dS(y)

Hydrodynamic interaction near elastic interfaces 4

and the double layer integral as

(Kpφ)β(x) :=∮Sp

φα(y)T (0)αβµ(y,x)nµ(y) dS(y) .

Here, ∆f is the traction jump, n denotes the outer nor-mal vector at the particle surfaces and F is the force act-ing on the rigid particle. The infinite-space Green’s func-tion is given by Eq. (5) and the corresponding Stresslet,defined as the symmetric part of the first moment of theforce density, reads85

T (0)αβµ(y,x) = − 3

4πsαsβsµs5 ,

with s := y − x and s := |s|. The traction jump acrossthe membrane ∆f is an input for the equations, deter-mined from the instantaneous deformation of the mem-brane. In order to solve Eqs. (13) numerically, the mem-brane and particles’ surfaces are discretized with flat tri-angles. The resulting linear system of equations for thevelocity v on the membrane and the density φ on therigid particles is solved iteratively by GMRES99. Thevelocity of each particle is determined from (14). Forfurther details concerning the algorithm and its imple-mentation, we refer the reader to Ref.55. Bending forcesare computed using Method C from100.In order to compute the particle self- and pair-

mobilities numerically, a harmonic oscillating forceFλ(t) = Aλe

iω0t of amplitude Aλ and frequency ω0is applied at the surface of the particle λ. After abrief transient time, both particles begin to oscillateat the same frequency as Vλ(t) = Bλe

i(ω0t+δλ) andas Vγ(t) = Bγe

i(ω0t+δγ). The velocity amplitudes andphase shifts can accurately be obtained by a fitting pro-cedure of the numerically recorded particle velocities. Forthat, we use a nonlinear least-squares algorithm based onthe trust region method101. Afterward, the αβ compo-nent of the frequency-dependent complex self- and pair-mobilities can be calculated as

µλλαβ = BλαAλβ

eiδλ , µγλαβ =BγαAλβ

eiδγ .

IV. RESULTS

For a single membrane, the corrections to the particlemobility can conveniently be split up into a correctiondue to shearing and area expansion together with a cor-rection due to bending55. In the following, we denote byµγγαβ = µλλαβ = µS

αβ (“self”) the components of the self-mobility tensor, and by µγλαβ = µλγβα = µP

αβ (“pair”) thecomponents of the pair-mobility tensor. Note that forα 6= β, µS

αβ = 0 and that µPαβ = −µP

βα.

A. Self-mobilities for finite-sized particles

Mathematical expressions for the translational parti-cle self-mobility corrections will be derived in terms of

ε = a/z0. The point-particle approximation presented inearlier work55 represents the first order in the perturba-tion series, valid when the particle is far away from themembrane.

1. Perpendicular to membrane

The particle mobility perpendicular to the membraneis readily obtained after plugging the correction ∆Gzz asdefined by Eq. (4) to the normal-normal component ofthe Green’s function from Eq. (12a) into Eq. (9). Aftercomputation, we find that the contribution due to shear-ing and bending can be expressed as

∆µSzz,S

µ0= eiβ

(− 9

16 E4(iβ)ε+ 34 E5(iβ)ε3

− 516 E6(iβ)ε5

), (15a)

∆µSzz,B

µ0= εf1 + ε3f3 + ε5f5 , (15b)

where the subscripts S and B stand for shearingand bending, respectively. The function En is thegeneralized exponential integral defined as En(x) :=∫∞

1 t−ne−xtdt102. Furthermore, β := 6Bz0ηω/κS is adimensionless frequency associated with the shearing re-sistance, whereas B := 2/(1 + C). Moreover, βB :=2z0(4ηω/κB)1/3 is a dimensionless number associatedwith bending. The functions fi, with i ∈ {1, 3, 5} aredefined by

f1 = −1516 + 3iβB

8

((β2

B12 + iβB

6 + 16

)φ+

+√

36 (βB + i)φ− +

(β2

B12 −

iβB

3 − 13

)ψ

),

f3 = 516 −

β3B

48

((βB

4 + i

)φ+ + i

√3βB

4 φ−

−(βB

2 − i)ψ + 3i

2

),

f5 = − 116 + β3

B384

(β2

B3

(√3

2 φ− + i

2φ+ − iψ)

+ i

),

with

φ± := e−izB E1 (−izB)± e−izB E1 (−izB) ,ψ := e−iβB E1(−iβB) ,

where zB := jβB and j := e2iπ/3 being the principalcubic-root of unity. The bar designates complex conju-gate.The total mobility correction is obtained by adding the

individual contributions due to shearing and bending, asgiven by Eqs. (15a) and (15b). In the vanishing frequency

Hydrodynamic interaction near elastic interfaces 5

limit, the known result for a hard-wall88 is obtained:

limβ,βB→0

∆µSzz

µ0= −9

8ε+ 12ε

3 − 18ε

5 . (16)

The particle mobility near an elastic membrane is de-termined by membrane shearing and bending properties.We therefore consider a typical case for which both effectmanifests themselves equally. For that purpose, we definea characteristic time scale for shearing as TS := 6z0η/κStogether with a characteristic time scale for bending asTB := 4ηz3

0/κB55. Then we take z2

0κS/κB = 3/2 suchthat the two time scales are equal and can be denotedby TS = TB =: T . In this case, the two dimensionlessnumbers β and βB are related by βB = 2(β/B)1/3. Thesituation for a membrane with the typical parameters ofa red blood cell is qualitatively similar as shown in theSupporting Information.103

In Fig. 2 a), we show the particle scaled self-mobilitycorrections versus the scaled frequency β, as stated byEqs. (15a) and (15b). The particle is set a distancez0 = 2a above the membrane. We observe that the realpart is a monotonically increasing function with respectto frequency while the imaginary part exhibits a bell-shaped dependence on frequency centered around β ∼ 1.In the limit of infinite frequencies, both the real andimaginary parts of the self-mobility corrections vanish,and thus one recovers the bulk behavior. For the per-pendicular motion we observe that the particle mobilitycorrection is primarily determined by the bending part.

