This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.The purpose of this paper is to explore, from a theoretical viewpoint, the mechanisms whereby
locomotion of low-Reynolds-number organisms and particles is affected by the presence of nearby no-slip
surfaces and free capillary surfaces. First, we explore some simple models of the unsteady dynamics of low-
Reynolds-number swimmers near a no-slip wall and driven by an arbitrarily imposed tangential surface slip.
Next, the self-diffusiophoresis of a class of two-faced Janus particles propelled by the production of gradients in
the concentration of a solute diffusing into a surrounding fluid at zero Reynolds and P´eclet numbers is studied,
both in free space and near a no-slip wall. The added difficulty now is that the tangential slip is not arbitrarily
chosen but is given by the solution of a separate boundary value problem for the solute concentration. Finally,
an analysis of a model system is used to identify a mechanism whereby a non-self-propelling swimmer can
harness the effects of surface tension and deformability of a nearby free surface to propel itself along it. The
challenge here is that it is a free boundary problem requiring determination of the surface shape as part of the
solution

4th Micro and Nano Flows Conference (MNF2014)

Crowdy, D

English

Brunel University Research Archive

4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014Particle self-diffusiophoresis near solid walls and interfacesDarren G. CROWDYTel.: +44(0)2075948587; Fax: +44(0)2075948517; Email: d.crowdy@imperial.ac.ukDepartment of Mathematics, Imperial College London, UKAbstract The purpose of this paper is to explore, from a theoretical viewpoint, the mechanisms wherebylocomotion of low-Reynolds-number organisms and particles is affected by the presence of nearby no-slipsurfaces and free capillary surfaces. First, we explore some simple models of the unsteady dynamics of low-Reynolds-number swimmers near a no-slip wall and driven by an arbitrarily imposed tangential surface slip.Next, the self-diffusiophoresis of a class of two-faced Janus particles propelled by the production of gradients inthe concentration of a solute diffusing into a surrounding fluid at zero Reynolds and Pe´clet numbers is studied,both in free space and near a no-slip wall. The added difficulty now is that the tangential slip is not arbitrarilychosen but is given by the solution of a separate boundary value problem for the solute concentration. Finally,an analysis of a model system is used to identify a mechanism whereby a non-self-propelling swimmer canharness the effects of surface tension and deformability of a nearby free surface to propel itself along it. Thechallenge here is that it is a free boundary problem requiring determination of the surface shape as part of thesolution.Keywords: self-diffusiophoresis, Janus particle, no-slip wall1. IntroductionAn important technological challenge, fordrug-delivery systems and in micromechanics,is to develop ways to enable small-scale ob-jects to perform autonomous controlled motion.A promising route for producing such artificialmicroswimmers, or “nanomotors”, is to endowa particle of fixed shape with anisotropicallypatterned physico-chemical properties. Amongthe many examples are so-called Janus col-loids which are chemically reactive beads orrods consisting of one of more portions of theirboundaries, or “faces”, having different chem-ical, electrical or thermomechanical properties.These particles take advantage of self-phoreticeffects where gradients of solute concentration,electric or temperature fields interact with theparticle’s surface properties to create slip ve-locities that lead to net propulsion and rota-tion by virtue of the constraints that the par-ticle is free of any net force or torque. Onthe other hand, micro-organisms or biological“swimmers”, such as bacteria, can propel them-selves in a qualitatively similar manner by us-ing the concerted action of surface-based ciliato produce a net surface slip velocity (or a sur-face stress).2. Wall-bounded motionThere has been much recent interest in the ef-fect of no-slip walls on the dynamics of low-Reynolds-number organisms. For example, themotility of sperm has been of particular in-terest and many investigators have been inter-ested in assessing how the vicinity of nearbywalls can affect it. In this spirit, simple math-ematical models based on singularity theory forStokes flows have been devised to explain thedynamics – observed both experimentally andby means of numerical simulations – of a classof artificial “swimmers” made up of networksof