Table Of Content7
Designing phoretic micro- and nano-swimmers
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R. Golestanian1∗, T.B. Liverpool2, A. Ajdari3,4†
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a 1 Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK
J
2 Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
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3 Physico-Chimie Th´eorique, UMR CNRS 7083, ESPCI, 10 rue Vauquelin, 75005 Paris, France
] 4 DEAS, Harvard University, 29 Oxford Street, Cambridge MA 02138, U.S.A
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February 6, 2008
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t
a
m Abstract
sis, [2]) are from this standpoint a natural avenue
- given the increased surface to volume ratio.
d
n Small objects can swim by generating around them Very recently, in line with earlier suggestions of
o fields or gradients which in turn induce fluid mo- self-electrophoresis of objects that would be able to
c tion past their surface by phoretic surface effects. generate an electric field around them [3, 2, 4] and
[
We quantify for arbitrary swimmer shapes and sur- our recent proposal of self-diffusiophoresis [5], many
1 face patterns, how efficient swimming requires both experimentalreportshaveappearedofheterogeneous
v surface “activity” to generate the fields, and surface objects (e.g. rods of hundreds of nanometers) swim-
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“phoretic mobility.” We show in particular that (i) ming using different catalytic or reactive chemistry
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swimming requires symmetry breaking in either or at their different ends [6, 9, 7, 8]. Although many
1
1 bothofthepatternsof”activity”and“mobility,”and mechanisms may contribute to the observed motion
0 (ii)foragivengeometricalshapeandsurfacepattern, [6], phoretic mechanisms have been shown to be es-
7 the swimming velocity is size-independent. In addi- sential for some systems [10].
0 tion,forgivenavailablesurfaceproperties,ourcalcu- In this Letter we consider generally self-phoretic
/
t lation frameworkprovides a guide for optimizing the motion, where the swimmer, generates through its
a
m design of swimmers. surface activity gradients of (at least) a quantity
(concentration of a dissolved species, electric po-
-
d tential, temperature), which in turn induce motion
Introduction. In the current miniaturization race
n through classical interfacial phoretic processes. Be-
o towardssmallmotorsandengines, a rapidlyexpand-
causeitisnowpossibletofabricatesuchobjectswith
c ing subdomain is the quest for autonomous swim-
: mers, able to move in fluids which appear very vis- controlled shape and patterns of surface properties
v
[6,7,11],we discusshow these designparametersaf-
i cousgiventhesmalllengthscales(lowReynoldsnum-
X fect the swimmer’s velocity. A general framework is
ber). Robotic microswimmers that generate surface
r distortions is an avenue (e.g. by mimicking sperms proposedtocomputethisvelocityforarbitraryswim-
a
mer size, shape and surface pattern. We then work
[1]), but it seems equally interesting to try to take
out explicitly a few examples for spheres and rods,
advantage of physical phenomena that become pre-
geometries amenable to analytical calculations, and
dominant at small scales. Interfacial ”phoretic” ef-
corresponding to existing means of fabrication [6, 7].
fects (electrophoresis, thermophoresis, diffusiophore-
From these examples we extract generic rules as to
∗r.golestanian@sheffield.ac.uk theroleofthepatternsymmetry,oftheswimmersize
†[email protected] and shape, before concluding with a few remarks.
1
General formalism. We start with a reminder of the surface of the object, and diffusing in the bulk.
what phoretic mechanisms are [2]. Consider e.g. a At steady-state, in the reference frame of the object,
colloid in a solution where the concentration of a so- andneglectingdistortionsinducedbythe flow(small
lutecexhibitsagradient. Thecolloidisthensetinto Pecletnumber),thesoluteconcentrationintheliquid
motion due to the induced imbalance of osmotic ef- is given by:
fects within the solid/liquid interfacial structure at D∇2c=0, (3)
the surface of the object. When the thickness of the
−Dn·∇c(r )=α(r ), (4)
interfacial layer is thin compared to the object, the s s
resulting flow is most conveniently described by an where D is the diffusion coefficient, and α(r ) mea-
s
effectiveslipvelocityoftheliquidpastthesolid,pro- suresthe “surfaceactivity”atpositionr onthe sur-
s
portional to the local gradient of c [2]: face, i.e. the generation or consumption of solute by
a chemical reaction. In general, describing this pro-
vs(rs)=µ(rs)(I−nn)·∇c(rs) (1) cess involves additional coupled transport problems
for other species involved in the surface reactions.