A very good agreement is obtained between the ana-lytical predictions and the numerical simulations over thewhole range of frequencies. Additionally, we assess theaccuracy of the point-particle approximation employedin earlier work55, in which only the first order correc-tion term in the perturbation parameter ε was consid-ered. While this approximation slightly underestimatesparticle mobilities, it nevertheless leads to a surprisinglygood prediction, even though the particle is set only onediameter above the membrane.

2. Parallel to membrane

We proceed in a similar way for the motion parallel tothe membrane. By plugging the correction ∆Gxx fromthe Green’s function in Eq. (12b) into Eq. (9) we find

∆µSxx,S

µ0= eiβ

(− 3

32

(3 E4(iβ)− 4 E3(iβ) + 2 E2(iβ)

+ 4eiCβ E2(i(1 + C)β))ε (17a)

+ 316 (2 E5(iβ)− E4(iβ)) ε3 − 5

32 E6(iβ)ε5),

∆µSxx,B

µ0= εg1 + ε3g3 + ε5g5 , (17b)

-0.6

-0.4

-0.2

0

0.2

10−6 10−4 10−2 100 102 104

∆µS zz

/ µ 0

β

a)

-0.3

-0.2

-0.1

0

0.1

10−4 10−2 100 102 104

∆µS xx

/ µ 0β

b)

Figure 2. (Color online) The scaled frequency-dependentself-mobility correction versus the scaled frequency for themotion perpendicular (a) and parallel (b) to the membrane.The particle is located at z0 = 2a. We take z2

0κS/κB = 3/2and C = 1 in the Skalak model. The analytical predictions areshown as dashed lines for the real part, and as solid lines forthe imaginary part. Symbols refer to BIM simulations. Theshearing and bending contributions are shown in green andred respectively. The dotted-dashed line in blue correspondsto the first order correction in the particle self-mobility, aspreviously determined in Ref.55. Horizontal dashed lines rep-resent the mobility corrections near a hard-wall as given byEqs. (16) and (18).

where we defined

g1 = − 332 + iβ3

B64 (φ+ + ψ) ,

g3 = 332 + β3

B64

(−i+ βB

3

(ψ − 1

2φ+ −i√

32 φ−

)),

g5 = − 132 + β3

B768

(i+ β2

B3

(i

2φ+ +√

32 φ− − iψ

)).

The well-known hard-wall limit, as first calculated byFaxén88,104, is recovered by considering the vanishing fre-quency limit:

limβ,βB→0

∆µSxx

µ0= − 9

16ε+ 18ε

3 − 116ε

5 . (18)

The mobility corrections in the parallel direction areshown in Fig. 2 b). We observe that the total correction

Hydrodynamic interaction near elastic interfaces 6

is mainly determined by the shearing part in contrast tothe perpendicular case where bending dominates.

B. Pair-mobilities for finite-sized particles

In the following, expressions for the pair-mobility cor-rections in terms of a power series in σ = a/h willbe provided. To start, let us first recall the particlepair-mobilities in an unbounded geometry. By applyingEq. (8) to the infinite space Green’s function Eq. (5), thebulk pair-mobilities for the motion perpendicular to andalong the line of centers read85 (p. 190)

µPzz

µ0= 3

4σ + 12σ

3 ,µPxx

µ0= 3

2σ − σ3 , (19)

and are commonly denominated the Rotne-Pragertensor26,105. Note that the terms with σ5 vanish for thebulk mobilities when considering only the first reflectionas is done here. The axial symmetry along the line con-necting the two spheres in bulk requires that µP

yy = µPzz

and that the off-diagonal components of the mobility ten-sor are zero. Physically, the parameter σ only takes val-ues between 0 and 1/2 as overlap between the two par-ticles is not allowed. In this interval, the pair-mobilityperpendicular to the line of centers µP

zz is always lowerthan the pair-mobility µP

xx, since it is easier to move thefluid aside than to push it into or to squeeze it out of thegap between the two particles.