rotating spheres connected by rigid rods.While it is common to model a low-Reynolds-number swimmer as a stresslet singularity, akey observation of the work of Crowdy & Or[Phys. Rev. E, 81, 036313, (2010)] is to iden-- 1 -4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014Ω UNo−slip wallfluid regionrd CDBA A′1ρζ−planeAA′B −1 −ρC DFigure 1: A circular swimmer translating withcomplex speed U and rotating with angular ve-locity Ω near a no-slip wall. The interior ofthe annulus ρ < |ζ | < 1 (shown right) is trans-planted, under the mapping z(ζ ) given in (5), tothe fluid region in the z-plane above the no-slipwall and outside the moving swimmer. Pointslabelled with the same letter correspond underthe mapping (5).tify the important role played by the irrotationalquadrupole singularity contribution. Althoughit is of higher order than the stresslet, and sogenerally expected to be less important, it isfound to play a decisive role in promoting lo-comotion along a wall. Indeed they show thatwhile a pure point stresslet will simply crashinto a wall, a superposition of a stresslet withan irrotational quadrupole can lead to nonlinearperiodic orbits involving net locomotion alongthe wall direction.The idea of posing a singularity distributioncomprising a stresslet-quadrupole combinationwas inspired by asking about the far-field ef-fective singularity distribution associated with apurely circular swimmer with a certain imposedtangential velocity distribution. This imposedtangential slip is intended to emulate the effectof the concerted motion of cilia on the swimmersurface. Crowdy & Or chose to study a par-ticular tangential slip velocity that would leadto no net locomotion of the swimmer if in freespace, away from any walls; that way, they iso-lated the net effect of the presence of the wallon the swimmer locomotion. Having found thefar-field singularity description associated withsuch a swimmer, they determined the motion ofsuch a stresslet-quadrupole singularity combi-nation near a wall and found excellent qualita-tive agreement with experiments and numericalsimulations.It was discovered, in later work [Crowdy, Int.J. Nonlin. Mech., 46, (2011)], that the prob-lem of a circular swimmer with an arbitrary im-posed tangential velocity can in fact be solvedin closed analytical form without any need toapproximate the swimmer by an effective sin-gularity description. This remarkable analyticaldiscovery resulted from combining two mathe-matical ideas: the reciprocal theorem of Stokesflow, and a known exact solution for the so-called “dragging problem” for a rigid circularcylinder near a wall. These new analytical in-sights have prompted further investigations intohow more complex swimmer types evolve neara nearby no-slip wall.3. Mathematical resultsA key ingredient for our analysis is a com-plex variable formulation, and use of confor-mal mapping techniques. These mathematicalideas are largely ignored by the low-Reynolds-number community, but they are neverthelessrelevant to two-dimensional Stokes flow mod-elling and have enormous technical advantages.In two dimensions it is known that an incom-pressible Stokes flow is describable in terms ofa streamfunction ψ which satisfies the bihar-monic equation∇4ψ = 0. (1)Complex analysis enters the analysis on notic-ing that the general solution of (1) can be writ-- 2 -4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014ten asψ = Im[z f (z)+g(z)], (2)where f (z) and g(z) are two functions that areanalytic functions of z = x+ iy in the fluid re-gion. These two so-called Goursat functions aredetermined by the boundary conditions on theflow. For a solid wall, these are no-slip condi-tions taking the form− f (z)+ z f ′(z)+g′(z) = 0, on the wall. (3)The usual fundamental singularities of Stokesflows, such as the stokeslet, stresslets androtlets, now manifest themselves as isolated sin-gularities of these two analytic functions. Forexample, as shown by Crowdy & Or , a pointstresslet of complex strength λ , say, at a pointz0 requires that, near this point, f (z) and g′(z)have the local formf (z) =λz− z0 + locally analytic,g′(z) =λ z0(z− z0)2 + locally analytic.(4)Conformal mapping theory can also aid theanalysis. For motion near an infinite straightwall it is convenient to consider a preimage do-main comprising the concentric annulus ρ <|ζ | < 1 in a parametric ζ -domain. The confor-mal mapping of this region to the doubly con-nected fluid region exterior to a circular swim-mer near to – but not touching – a plane wall isgiven by the linear-fractional mappingz(ζ ) = iR[ζ +1ζ −1](5)where both parameters R and ρ depend on theradius of the swimmer s and its distance d fromthe wall according toρ =ds−√(ds)2−1,R= d[ρ2−1ρ2+1].