where n is the local normal to the surface, I is the
For simplicity, we nevertheless proceed with a fixed
identity tensor. µ(r ) is the local ”surface phoretic
s local value of α. This should well describe situations
mobility”,positiveornegativedependingonspecifics
where the solventcontains anexcess of the reactants
of the solute/surface interactions [2].
necessary to produce and destroy the solute of inter-
This distribution of slip on the object’s surface
est, so that the density of active sites on the surface
induces a net drift velocity V, which can be com-
is whatlimits the fluxes. α is then this density times
putedusingthereciprocaltheoremforlowReynolds-
the rate of the chemical reaction per site [5].
numberhydrodynamics[12]. Itscomponentsinaref-
Forself-electrophoresis,similarformulaeholdupon
erential (eˆ ) are:
i replacing c by φ and D by the solution electrical
el
conductivity. α isthen the electricalcurrentinjected
V·eˆi =− drs n·σi·vs (2) by the surface. Existing realizations [9, 10], corre-
ZZS
spondtononetcurrentgeneration dr α(r )=0,
S s s
where σi is the hydrodynamic stress tensor at the with the ions produced on one sideRRof the swimmer
surface of an object of similar shape dragged by an and consumed on the other one, while electrons are
applied unit force exactly equal to eˆ , in the absence transported through the body of the swimmer. Sim-
i
of slip. ilarly, for thermophoresis, beyond replacing c by T
Similar principles describe the motion of the ob- and D by the thermal conductivity, the boundary
ject in a gradient of electric potential (electrophore- condition (4) must be adjusted to account for heat
sis), or in a temperature gradient (thermophoresis). transport through the object.
The above formalism can be applied upon simple re- The general formal description (1-4) predicts a
placement of c by the potential φ or by the tem- swimming velocity V ∼ αµ/D, linear in the surface
el
perature T. For electrophoresis µ is up to a con- propertiesαandµ,andinverselyproportionaltothe
stant the so-called zeta-potential [2]. For all these mediumconductivityD. Moreinterestingly,oncethe
situations, achieving large values of µ requires en- patternofsurfacepropertiesandtheobjectshapeare
gineering of the surface in relation to the solvent given, V is independent of the size R of the swim-
(e.g. nanometer scale roughness and solvophibicity mer. Thiscentralpointemphasizesthe robustnessof
do matter [13, 14]). this swimming strategy against downsizing. In con-
We now focus on objects that generate the gradi- trast, anexternalbody forceonthe objectscalingas
ents themselves, e.g. using the chemical free-energy F ∼R3 (dielectrophoresis,magnetophoresis)leadsin
available in the solvent [6, 7]. Returning to self- this viscous realm to velocities ∼R2, which decrease
diffusiophoresis, the simplest situation corresponds rapidly with length scale. Further, our approach (1-
to a solute generated by active sources and sinks on 4)allowsustorelatequantitativelythevelocityV to
2
the design of the swimmer (shape, surface patterns (a) z (b) (c)
of α and µ), as we show analytically for spheres and (cid:68)> 0 (cid:68)= 0 (cid:68)> 0
rods. (cid:80)(cid:3)> 0 (cid:80)> 0 (cid:80)> 0
(cid:68)> 0 (cid:68)= 0 (cid:84)
