Table Of ContentLeidenforst gas ratchets driven by thermal creep
Alois Wu¨rger
Laboratoire Ondes et Mati`ere d’Aquitaine, Universit´e de Bordeaux & CNRS,
351 cours de la Lib´eration, 33405 Talence, France
WeshowthatthermalcreepisattheoriginoftherecentlydiscoveredLeidenfrostratchet,where
liquiddropletsfloatonavaporlayeralongaheatedsaw-toothsurfaceandacceleratetovelocitiesof
up to 40 cm/s. As the active element, the asymmetric temperature profile at each ratchet summit
rectifies the vapor flow in the boundary layer. This mechanism works at low Reynolds number and
provides a novel tool for controlling gas flow at nanostructured surfaces.
PACS numbers: 47.61.-k, 47.15.Rq, 44.20.+b, 68.03.-g
PACSnumbers:
4
1
0 Liquid spilled on a hot surface rapidly evaporates. At
2
theLeidenfrosttemperaturewellabovetheboilingpoint,
n however, one observes long-lived droplets that levitate
a due to the excess pressure resulting from the permanent
J
feed of vapor at the bottom. Their contact-free suspen-
9
sion makes such Leidenfrost droplets very mobile. Linke
2
et al. observed that, when placed on a millimeter-sized
brassratchet,thedropletsrapidlyacceleratetoaspeedof
]
t about 10 cm/s [1]. Very recently, Lagubeau et al. found
f
FIG.1: Leidenfrostdropletonaratchet. Theleftpanelshows
o that the same effect occurs for a piece of solid dry ice,
s and thus is not related to properties of the liquid phase thegasflowduetotheevaporationatthebottomofadroplet
. of radius R; the temperature of the solid T is significantly
t [2]. Even more surprisingly, Ok et al. reported that re- S
a abovethedroplet’sboilingtemperatureT . Thearrowsinthe
m ducing the ratchet profile to 200 nm, has little effect on right panel indicate the thermal creep floBw along the ratchet
the droplet velocity [3].
of period L and height D. The dashed line indicates the
-
d Unlike other self-propulsion mechanisms based on boundary layer; its thickness (cid:96) corresponds to the molecular
n chemical or thermal gradients [4–8], this motion is not mean free path. In recent experiments the ratchet height
o directed along an applied fieldbut ratherarises fromthe D varies from 200 nm to about 1 mm; for small D one has
c asymmetric surface structure of the solid support. This h0 =10...100µm.
[
sawtooth profile acts as a rectifier transferring momen-
1 tum on the interstitial vapor; the resulting gas flow ad-
v vects the floating Leidenfrost droplet. The experimental where ν is the kinematic viscosity of the vapor and ∇ T
(cid:107)
2 findings [1–3] suggest that there is a common principle the parallel component of the temperature gradient. Ki-
5
thatworksforbothliquidsandsolids,andindependently netictheoryrelatesthisgasflowtothenon-uniformden-
4
of the height of the ratchet profile. Several ideas have sity and velocity distribution: Molecules coming from
7
. been put forward, relying on non-uniform Laplace pres- the cold side and hitting the surface at a given point,
1
sure and Marangoni forces in the droplet, surface vibra- aremorefrequentbutcarrylowermomentumthanthose
0
tions,orrectificationoftheradialvaporflowthroughthe from the hot side, thus resulting in an off-diagonal com-
4
1 non-linear term of the Navier-Stokes equation [1–3, 9]; ponentofthesurfacestressandtheboundaryvelocityvC
: yet none of them explains all of the mentioned exper- [11]. Thermal creep drives aerosol thermophoresis [12],
v
iments. In particular, the submicron ratchets of Ok et repelsair-suspendedparticlesfromahotsurface[13],and
i
X al. [3] exclude non-linear hydrodynamics as the domi- operatesinsmall-scalegasflowdevicessuchasthermally
r nant mechanism, as illustrated in the left panel of Fig. actuated microcantilevers and Knudsen pumps [14–19].
a 1: For small profile D, the gas velocity and the effec- ThepresentLetterpointsouttheroleofthermalcreep
tive Reynolds number Re in the ratchet layer are much forself-propellingLeidenfrostdropletsand,inparticular,
smaller than at midheight where Re∼1 [2]; thus rectifi- analyzestheflowaroundaratchetsummit. Theessential
cation is expected to disappear for D (cid:28)h , whereas the argument is illustrated in the right panel of Fig. 1. In
0
data of Ok et al. rather show the opposite behavior. thecleftbelowthedroplet,thereisastrongtemperature
Inhis1879attempttoexplainCrookes’radiometerex- gradientofseveraltensofKelvinpermicron. Becauseof
periment and building on Reynolds’ theory for thermal the asymmetric profile, the horizontal component of the
transpiration[10], Maxwellshowedtheexistenceofther- creep velocity has a finite mean value; the resulting gas
mal creep velocity along a solid-gas interface [11], flow along the ratchet surface drags the droplet toward
the right. Note that this argument does not rely on the
3 ∇ T existence of the outward gas flow illustrated in the left
(cid:107)
vC = 4ν T , (1) panel of Fig. 1.