Consider next the pair-mobilities near an elastic mem-brane. By applying Eq. (8) to Eqs. (12a) through (12d),we find that the corrections to the pair-mobilities canconveniently be expressed in terms of the following con-vergent integrals,

∆µPzz

µ0=∫ ∞

0− iσu

3

3ξ5/2

(Λ2

2iu− β +4Γ2−

8iu3 − β3B

)× χ0e

−2udu , (20a)∆µP

xx

µ0=∫ ∞

0

(iσ

6ξ5/2

(Γ2

+2iu− β + 4u4Λ2

8iu3 − β3B

)×(ξ1/2χ1 − 2uχ0

)− 3σB

2χ1

Bu+ iβ

)e−2udu ,

(20b)∆µP

yy

µ0=∫ ∞

0

(− iσ

6ξ2

(Γ2

+2iu− β + 4u4Λ2

8iu3 − β3B

)χ1

+ 3σB2ξ1/2

ξ1/2χ1 − 2uχ0

Bu+ iβ

)e−2udu , (20c)

∆µPxz

µ0=∫ ∞

0

iσu2

3ξ5/2 Λ(

Γ+

2iu− β + 4u2Γ−8iu3 − β3

B

)× χ1e

−2udu , (20d)

where ξ := 4z20/h

2 = 4σ2/ε2 and

Λ := 4σ2u− 3ξ ,Γ± := 4σ2u2 − 3uξ ± 3ξ ,

χn := Jn

(2uξ1/2

).

The terms involving β and βB in Eqs. (20a) through(20d) are the contributions coming from shearing andbending, respectively. Due to symmetry, µP

αy = 0 forα ∈ {x, z}.For future reference, we note that each component of

the frequency-dependent particle self- and pair-mobilitytensor can conveniently be cast in the form

µ(ω)µ0

= b+∫ ∞

0

ϕ1(u)ϕ2(u) + iωT

du , (21)

where indices and superscripts have been omitted. Hereb denotes the scaled bulk mobility (cf. Eq. (1)), and theintegral term represents either shearing or bending re-lated parts in the mobility correction. Note that ϕ1 andϕ2 are real functions which do not depend on frequency.Moreover, ϕ2(u) = 2u/B or ϕ2(u) = u for the shear-ing related parts and ϕ2(u) = u3 for bending such thatϕ2(u) ≥ 0, ∀u ∈ [0,∞).In the vanishing frequency limit, i.e. for β, βB both

taken to zero we recover the pair-mobilities near a hard-wall with stick boundary conditions, namely

∆µPzz

µ0= −3

43ξ2 + 5

2ξ + 1(1 + ξ)5/2 σ +

4ξ2 − 4ξ − 12

(1 + ξ)7/2 σ3

−4ξ2 − 12ξ + 3

2(1 + ξ)9/2 σ5 , (22a)

∆µPxx

µ0= −3

21 + ξ + 3

4ξ2

(1 + ξ)5/2 σ +ξ2 − 11

2 ξ + 1(1 + ξ)7/2 σ3

−2ξ2 − 27

2 ξ + 2(1 + ξ)9/2 σ5 , (22b)

∆µPyy

µ0= −3

41 + 3

2ξ

(1 + ξ)3/2σ +ξ − 1

2(1 + ξ)5/2σ

3 −2ξ − 1

2(1 + ξ)7/2σ

5 ,

(22c)∆µP

xz

µ0= 9

8ξ3/2

(1 + ξ)5/2σ −32

(4ξ − 1)ξ1/2

(1 + ξ)7/2 σ3

+ 52

(4ξ − 3)ξ1/2

(1 + ξ)9/2 σ5 , (22d)

in agreement with the results by Swan and Brady76.In Fig. 3 we plot the particle pair-mobilities as given

by Eqs. (20a) through (20d) as functions of the dimen-sionless frequency β for h = 4a. We observe that the realand imaginary parts have basically the same evolution asthe self-mobilities. Nevertheless, two qualitatively differ-ent effects are apparent from Fig. 3: First, the amplitudeof the normal-normal pair-mobility

∣∣µPzz

∣∣ in a small fre-quency range even exceeds its bulk value. This enhanced

Hydrodynamic interaction near elastic interfaces 7

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

10−6 10−4 10−2 100 102 104

∆µP zz

/ µ 0

β

a)

0.16

0.22

100 102

|µP zz|/ µ 0

β-0.02

0

0.02

0.04

0.06

10−4 10−2 100 102

∆µP xz

/ µ 0

β

b)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

10−4 10−2 100 102

∆µP xx

/ µ 0

β

c)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

10−4 10−2 100 102

∆µP yy

/ µ 0

β

d)

Figure 3. (Color online) The scaled pair-mobility corrections versus the scaled frequency β. The two particles are locatedabove the membrane at z0 = 2a with a distance h = 4a. The real and imaginary parts of the mobility correction are shownas dashed and solid lines, respectively. The shearing and bending related parts are shown in green and red, respectively. Thehard-wall limits are shown as horizontal dashed lines. The inset in a) shows that the amplitude of the total pair-mobilitycomponent zz exceeds its bulk value (dotted line) in a small frequency range around β ∼ 1.

mobility results in a short-lasting superdiffusive behavioras will be described in Sec. V.

Secondly, for the components xx and xz in Fig. 3 wefind that, unlike the self-mobilities, shearing and bend-ing may have opposite contributions to the total pair-mobilities. For the xz component this implies the in-teresting behavior that hydrodynamic interactions canbe either attractive or repulsive depending on the mem-brane properties. This will be investigated in more detailin the next subsection.