Spheres. For a sphere of radius R, as n·σi = (cid:68)(cid:80)>< 00 (cid:80)(cid:68)== 00 (cid:80)(cid:19)> 0(cid:68) la<r g0e 0
(1/4πR2)eˆ , equation (2) becomes (cid:80)< 0 (cid:80)> 0
i
1
V=− dr µ(r )(I−nn)·∇c(r ), (5)
4πR2 ZZS s s s Figure 1: Three sphericalswimmers described in the
Wefocusfurtheronazimuthallysymmetricpatterns, text. The flux of c (opposite to the gradient) is de-
with surface quantities depending only on the “lat- pictedwiththedashedarrows,whiletheresultingslip
itude” angle θ with the polar axis eˆ . We ex- velocityoftheliquidalongthesphereisdescribedby
z
pand in Legendre polynomials. The surface activity openarrows. Thenetswimmingdirectionisgivenby
∞
α(θ) = α P (cosθ), generates a concentration the thick dark arrow. (a) a Janus swimmer, with
ℓ=0 ℓ ℓ
variationPgiven by Eqs. (3) and (4): a homogeneous µ and a broken top-down symme-
try in α: one hemisphere produces c while the other
∞ ℓ+1
R α R
ℓ consumes it. (b) a Saturn swimmer with an equa-
c(r,θ)=c∞+ Pℓ(cosθ), (6)
D ℓ+1(cid:18)r(cid:19) torial belt producing c, and a broken symmetry in
Xℓ=0
the phoretic mobility between the two hemispheres
where c∞ is the concentration at infinity. For a sur- (here even opposite signs), which induce swimming
face mobility µ(θ) = µ P (cosθ), the corresponding
ℓ ℓ in the same direction from the (opposite) equator
gradient yields a swimming velocity (5):
to poles gradients. (c) a three-slice design in with
∞ polar slices efficient in producing and consuming c,
V=−eˆz ℓ+1 αℓ+1 µℓ − µℓ+2 . while the equatorial slice is chosen for its large mo-
D (cid:18)2ℓ+3(cid:19) (cid:20)2ℓ+1 2ℓ+5(cid:21)
Xℓ=0 bility. Properlychoosing θ0 permits maximizationof
(7) the swimming velocity (Fig. 2)
As a first check, we recover [5] that a single ac-
tive site at the pole α = τ−1δ2D(rs) of a sphere of
uniform mobility µ yields V = µ . By linear- This shows that symmetry breaking in the chemical
4πR2Dτ
ity, an emitter at the back pole with rate 1/τ and a activity (α− 6= α+) is essential to propulsion, and
e
that the swimming velocity is larger if the two mo-
consumer at the front pole with rate 1/τ , results in
c
bilitiesareofthesamesign(whichdictatestheswim-
V = µ 1 + 1 .
4πR2D (cid:16)τe τc(cid:17) ming direction).
Toaccessusefulgeneralprincipleswenowfocuson
The second design is a “Saturn” particle, where
three simple designs (Figure 1), within reach of cur-
the surface activity is concentratedaroundthe equa-
rentfabricationtechniques[11]. ThefirstistheJanus
tor, and where symmetry is now broken by choos-
sphere,withtwohemispherescoveredwithadifferent
inghemispheresofdifferentmobilities. Forcomputa-
enzymatic or catalytic material. Generically, their
tional convenience we solve the very similar problem
mobilities µ also differ, so we take:
defined by:
(α+,µ+), 0<θ < π2, α(θ)=α0(1−cos2θ), µ(θ)=µ0cosθ, (10)
(α(θ),µ(θ))= (8)
(α−,µ−), π2 <θ <π, and find 4 µ0α0
and find analytically V= 45 D eˆz. (11)
So symmetry breaking in mobility alone also leads
1 1
V= (α−−α+)(µ++µ−)eˆz. (9) to swimming: the activity at the equator generates
8 D
3
gradients of c towards the poles that generate slip |V|/(αµ0/D)
velocities in the samedirection alongz as the signof
2 Θ0(cid:144)Π
µ is opposite on the two hemispheres (Figure 1b).