2
Our detailed analysis relies on Stokes hydrodynamics.
In anology to thermal transport in colloidal dispersions
[20, 21], the droplet velocity is derived from the over-
all force balance on a closed surface. In the absence of
external forces in horizontal direction one has
(cid:73)
σ dS =0, (2)
xn
where σ is the stress pulling in x-direction on the area
xn
elementdS withnormaln. Thestresstensorσ =σ(cid:48) −
ij ij
Pδ comprises a viscous part σ(cid:48) = η(∂ v +∂ v ) and FIG. 2: Temperature profile close to a solid-vapor interface.
ij ij i j j i
the excess pressure P. A non-uniform flow velocity v in Due to the small conductivity ratio κV/κS, the isotherms
the cleft of width h creates a stress of the order ηv/h, (solidlines)arestronglydistorted. Atadistanceofonemolec-
which by far exceeds the viscous drag at the remaining ular mean free path from the hot surface (dashed line), the
temperature gradient in the vapor has a significant compo-
part of the droplet surface ∼ ηv/R. Thus the surface
nentparalleltothesurface,∇T ,whichislargestclosetothe
integral may be limited to the part between droplet and (cid:107)
upper corner.
support; it closely follows the ratchet profile beyond the
boundary layer, as indicated by the dashed line in Fig.
1.
limit κ /κ → 0 the brass surface is at constant tem-
The velocity profile in the cleft comprises two contri- V S
perature T , and the profile in the cleft is given by
butionsofdifferentorigin. Thefirstone,duetoevapora- S
T(x,z) = T − (z/h)∆T, where ∆T = T − T and
tion at the bottom of the droplet, is the outside gas flow B S B
h = h +xD/L. The resulting temperature gradient is
in the left panel of Fig. 1; in the framework of Stokes 0
perpendicular on the solid-vapor and droplet-vapor in-
hydrodynamics it does not contribute to the stress in-
terfaces. The velocity distribution of the molecules hit-
tegral. Thus in the following, we consider the second
ting the brass surface is given by the temperature profile
velocity term, which arises from the thermal creep along
evaluated at one mean-free path from the brass surface,
theratchetprofile,asindicatedbythearrowsintheright
z =(cid:96)−h. Atthisfinitedistance,thegradienthasacom-
panel.
ponentparalleltothesurface∇T ∼∆T(cid:96)/h2. Replacing
The rectification mechanism is most obvious when (cid:107)
hwithh anddiscardingnumericalfactorsgivesarough
comparing the viscous stress at the two slopes of the 0
estimate for the drift velocity,
ratchet. The normal on the vertical part points in x
direction; the corresponding diagonal element σ(cid:48)xx = u∼ν(cid:96)/h2. (5)
2ηdv /dx vanishes since v and its derivative are zero. 0
x x
On the opposite side of slope m = D/L, the stress σ(cid:48)
xn With the mean-free path (cid:96)=130 nm and the kinematic
is finite. The hydrostatic pressure varies little along the viscosity ν = 60 mm2/s of vapor at 300◦ C, and h ∼
0
profileandwillbediscarded; thenEq. (2)reducestothe
10µm [22], one finds u∼10 cm/s, which is in qualitative
condition
agreement with experiment [1–3].
1 (cid:90) L For the sake of a more quantitative description we
(cid:104)σ(cid:48)xz(cid:105)= L σ(cid:48)xzdx=0 (3) refine the vapor temperature profile close to the up-
0 per corner of the ratchet, which turns out to domi-
on the viscous drag on the slope of the ratchet tooth. If nate the creep flow. In analogy to the electrostatic
the droplet is immobile, the shear stress reads as σ(cid:48) ≈ potential of a charged polygon, a simple conformal
xn
−ηv /h, where h is the width of the cleft. In order to transformation provides the expression T(r,ϕ) = T −
C S
satisfy (3) the droplet moves at a velocity u, leading to ∆T(r/h )π/αsin(πϕ/α)[24],wherer,ϕarepolarcoordi-
0
σ(cid:48) = η[u−v (x)]/h. Inserting σ(cid:48) in (3) one readily nateswithrespecttothecorner.Theangleαisrelatedto
xz C xz
obtains the expression for the drift velocity, the aspect ratio m = D/L = −cotα; for the ratchets of
Ref. [3] one finds the exponent π/α ≈ 0.63. The result-
(cid:104)v /h(cid:105)
u= C . (4) ing parallel component of the gradient along the dashed
(cid:104)1/h(cid:105)
line close to point A reads [23]
Thetemperatureprofileisdeterminedbytheboundary
conditionsatthesolid-gasinterface,imposingcontinuous ∇T =ˆξ∆T(cid:96)h−0π/α, (6)
temperatureandheatflowthroughtheinterface.Because (cid:107) r2−π/α
oftheimportantdifferenceinthermalconductivityofthe
brass support and the vapor layer, κ /κ ∼ 10−4, the whereˆξ =−π2 cosα. Itsessentialfeatureistheweaksin-
V S α2 2
temperature profile is strongly distorted, and the gradi- gularityattheratchetsummit,verysimilartotheelectric
ent is much larger in the vapor phase. field close to a charged cusp. The molecular mean-free
For a first estimate we calculate ∇T far from the path (cid:96) provides a physical cut-off for the divergency.