(6)Figure 1 shows a schematic illustrating the fluiddomain and the preimage ζ annulus.Suppose a smooth tangential velocity profileimposed on the boundary of the particle (i.e. on|ζ |= ρ) has the formbnζ n+bnρ2nζ n, n≥ 0 (7)for some complex coefficient bn. This quantityis real on |ζ |= ρ and it is important to note thatany smooth imposed tangential velocity fieldcan be represented as an infinite sum of suchterms – the result is just the Laurent expansionof the velocity profile valid on |ζ |= ρ . On useof the reciprocal theorem, and computation ofthe resulting integrals by means of the residuetheorem, we find the remarkable result that(U ′,V ′,Ω′) =(0,0,−b0/s), (n= 0)(ρRe[b1],−ρ(1−ρ2)(1+ρ2)Im[b1],− 2ρ2s(1+ρ2)Re[b1]),(n= 1)(0,0,0), (n> 1).(8)Therefore, only the two modes with n = 0 and1 of the tangential velocity profile, expressed asa Laurent expansion in ζ , lead to any nontrivialparticle velocities U ′,V ′ and Ω′ with the n = 0term serving only to alter the particle’s angularvelocity. We believe this observation is signifi-cant. Mathematically it means that, for an arbi-trary imposed tangential slip, only its projectiononto these two modes are relevant to its locomo-tive properties.4. Janus particle modelsWe now give details of a theoretical inves-tigation of the self-diffusiophoresis of a classof two-faced, two-dimensional Janus particlespropelled by the production of gradients in theconcentration of a solute diffusing into a sur-rounding fluid at zero Reynolds and Pclet num-bers. Those concentration gradients produce atangential boundary slip resulting in translationand rotation of the particle, as a consequence of- 3 -4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014the fact that it is free of both force and torque.Mathematically, the additional complication isthat it is now necessary to simultaneously solvea boundary value problem for the solute con-centration and then couple it to the Stokes flowproblem.Our model Janus particle as an isolated two-faced circular particle which is a zero net sourceof solute; that solute diffuses around the particleat zero Pe´clet number with diffusion coefficientD. Let the solute concentration exterior to theparticle be denoted by c(x,y). In the zero Pe´cletnumber limit the boundary value problem forc(x,y) is to solve Laplace’s equation∇2c= 0 (9)outside the particle with the Neumann boundarycondition−D n.∇c= A, (10)where D is the diffusion coefficient and A isthe local surface activity. Positive values ofsurface activity A correspond to solute emittingsurfaces. We assume that there is no solute inthe far-field. The particle is situated in fluid ofviscosity µ assumed to be in the zero Reynoldsnumber re´gime. Once the solute concentrationhas been found from the above boundary valueproblem the local phoretic slip velocity us onthe boundary isus =M(I−nn).∇c, (11)where M is the surface mobility, and it is thissurface velocity that will drive a local Stokesflow around the particle. We therefore under-stand that, unlike the swimmer problems con-sidered previously where the tangential surfaceslip was simply imposed arbitrarily, now thetangential slip is given by the solution of an-other boundary value problem.We make the special choice of piecewise con-stant surface activity A and mobility M given by(A,M) ={(A1,M1), on C1,(A2,M2), on C2,(12)where Ai,Mi for i = 1,2 are constants; the por-tion of the boundary with arg[z] ∈ [−θ ,θ ] is de-noted by C1 while the remainder of the bound-ary is C2 as shown in Figure 2.A natural first step is to consider such a swim-mer in isolation (away from any walls). Weanalyze this case and show that if the particleis actuated by a ratio r of its surface havinguniform surface mobility M1 and surface activ-ity A1, with the remainder having mobility M2,then the (reduced) particle speed isUU0=−sinpirpi[r+λ (1− r)1− r], (13)whereU0 =A1M1D, λ =M2M1. (14)The direction of travel is shown in Figure 2.Notice already that the locomotion speed is acomplicated nonlinear function of the ratio r.The mathematical details here are complicatedby the presence of discontinuities in the sur-face properties, and these have to be properlyaccounted for. But, again, complex variablemethods can be used to find the explicit resultsabove.Confinement effects are then investigated byplacing the same Janus particle just describednear a straight no-slip wall. The mathematicaldifficulties associated with discontinuities in thesurface properties have to be accounted for, but,in combination with the result (8) cited above,it has been found