0.5
The third design is that of a three-slice sphere,
1.5 0.4
thatillustratestypicalconsiderationsthatarisewhen 0.3
trying to increase by design the swimming velocity. 0.2
Fromthescalinganalysisaboveoneshouldobviously 1 0.1
aim for surfaces with both large values of α and µ. 0.5 1 1.5 2 Μ(cid:144)Μ0
However, in many cases these goals will not be met 0.5
by a single surface, so one may instead look for ways
tocombineinanefficientdesignsurfaceswithlargeα θ /π
0
and surfaces with large µ. As an example, let us try 0.1 0.2 0.3 0.4 0.5
toupgradetheJanusdesignofFigure1abyaddinga
passivebeltoflargephoreticmobilityµ0,whichleads Figure 2: Reduced swimming velocity V/(αµ0/D) of
to the swimmer of Figure 1c: the three-sliceswimmerofFigure1c,asafunctionof
”polar” angle θ0 for various values of µ/µ0: 0, 1, 4
(α+,µ+), 0<θ <θ+, (frombottomtotop). Theoptimalvalueatlowµ/µ0
reflects the compromise between large poles to have
(α(θ),µ(θ))= (0,µ0), θ+ <θ <π−θ−, sfotrrotnhgescehgermaidciaelngtsratdoiternatns,slaantdeiantlaorlgaergeequflaotwormiaoltbioenlt.
(α−,µ−), π−θ− <θ <π,(12) Inset: optimal angle θ0 against µ/µ0, .
To reach the largest swimming velocity for given
ming velocity along z:
surface properties, one can adjust the values of θ+
and θ− in a compromise between increasing the mo- 1 L
bility by enlarging the belt without shrinking too V ≃− dz µ(z)∂zcs, (13)
2LZ−L
much the poles that generate the gradient. Figure
2 shows results for the model case α+ = −α− = α, In the same limit, the surface concentration cs is re-
µ+ = µ− = µ of same sign than µ0, θ+ = θ− = θ0: lated to the surface activity α(z) by [6]
for µ0 > µ there is a value of θ0 that maximizes the b L α(z′)
speed,whichisalwayslargerthanπ/3,i.e. nomatter ′
c (z)≃ dz . (14)
howlargeµ0,the ”optimal”beltshouldnotbe made s 2D Z−L |z−z′|
largerthan a fixed value. The relativeflatness of the
Thesetwoequationsgivethevelocityofarodforany
bottom curve in Figure 2 shows a moderate depen-
”bar-coded”surface properties α and µ. For a Janus
dence onθ0 aroundthis value, so that the swimming rod
velocityshouldberatherrobusttosmallimprecisions
in the fabrication. (α−,µ−), −L<z <0,
(α(z),µ(z)) = (15)
(α+,µ+), 0<z <L,
Thin rods. Because of the many experimental re-
alizations of such systems [6, 7], we discuss the case we find
oflongcylindricalrods(length2Landradiusb≪L) 1 b L
with surface properties varying only along their axis V = ln (µ+α−−µ−α+) (16)
4D (cid:18)L(cid:19) (cid:18)4b(cid:19)
−L ≤ z ≤ L, i.e. µ = µ(z),α = α(z). In the limit
L≫bslender body theory canbe used,andremark- which illustrates a few important points: (i) again
ably equation (2) yields a simple form for the swim- swimming is obtained by symmetry breaking in the
4
∞
chemicalactivityor/andinthephoreticmobility,(ii) c the reaction becomes diffusion limited so that
fuel
swimming with uniform activity (α =constant) is α depends on the global geometry.
possible - in contrast to the special case of spheres - Due to thermal rotational diffusion, if no exter-
for which this leads to no tangential gradients of c nal field orients the objects, the direction of mo-
-, (iii) the velocity scale is decreased by a factor of tion persists only for a time τ ≃ 1/D , where
rot
∼ b/L when compared to a sphere of radius ∼ L, D ≃ k T/ηR3 and R is the largest dimension
rot B
because the surface generated flux of c is reduced of the object. Note however that the field c typ-
by a similar factor, and (iv) V is up to logarithms ically adjusts much faster to a change of orienta-
proportional to 1/L at fixed rod diameter, but scale tion (R2/D ≪ 1/D ), so our steady-state esti-
rot
independent if length and diameter are varied in the mates still hold for the instantaneous velocity. A
same proportion. simpleLangevindynamicswithadiffusioncoefficient
D ∼ k T/ηR and drift at V along a stochasti-
trans B
cally changing direction (D = 1/τ) leads to diffu-
rot
Discussion. The swimming velocity V of a sivebehaviouratlongtimes(t≫τ)withanactivity-
phoretic swimmer, is independent of its size and enhanceddiffusioncoefficientD =D +1V2τ.