C. Perpendicular steady motion

A situation in which hydrodynamic interactions areparticularly relevant is the steady approach of two par-ticles towards an interface, such as e.g. drug moleculesapproaching a cell membrane, reactant species approach-

ing a catalyst interface, charged colloids being attractedby an oppositely charged membrane, etc. For hard walls,it is known that hydrodynamic interactions in this caseare repulsive17,76,79 leading to the dispersion of particleson the surface. Near elastic membranes, the differentsigns of the bending and shear contributions to the pair-mobility in Fig. 3 b) point to a much more complex sce-nario including the possibility of particle attraction.The physical situation of two particles being initially

located at z = z0 and suddenly set into motion towardsthe interface is described by a Heaviside step functionforce F (t) = Aθ(t). Its Fourier transform to the fre-quency domain reads106

F (ω) =(πδ(ω)− i

ω

)A .

Using the general form of Eq. (21), the scaled particle

Hydrodynamic interaction near elastic interfaces 8

-1.4

-1.2

-1

-0.8

-0.6

-0.4

10−2 10−1 100 101 102 103

Vγz

/ µ 0A=Vλz

/ µ 0A

τ

a)

-0.04

0

0.04

0.08

0.12

10−2 10−1 100 101 102 103

( V λ x−Vγx

)/ µ 0A

τ

b)

Figure 4. (Color online) The scaled particle velocities per-pendicular to the membrane (a) and relative to each other (b)versus the scaled time for a constant force acting downward onboth particles near a membrane endowed with only shearing(green), only bending (red) or both rigidities (black). Solidlines are the analytical predictions as given by Eq. (23), sym-bols are obtained by boundary-integral simulations. Horizon-tal dotted and dashed lines stand for the bulk and vanishingfrequency limits respectively.

velocity in the temporal domain is then given by

V (τ)µ0A

=(b+

∫ ∞0

ϕ1(u)ϕ2(u)

(1− e−ϕ2(u)τ

)du)θ(τ) ,

(23)where τ := t/T is a dimensionless time. At larger times,the exponential in Eq. (23) can be neglected comparedto one. In this way, we recover the steady velocity neara hard-wall.

In corresponding BIM simulations, a constant force ofsmall amplitude towards the wall is applied on both par-ticles in order to retain the system symmetry. At the endof the simulations, the vertical position of the particleschanges by about 8 % compared to their initial positionsz0.In Fig. 4 a) we show the time dependence of the vertical

velocity which at first increases and then approaches itssteady-state value. Figure 4 b) shows the relative velocitybetween the two particles: clearly, the motion is attrac-tive for a membrane with negligible bending resistance(such as a typical artificial capsule) which is the oppo-site of the behavior near a membrane with only bendingresistance (such as a vesicle) or a hard wall.

In order to illustrate more clearly for which wall andparticle distances a repulsion/attraction is expected weshow in Fig. 5 the pair-mobility correction for the shear

Figure 5. (Color online) Contour diagram (ε, σ) of the shear-ing (a) and bending (b) contribution in the vanishing fre-quency limit in the xz pair-mobility as stated by Eq. (B1d)for shearing and by Eq. (B2d) for bending. c) is the samecontour near a hard-wall as given by Eq. (22d). The pertur-bation solution given by Eq. (24) is presented as circles in(a). Contrary to a membrane with bending resistance andto a hard wall, the mobility changes sign near a membranewith shearing resistance. This sign change directly reflectsthe physically observable situation as the bulk contributionfor the xz pair-mobility is zero.

∆µPxz,S and bending ∆µP

xz,B contributions in the (ε, σ)plane. To reduce the parameter space and to bring outthe considered effects most clearly, we consider the ideal-ized limit ω → 0. In this limit, the contributions becomeindependent of the elastic moduli since ω → 0 directlyimplies that β, βB → 0 meaning that even infinitesi-mally small shearing and bending resistances would makethe membrane behave identical to the hard wall. Thisunphysical behavior is remedied in a realistic situationwhere a small bending resistance will lead to a corre-spondingly large time scale TB and thus to a long-livedtransient regime as given by Eq. (23) and shown in theSupporting Information. Therefore, the contours shownin Fig. 5 faithfully represent the behavior of membraneswith small bending (Fig. 5 a)) or small shear (Fig. 5 b))resistance. The corresponding equations can be found inAppendix B.By equating Eq. (B1d) to zero and solving the result-

ing equation perturbatively, the threshold lines where theshearing contribution changes sign are given up to fifth

Hydrodynamic interaction near elastic interfaces 9

order in σ by

εth =√

2(σ − 4

3σ3 + 17

27σ5)

+O(σ7) . (24)

Eq. (24) is shown as circles in Fig. 5. The bending con-tribution in Fig. 5 b) always has a positive sign corre-sponding to a repulsive interaction similar as the hardwall.

Similar changes in sign are observed for ∆µPzz,S for

shear and ∆µPxx,B for bending. The corresponding con-

tours are given in the Supporting Information. Theirphysical relevance, however, is less important than for∆µP

xz shown in Fig. 5 as the effects may be overshad-owed by the bulk values of the mobilities (which is zeroonly for µP

xz).

V. DIFFUSION

The diffusive dynamics of a pair of Brownian particlesis governed by the generalized Langevin equation writtenfor each velocity component of particle γ as107

mdVγα

dt = −∫ t

−∞ζγγαβ(t− t′)Vγβ(t′)dt′

−∫ t

−∞ζγλαβ(t− t′)Vλβ(t′)dt′ + Fγα(t) .