that the governing nonlineardynamical system can be established in com-pletely explicit form. The resulting nonlinearsystem is used to study how the geometry, loca-tion and orientation of the particle relative to thewall affect its motion. It is found that if the par-ticles do not hit the wall in finite time they areeventually repelled away from it. In particular,no steadily translating or periodic orbits alongthe wall could be found suggesting that it is un-likely that such swimmers will be able to movealong the wall.D.G. Crowdy, Wall effects on self-diffusiophoretic Janus particles: a theoreticalstudy, J. Fluid Mech., 735, 473-498, (2013).5. Motion near a free surfaceOther types of low-Reynolds-number swim-mers have been found to locomote parallel to- 4 -4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014βθU(A1, M1)(A2, M2)Figure 2: A circular Janus particle C1 occupy-ing r= θ/pi of the surface andC2 (the thick lineportion) occupying the remainder. Its locomo-tion speed is given in (13) as a function of r andsurface mobility ratio λ .free surfaces. What is surprising here is that,unlike no-slip surfaces, free surfaces cannotsupport any shear stress so the way in whichan organism makes use of the nearby surfaceto assist its locomotion must be fundamentallydifferent. We have attempted to investigate thisusing the same two-dimensional models alreadydiscussed in the context of no-slip walls.The analysis is, however, significantly morecomplicated owing to the deformation of thefree surface: it is a free boundary problem re-quiring that the shape of the free surface be de-termined as part of the solution. Given this dif-ficulty we have only investigated the existenceof possible steady states in which the swimmeris travelling uniformly at constant speed in thedirection parallel to the undisturbed free sur-face. Moreover, given these complications withthe free surface, it is natural to resort back to asingularity model of the swimmer comprisinga stresslet and superposed quadrupole (ratherthan thinking of a circular swimmer with an im-posed tangential slip).Similar mathematical techniques based oncomplex analysis are useful here too. By adapt-ing mathematical techniques used by Jeong &Moffatt [J. Fluid Mech., 241, (1992)] to studyfree surface cusps induced by counter-rotatingrollers, we have been able to analyze this prob-lem and find the required steadily translatingfluidairinterfacezd fl idint faceair1zdsFigure 3: Model of a swimmer, as a point singu-larity at zd , beneath a deformable free surface.state in terms of closed form formulas. Themethods again rest on strategic use of a complexvariable formulation of the Stokes flow prob-lem with conformal mapping theory used to de-scribe deformation of the free surface. Owing tothe fact that the swimmer is modelled as a sin-gularity at a single point, without any spatial ex-tent, the fluid region is now simply connected,implying that the required conformal mappingfunction can be taken from the unit disc, ratherthan the doubly connected annulus (i.e. thisis the limit ρ → 0 for the preimage ζ annulusshown in Figure 1). This affords a major sim-plication since now the flow is bounded only bythe free surface with a point singularity sittingin the fluid region at some point zd beneath it.Figure 3 shows a schematic.A non-zero surface tension T is taken to beactive on the free surface and, in this case, thestatement of the stress balance on the free sur-face takes the modified formf (z)+ z f ′(z)+g′(z) =iT2dzds, on the surface,(15)where s denotes arclength along the free sur-face as shown in Figure 3. This relation de-termines the unknown Goursat functions f (z)and g(z). In this case, a non-dimensional capil-lary number Ca can be defined that governs thebalance between surface tension and viscous ef-fects generated by the local motion induced bythe swimmer. By virtue of an asymptotic analy-sis of the system as Ca→ 0, a new mechanismof locomotion is identified. Its origin is a sub-tle balance of surface tension and free surface- 5 -4th Micro and Nano Flows ConferenceUCL, London, UK, 7-10 September 2014deformation.Full details of this analysis have been publishedin:D.G. Crowdy, S. Lee, O. Samson, E. Lauga andA.E. Hosoi, A two-dimensional model of low-Reynolds number swimming beneath a free sur-face, J. Fluid. Mech., 681, 24-47, (2011).- 6 -

Particle self-diffusiophoresis near solid walls and interfaces

http://bura.brunel.ac.uk/bitstream/2438/9464/1/MNL-submitted.pdf

Particle self-diffusiophoresis near solid walls and interfaces

Abstract

Similar works

Full text

Available Versions

Brunel University Research Archive