eff trans 6
scales as V ∼ αµ/D, where α and µ are some aver- Thisenhancementmaybesignificantandmeasurable
ages quantifying the surface activity generating the for not too small swimmers, i.e. R > (k T/ηV)1/2,
B
gradientandthesurfacephoreticmobility. Theexact oforder∼afew100nmforV ∼10µm.s−1 inwater.
value of V depends onthe shape andsurface pattern For nano-rods of radius b fixed due to fabrication
in a computable way. For example, spheres swim constraints [6], the scaling of the previous quantities
faster than rods of same longest dimension, and im- become (using the shorthandV0 =α/µD andforget-
provement can be obtained by properly combining ting logarithms) V ≃ V0(b/L), τ ∼ ηL3/kBT, and
various types of surfaces. an activity-induced enhancement of the bare trans-
- This type of formalism holds for self- lational coefficient D ∼ k T/ηL by a quantity
trans B
diffusiophoresis (with α the surface reaction density, ∼(ηV02/kBT)b2L, significant if L/b>(kBT/ηV0b2).
µ the diffusiophoretic mobility, and D the solute - Beyond the “linear” swimmers described here,
diffusion coefficient), for self-electrophoresis (with α our approach can also describe active “spinners,”
the electricalcurrentdensityinjectedinthe solution, whichrotateusingthesamemechanisms[6,7]. Their
µ the electrophoretic mobility, and D the solution angularvelocitycanbecomputedbyreplacing(4)by
electrical conductivity), and for self-thermophoresis theformulaforangularswimmingspeedgiveninRef.
(withαthe surfaceheatfluxinjectedinthe solution, [12], and scales as Ω ∼ αµ/DR, decreasing with ob-
µ the thermophoretic mobility, and D the solution ject size for a given shape and geometrical surface
thermal conductivity). As for motion driven by ex- pattern.
ternal gradients, the largest velocities are expected - There is a growing interest in the collective be-
for highly charged surfaces at low ionic strengths, haviourofswimmingorganisms(seee.g. [15]andref-
which yield large mobilities µ because of the thick erencestherein). Phoreticswimmersarespecialwith
interfacial electrical double-layer. regards to these questions as they interact through
- The present line of reasoning can be adapted to both their hydrodynamic fields (decaying as the in-
more accurate descriptions of surface chemistry for verse of the cube of their relative distance r−3) [2]
specificexperimentalconditions. Whatisrequiredin and the c-field gradients they generate (which decay
a given situation is an appropriate scheme for com- as r−2 for a net production/absorption of c per ob-
puting α(r ). For example if c is produced at the ject, and as r−3 otherwise).
s
∞
surfacefromsomechemicalatconcentrationc far In conclusion,we have providedgeneric considera-
fuel
away from the object, various regimes show up. We tions for the design of small phoretic swimmers (size
have focused on kinetics limited by the number of (in)dependence of the swimming velocity, necessary
catalytic sites on the surface, but for low values of symmetry breaking in the surface pattern), as well
5
as quantitative procedures to estimate the influence [12] H.A. Stone, A. Samuel, Phys. Rev. Lett. 77,
of their velocity. Beyond these considerations, fast 4102 (1996).
motion of course relies on clever surface engineer-
[13] Y.W. Kim, R.R. Netz, Europhys.Lett., 72, 837
ing to obtain large surface “activity” and “phoretic
(2005).
mobility.” We hope that our calculations will stim-
ulate experimental studies of swimmers with given
[14] A. Ajdari, L. Bocquet, Phys. Rev. Lett. 96,
surface chemistries but various surface patterns, e.g.
186102 (2006).
nanorods [6, 7] and spheres [11, 16], keeping in mind
that other effects (e.g. bubble generation [7]) may [15] J. Hernandez-Ortiz,C.G. Stoltz, M.D. Graham,
compete with the ones described here. Phys. Rev. Lett. 95, 204501(2005).
[16] J.R. Howse et al. preprint 2006.
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