(25)

A similar equation can be written for the velocity com-ponents of the other particle λ. Here, m denotes the par-ticles’ mass, ζγλαβ(t) stands for the time-dependent two-particle friction retardation tensor (expressed in kg/s2)and Fγα is a random force which is zero on average.By evaluating the Fourier transform of both membersin Eq. (25) and using the change of variables w = t − t′together with the shift property in the time domain ofFourier transforms we get

imωVγα(ω) + ζγγαβ [ω]Vγβ(ω) + ζγλαβ [ω]Vλβ(ω) = Fγα(ω) ,(26)

where ζγλαβ [ω] and ζγγαβ [ω] are the Fourier-Laplace trans-forms of the retardation function defined as

ζγλαβ [ω] :=∫ ∞

0ζγλαβ(t)e−iωtdt , (27)

and analogously for ζγγαβ [ω].In the following, we shall consider the overdamped

regime for which the particles are massless (m = 0). Solv-ing Eq. (26) for the particle velocities and equating withthe definition of the mobilities,

Vγα(ω) = µγγαβ(ω)Fγβ(ω) + µγλαβ(ω)Fλβ(ω) , (28a)

Vλα(ω) = µλλαβ(ω)Fλβ(ω) + µλγαβ(ω)Fγβ(ω) , (28b)

leads to expressions of the mobilities in terms of the fric-tion coefficients:

µSxx(ω) = ζS

xxζPzz

(ζSxx

2 − ζPxx

2)ζPzz − ζP

xxζPxz

2 ,

µPxx(ω) = − ζP

xxζPzz

(ζSxx

2 − ζPxx

2)ζPzz − ζP

xxζPxz

2 ,

µSyy(ω) =

ζSyy

ζSyy

2 − ζPyy

2 ,

µPyy(ω) = −

ζPyy

ζSyy

2 − ζPyy

2 .

µSzz(ω) = ζS

xxζSzz

(ζSzz

2 − ζPzz

2)ζSxx − ζS

zzζPxz

2 ,

µPzz(ω) = − ζS

xxζPzz

(ζSzz

2 − ζPzz

2)ζSxx − ζS

zzζPxz

2 ,

µPxz(ω) = − ζS

zzζPxz

(ζSzz

2 − ζPzz

2)ζSxx − ζS

zzζPxz

2 ,

where the brackets [ ] are dropped out for the sake ofclarity. Similar as for the mobilities, the self- and paircomponents of the retardation function are denoted byζγγαβ = ζλλαβ = ζS

αβ and ζγλαβ = ζλγβα = ζPαβ , respectively.

Note that ζPxxζ

Szz = ζS

xxζPzz so that µS

xz = 0 as required bysymmetry.According to the fluctuation-dissipation theorem, the

frictional and random forces are related via108[p. 33]109

〈Fγ(ω)Fλ(ω′)〉 = kBT(ζγλαβ [ω] + ζγλαβ [ω]

)δ(ω−ω′) , (30)

and analogously for the γγ component, where kB is theBoltzmann constant and T is the absolute temperatureof the system.110Multiplying Eq. (28a) by its complex conjugate, taking

the ensemble average and using Eq. (30), it can be shownthat the velocity power spectrum obeys the relation

PVSαβ(ω) = kBT

(µSαβ(ω) + µS

αβ(ω)). (31)

Next, by considering both Eqs. (28a) and (28b) togetherwith Eq. (30) we find in a similar fashion

PVPαβ(ω) = kBT

(µPαβ(ω) + µP

αβ(ω)). (32)

According to the Wiener-Khinchin-Einstein theorem,the velocity auto/cross-correlation functions can directlybe obtained from the temporal inverse Fourier transformas108

φγλαβ(t) = kBT

2π

∫ ∞−∞

(µγλαβ(ω) + µγλαβ(ω)

)eiωtdω . (33)

It can be shown using the residue theorem108 (p. 34)that the integral over the second term in Eq. (33) vanishesif the mobility is an analytic function for Im(ω) < 0. The

Hydrodynamic interaction near elastic interfaces 10

present mobilities all fulfill this condition as can be seenby their general form in Eq. (21).

Most commonly, diffusion is studied in terms of themean-square displacement (MSD) which can be calcu-lated from the correlation function as108 (p. 37)

〈∆rγα(t)∆rλβ(t)〉 = 2∫ t

0(t− s)φγλαβ(s)ds , (34)

where ∆rγα denotes the displacement of the particle γin the direction α. Furthermore, we define the time-dependent pair-diffusion tensor as

Dγλαβ(t) :=

〈∆rγα(t)∆rλβ(t)〉2t . (35)

Analogous relations to Eqs. (33)-(35) hold for the γγcomponent. We now consider the collective motions ofthe center of mass ρ := rλ + rγ as well as the relativemotion h := rλ − rγ with the corresponding diagonalpair-diffusion tensor

DC,Rαα = 2

(DSαα ±DP

αα

), (36)

where the positive sign applies for the collective mode ofmotion and the negative sign to the relative mode. Inthe absence of the membrane, Eqs. (36) reduces to thegeneralization of the Einstein relation as calculated byBatchelor25 for the relative mode, namely

DRzz

2D0= 1− 3

4σ −σ3

2 ,DRxx

2D0= 1− 3

2σ + σ3 , (37)

where D0 := µ0kBT is the diffusion coefficient. The col-lective diffusion coefficients read

DCzz

2D0= 1 + 3

4σ + σ3

2 ,DCxx

2D0= 1 + 3

2σ − σ3 , (38)