(cid:107)
corners, in the middle part of a ratchet tooth. In the NowthedriftvelocityisevaluatedintermsofEq. (4),
3
FIG. 4: Thermal creep below a piece of dry ice (solid CO )
2
floating above a hot metal surface due to sublimation [2].
Due to the ratchet profile printed at its lower side, there is
FIG.3: Driftvelocityuasafunctionoftheratchetparameter
a parallel temperature gradient ∇T as indicated by dashed
Dforh =10,30,100µm. ThecurvesarecalculatedfromEq. (cid:107)
0 arrows; the rectified thermal creep flow propels the disk to
(7) with ν = 60mm2/s, (cid:96) = 130 nm, π/α = 0.63, L/D = 4,
the left. The mean velocity of the vapor in the cleft is zero.
and ∆T/T =1/2.
nent β ≈ 1.5. In the present work, this force is given
resulting in [23]
by the integral of the shear stress over the contact area,
∆T ν (cid:18) (cid:96) (cid:19)π/α D/L F = πR2η(cid:104)vC/h(cid:105). With the above expression for the
u=ξ (7) thermal creep velocity one finds
T h h ln(1+D/h )
S 0 0 0
ηR2D ∆T (cid:96)π/α
with the numerical prefactor ξ ≈ 0.6 [23]. This expres- F =πξ ν . (8)
sion confirms the estimate (5) yet shows additional de- L2 T h1+π/α
0
pendencies on the ratchet parameters. Fig. 3 reveals a
√
striking variation of u with D: the smaller the ratchet With the relation h ∝ R [2], the force varies with the
0
profile, the larger the droplet velocity. This at first sight droplet size as β = 3 − π ≈ 1.2; within the experi-
2 2α
counterintuitive result is confirmed by the experiment of mental uncertainities, this compares favorably with the
Ok et al. [3]: Their data at intermediate temperatures measured value.
are well fitted by a logarithmic variation, similar to (7). The thermal-creep mechanism described here is not
Though this comparison does not account for the im- limited to the motion of Leidenfrost droplets. As a
plicit dependence of h on D, one may safely conclude straightforward application we discuss the gas pump
0
on a qualitative agreement of (7) with the data. Note shown in Fig. 5, which consists of two nanostructured
that for a ratchet driven by non-linear hydrodynamics, plates at temperatures differing by ∆T. Both solid in-
oneexpectstheoppositebehavior,i.e.,asmallervelocity terfaces show thermal creep and thus impose a unifom
for small D. Indeed, from the left panel of Fig. 1 it is gas flow across the cleft. For a sufficiently small ratchet
clear that for D (cid:28) h , the gas velocity in the ratchet profile, D <h , Eq. (7) simplifies to
0 0
layer is small and the non-linear term (v ·∇)v of the
Navier-Stokes equation insignificant. ν ∆T (cid:18) (cid:96) (cid:19)π/α
u =ξ . (9)
So far we have considered liquid droplets on a ratchet. 0 L T h
0
The same mechanism holds for a piece of dry ice (solid
CO ) above its sublimation temperature floating above This velocity may attain several meters per second. It
2
a hot metal surface; when graving a saw tooth profile at turns out instructive to compare this ratchet with a
its lower face, Lagubeau et al. observed motion similar Knudsenpump;inthepresentcase,thethermalgradient
to the droplets discussed so far. Since the creep velocity is perpendicular to the gas flow, whereas both are par-
occurs at the bottom of the dry ice, the mean velocity of allel in the latter. Moreover, a Knudsen pump requires
the vapor in the cleft is zero, as illustrated in Fig. 4. the system size to be comparable to or smaller than the
Lagubeau et al. measured the force F which required mean-freepath,andthusisrestrictedtoverydilutegases.
to immobilize a droplet floating on a ratchet [2]. In the Although the ratchet mechanism does depend on the ra-
range R = 1...7 mm, they found values from 3 to 30 tio(cid:96)/h ,itworksforfilmsthatarehundredtimesthicker
0
microNewton, and a power law F ∝ Rβ, with an expo- than the mean free path.
4
FIG. 5: Gas pump driven by thermal creep. The thermal
gradientacrossthechannelisgivenbytheirtemperaturedif-
ference ∆T and spacing h . At the ratchet corners, there
0
is a parallel component ∇T , as indicated by dashed arrows.
(cid:107)
Thermalcreepgivesrisetoauniformgasflowatvelocityu ,
0
as given in Eq. (4). Note the opposite orientation of the saw
teeth on the cold and hot side.
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