A. Self-diffusion for finite-sized particles

From Eqs. (33)-(35) we first obtain the scaled self-diffusion coefficient for the motion of a single particle

perpendicular to the membrane,

DSzz

D0= 1− 3

32τS(3B + 2τS)

(B + τS)2 ε+ τS

163τ2

S + 8BτS + 6B2

(B + τS)3 ε3

− τS

644τ3

S + 15Bτ2S + 20B2τS + 10B3

(B + τS)4 ε5

− ε

12

∫ ∞0

(3 + 3u− ε2u2)2

(1− 1− e−τBu

3

τBu3

)e−2udu ,

(39)

where τS := t/TS and τB := t/TB are dimensionless timesfor shearing and bending, respectively.For motion parallel to the membrane the scaled self-

diffusion coefficient reads

DSxx

D0= 1− 3

64

((2τS + 3B)(5τS + 4B)

(τS +B)2 − 4BτS

ln(

1 + τS

B

)− 16τS

ln(

1 + τS

2

))ε+ τS

32τ2

S + 3BτS + 3B2

(τS +B)3 ε3

− τS

1284τ3

S + 15Bτ2S + 20B2τS + 10B3

(τS +B)4 ε5

− ε

12

∫ ∞0

(3− ε2u

)2(u2 − 1− e−τBu

3

τBu

)e−2udu .

(40)

We mention that Eqs. (39) and (40) correspond to leadingorder in ε to the ones reported in our earlier work55. Forlong times, the perpendicular velocity auto-correlationfunction φS

zz,S decays as t−4 whereas the bending partφSzz,B as t−4/3. For parallel motion, both the shearing and

bending parts in the velocity auto-correlation functionhave a long-time tail of t−2.

B. Pair-diffusion for finite-sized particles

The pair-diffusion coefficients are readily obtained byplugging Eqs. (20a) through (20d) into Eqs. (33)-(35):

DPzz

D0= 3σ

4 + σ3

2 −σ

12ξ5/2

∫ ∞0

(uΛ2ΠS +

2Γ2−

u3 ΠB

)χ0e−2udu , (41a)

DPxx

D0= 3σ

2 − σ3 − σ

∫ ∞0

(1

24ξ5/2

(−ξ1/2χ1 + 2uχ0

) (Γ2

+ΠS + 2Λ2ΠB)

+ 3χ1

2 Π′S)e−2u

u2 du , (41b)

DPyy

D0= 3σ

4 + σ3

2 − σ∫ ∞

0

(χ1

24ξ2

(Γ2

+ΠS + 2Λ2ΠB)

+ 3Π′S2ξ1/2

(−ξ1/2χ1 + 2uχ0

)) e−2u

u2 du , (41c)

DPxz

D0= σ

12ξ5/2

∫ ∞0

(Γ+ΠS + 2Γ−ΠB

u2

)χ1Λe−2udu , (41d)

Hydrodynamic interaction near elastic interfaces 11

where we define

ΠS := Be−2uτSB + 2uτS −B

τS, Π′S := e−τSu + τSu− 1

τS, ΠB := e−τBu

3 + τBu3 − 1

τB.

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

10−4 10−2 100 102 104 106 108

DP zz

/ D 0

τ

σ = 13

σ = 14

σ = 15

σ = 17

σ = 19

σ = 111

Figure 6. (Color online) The zz component of the scaled pair-diffusion tensor versus the scaled time as given by Eq. (41a)for different values of σ with the parameters of Fig. 3. Hori-zontal dotted and dashed lines represent the bulk and hard-wall limits, respectively. For large inter-particle distances(small σ) a short superdiffusive regime is observed.

We observe that the xx, yy and zz cross-correlationfunctions have the same large time behavior as their cor-responding auto-correlation functions. For the compo-nent φP

xz, the shearing and bending related parts havelarge-time tails of t−4 and t−2, respectively.

Fig. 6 shows the variations of the zz component of thescaled pair-diffusion coefficient as stated by Eq. (41a)upon varying σ. We observe that as σ decreases, i.e.when the two particles stand further apart, the pair-diffusion coefficient can rise and exceed the bulk valuefor intermediate time scales as hinted on already by thepair-mobility around β ∼ 1 (cf. inset of Fig. 3 a). Suchbehavior is a clear signature of a short-lived superdiffu-sive regime.

In Fig. 7 we show the variations of the scaled collectiveand relative diffusion coefficients as defined by Eq. (36),versus the scaled time τ , using the parameters of Fig. 3.At shorter time scales, the particle pair exhibits a normalbulk diffusion, since the motion is hardly affected by thepresence of the membrane. As a result, the diffusion coef-ficients are the same as calculated by Batchelor and givenby Eq. (37). As the time increases, both diffusion coeffi-cients’ curves bend down substantially to asymptoticallyapproach the diffusion coefficients near a hard-wall.

0.5

0.6

0.7

0.8

0.9

10−4 10−2 100 102 104 106 108

DR αα

/ 2D0

τ

b)

0.4

0.6

0.8

1

1.2

1.4

10−4 10−2 100 102 104 106 108

DC αα

/ 2D0

τ

a)zzxxyy

Figure 7. (Color online) The scaled collective (a) and rel-ative (b) diffusion coefficients as defined by Eq. (36) versusthe scaled time. The horizontal dotted and dashed lines cor-respond to the bulk and hard-wall limits, respectively.

VI. CONCLUSIONS

We have investigated the hydrodynamic interaction ofa finite-size particle pair nearby an elastic membrane en-dowed with shear and bending rigidity. Using multipoleexpansions together with Faxén’s law, we have providedanalytical expressions for the frequency-dependent self-and pair-mobilities. We have demonstrated that shearingand bending contributions may give positive or negativecontributions to particle pair-mobilities depending on theinter-particle distance and the pair location above themembrane. Most prominently, we have found that twoparticles approaching a membrane with only shearing re-sistance (as is typically assumed for elastic capsules) mayexperience hydrodynamic attraction in contrast to thewell-known case of a hard wall where the interaction isrepulsive. This unexpected effect will facilitate chemicalreactions near the surface and may possibly even lead tothe formation of particle clusters near elastic membranes.On the other hand, membranes with bending resistance

Hydrodynamic interaction near elastic interfaces 12

(such as vesicles) induce repulsive interactions similar tothe hard wall. All our theoretical mobilities are validatedby detailed boundary integral simulations.

Using the frequency-dependent particle mobilities, wehave computed self- and pair-diffusion coefficients. Mostcommonly, relative and collective pair-diffusion is subdif-fusive on intermediate time scales similar to earlier obser-vations on the diffusion of a single particle55. A notableexception is the zz-component of the pair-mobility tensorwhich for certain parameters and frequencies surpassesits corresponding bulk value. This induces a short-lastingsuperdiffusive regime in the corresponding mean-square-displacement.

ACKNOWLEDGMENTS

The authors gratefully acknowledge funding fromthe Volkswagen Foundation as well as computing timegranted by the Leibniz-Rechenzentrum on SuperMUC.

Appendix A: Derivation of Green’s functions

In this appendix, we briefly sketch the derivation ofthe Green’s functions in the presence of an elastic mem-brane, as stated by Eqs. (12a) through (12d) of the maintext. For the solution of the steady Stokes equationsEqs. (10) and (11), we use a two-dimensional Fouriertransform technique. The variables x and y are trans-formed into the wavevector components qx and qy. Herewe use the convention with a negative exponent for theforward Fourier transforms. The transformed equationsread

−q2vx + vx,zz + iqxp+ Fxδ(z − z0) = 0 ,−q2vy + vy,zz + iqyp+ Fyδ(z − z0) = 0 ,−q2vz + vz,zz − p,z + Fzδ(z − z0) = 0 ,

−iqxvx − iqy vy + vz,z = 0 ,

where a comma in indices denotes the spatial derivativewith respect to the following coordinate.

We introduce a new orthonormal system in which theFourier transformed vectorial quantities are decomposedinto longitudinal, transverse and normal components, de-noted by vl, vt and vz respectively. The correspondingorthonormal in-plane unit vector basis are

ql := qxqex + qy

qey , qt := qy

qex −

qxqey , (A1)

where q :=√q2x + q2

y is the wavenumber. After transfor-

mation, the momentum equations become48

q2vt − vt,zz = Ftηδ(z − z0) , (A2a)

vz,zzzz − 2q2vz,zz + q4vz = q2Fzη

δ(z − z0)

+ iqFlηδ′(z − z0) , (A2b)

where δ′ is the derivative of the Dirac delta function. Thelongitudinal component vl is readily determined from vzvia the incompressibility equation (11) such that

vl = ivz,zq

. (A3)

According to the Skalak89 and Helfrich93 models, thelinearized tangential and normal traction jumps acrossthe membrane are related to the membrane displacementfield u at z = 0 by55

[σzα] = −κS

3(∆‖uα + (1 + 2C)e,α

), α ∈ {x, y} ,

(A4a)[σzz] = κB∆2

‖uz , (A4b)

where the notation [w] := w(0+)−w(0−) designates thejump of the quantity w across the membrane. Here C :=κA/κS is a dimensionless number representing the ratioof the area expansion modulus to shear modulus, and κBis the membrane bending modulus. ∆‖ := ∂,xx+∂,yy de-notes the Laplace-Beltrami operator along the membraneand e := ux,x+uy,y is the dilatation function, mathemat-ically defined as the trace of the in-plane strain tensor.The membrane displacement u as appearing in

Eqs. (A4a) and (A4b) is related to the fluid velocity bythe no-slip boundary condition at the undisplaced mem-brane which reads

vα = iωuα|z=0 . (A5)

After solving the transformed equations (A2a), (A2b)and (A3) and properly applying the boundary conditionsat the membrane, we find that the diagonal componentsof the Green’s function for z ≥ 0 read

Gzz = 14ηq

((1 + q|z − z0|) e−q|z−z0|

+(iαzz0q

3

1− iαq + iα3Bq

3(1 + qz)(1 + qz0)1− iα3

Bq3

)e−q(z+z0)

),

Gll = 14ηq

((1− q|z − z0|)e−q|z−z0|

+(iαq(1− qz0)(1− qz)

1− iαq + izz0α3Bq

5

1− iα3Bq

3

)e−q(z+z0)

),

Gtt = 12ηq

(e−q|z−z0| + iBαq

2− iBαq e−q(z+z0)

),

Hydrodynamic interaction near elastic interfaces 13

and the off-diagonal component Glz reads

Glz = i

4ηq

(− q(z − z0)e−q|z−z0|

+(iαz0q

2(1− qz)1− iαq − iα3

Bzq4(1 + qz0)

1− iα3Bq

3

)e−q(z+z0)

),

where α := κS/(3Bηω) is a characteristic length scale forshearing and area expansion with B := 2/(1 + C), andαB := (κB/(4ηω))1/3 is a characteristic length scale forbending. Furthermore, Gtz = Gzt = 0 because of the de-coupled nature of Eqs. (A2a) and (A2b). Employing thetransformation equations (A1) back to the usual Carte-sian basis, we obtainGxx(q, z, ω) = Gll(q, z, ω) cos2 φ+ Gtt(q, z, ω) sin2 φ ,

Gyy(q, z, ω) = Gll(q, z, ω) sin2 φ+ Gtt(q, z, ω) cos2 φ ,

Gxz(q, z, ω) = Glz(q, z, ω) cosφ ,where φ := arctan(qy/qx).

The components Gyz and Gzy are irrelevant for our dis-cussion because the resulting mobilities vanish, thus theyare omitted here. In addition, the component Gzx leadsto the same mobility as Gxz because of the symmetry ofthe mobility tensor. Furthermore, we define

G±(q, z, ω) := Gtt(q, z, ω)± Gll(q, z, ω) .Eqs. (12a)-(12d) of the main text follow immediatelyafter performing the two dimensional inverse spatialFourier transform of the Green’s function106.

Appendix B: Vanishing frequency behavior

In the following, analytical expressions of the shearingand bending related parts in the particle self- and pair-mobilities are provided in the vanishing frequency limit.

1. Self mobilities

By taking the vanishing frequencies limit in Eqs. (15a)and (15b), the shearing and bending related correctionsfor the perpendicular motion read

limβ→0

∆µSzz,S

µ0= − 3

16ε+ 316ε

3 − 116ε

5 ,

limβB→0

∆µSzz,B

µ0= −15

16ε+ 516ε

3 − 116ε

5 ,

leading to the hard-wall limit Eq. (16) after summingup both contributions term by term. Similarly, for theparallel motion, by taking the vanishing frequency limitin Eqs. (17a) and (17b) we get

limβ→0

∆µSxx,S

µ0= −15

32ε+ 132ε

3 − 132ε

5 ,

limβB→0

∆µSxx,B

µ0= − 3

32ε+ 332ε

3 − 132ε

5 ,

which also give the hard-wall limit Eq. (18) when sum-ming up both parts.

2. Pair mobilities

By considering independently the shearing and bend-ing related parts in the pair-mobility corrections as givenby Eqs. (20a) through (20d), and taking the vanishingfrequency limit, we obtain for the shearing part

limβ→0

∆µPzz,S

µ0= − 3

16ξ(2ξ − 1)(1 + ξ)5/2σ + 3

4ξ(2ξ − 3)(1 + ξ)7/2σ

3

−2ξ2 − 6ξ + 3

4(1 + ξ)9/2 σ5 , (B1a)

limβ→0

∆µPxx,S

µ0= − 3

165ξ2 + 10ξ + 8

(1 + ξ)5/2 σ + 14ξ2 − 10ξ + 4

(1 + ξ)7/2 σ3

−ξ2 − 27

4 ξ + 1(1 + ξ)9/2 σ5 , (B1b)

limβ→0

∆µPyy,S

µ0= − 3

165ξ + 4

(1 + ξ)3/2σ + 14

ξ − 2(1 + ξ)5/2σ

3

−ξ − 1

4(1 + ξ)7/2σ

5 , (B1c)

limβ→0

∆µPxz,S

µ0= 3

16(ξ − 2)ξ1/2

(1 + ξ)5/2 σ −34

(3ξ − 2)ξ1/2

(1 + ξ)7/2 σ3

+ 54

(4ξ − 3)ξ1/2

(1 + ξ)9/2 σ5 , (B1d)

and for the bending part

limβB→0

∆µPzz,B

µ0= − 3

1610ξ2 + 11ξ + 4

(1 + ξ)5/2 σ + 14

10ξ2 − 7ξ − 2(1 + ξ)7/2 σ3

−2ξ2 − 6ξ + 3

4(1 + ξ)9/2 σ5 , (B2a)

limβB→0

∆µPxx,B

µ0= − 3

16ξ(ξ − 2)

(1 + ξ)5/2σ + 34ξ(ξ − 4)

(1 + ξ)7/2σ3

−ξ2 − 27

4 ξ + 1(1 + ξ)9/2 σ5 , (B2b)

limβB→0

∆µPyy,B

µ0= − 3

16ξ

(1 + ξ)3/2σ + 34

ξ

(1 + ξ)5/2σ3

−ξ − 1

4(1 + ξ)7/2σ

5 , (B2c)

limβB→0

∆µPxz,B

µ0= 3

16(5ξ + 2)ξ1/2

(1 + ξ)5/2 σ − 154

ξ3/2

(1 + ξ)7/2σ3

+ 54

(4ξ − 3)ξ1/2

(1 + ξ)9/2 σ5 . (B2d)

The total correction as given by Eqs. (22a) through(22d) is recovered by summing up term by term bothcontributions.

Hydrodynamic interaction near elastic interfaces 14

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