Table Of ContentDynamics of microdroplets over the surface of hot water
Takahiro Umeki,1 Masahiko Ohata,1 Hiizu Nakanishi∗,2,∗ and Masatoshi Ichikawa1
1Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2Department of Physics, Kyushu University, Fukuoka 812-8581, Japan
(Dated: January 6, 2015)
When drinking a cup of coffee underthe morning sunshine, you may notice white membranes of
steamfloatingonthesurfaceofthehotwater. Theystaynotablyclosetothesurfaceandappearto
almoststicktoit. Althoughthemembraneswhifflebecauseoftheairflowofrisingsteam,peculiarly
fast splitting events occasionally occur. They resemble cracking to open slits approximately 1 mm
5 wideinthemembranes,andleavecuriouspatterns. Westudiedthisphenomenonusingamicroscope
1 withahigh-speedvideocameraandfoundintriguingdetails: i)thewhitemembranesconsistoffairly
0 monodispersed small droplets of the order of 10 µm; ii) they levitate above the water surface by
2 10∼100µm;iii)thesplittingeventsareacollectivedisappearanceofthedroplets,whichpropagates
as a wave front of thesurface wave with a speed of 1∼2 m/s; and iv) these events are triggered by
n
a surface disturbance, which results from thedisappearance of a single droplet.
a
J
3
As a Japanese physicist Torahiko Terada wrote in his
fascinating essay “Chawan no yu (A cup of hot tea)” in
]
t 1922(in “The Complete Works of Terada Torahiko”, vol
f
o 2, pp.3-9, Iwanami, 1997), many interesting phenomena
s occur in a teacup filled with hot tea, including convec-
.
t tion, condensation, and vortex flow. Many are related
a
to a wide range of natural phenomena in larger scales
m
and have been subjects of active study in physics dur-
-
ing the last few decades. One phenomenon in his es-
d
n saythathasnotbeensystematicallystudiedisthe misty
o thin skin that covers a hot water surface(Fig. 1). He
c also mentioned the crack patterns that run through the
[
skins, and conjecturedthat the patterns must be related
1 tothetemperaturevariationdevelopedbytheconvection
v flow in hot water. Nearly 50 years later, another essay
3 on this phenomenon was written by Schaefer in 1971 [1].
2
He performed some simple experiments and conjectured
5
thatthewhiteskinsweremadeofsmallchargeddroplets
0
thatlevitatedbecauseoftherisingevaporationflowfrom
0
. the surface. He contendedthatthe patterns onthe skins
1 FIG.1. Mistyskinsoverhotcoffee. Peculiarpatternsareob-
delineate the Rayleigh-B´enardconvectionpattern in hot
0 servedinwhitemembranesonthecoffeesurface. Thephotois
waterandsuggestedthattheslitssuddenlyappearedbe-
5 providedthroughthecourtesyofMachida-Sagamiharaportal
cause of micro whirlwinds that develop in the rising hot
1 site Vita (http://vita.tc) and Tamagawa Coffee Club and is
: moist air flow. allowedforuseinthispaperwithreferencetothecompanies.
v
Recently, a similar phenomenon was accidentally dis-
i
X covered by Fedorets in a different setting when he was
r examining the surface of water layer on an ebonite sub- In this work, we examine the surface of bulk hot wa-
a strate heated with a lamp [2]. He found a layer of mi- ter using a microscope with a high-speed video camera,
crodropletsofapproximately10µmformingatriangular and report what the white membranes are and how the
lattice and observed that a segment of a droplet cluster “cracking events” actually occur.
suddenly disappears within 0.04 sec, i.e., a single-frame
interval of his video camera. In a series of works [2–
5],Fedoretsandco-workersexaminedseveralconceivable RESULTS
mechanisms of the droplet levitation and suggested two
possibilities: the force from droplet spinning because of Weperformedtwosetsofexperiments: preliminaryob-
the Marangoni effect and the Stokes drag force because servations on several types of hot water/beverages and
of the rising evaporation flow. detailedobservationsonhottapwaterwithahigh-speed
video camera. The preliminary experiments were per-
formed with a simple setting; the hot water/beverage in
a beaker was examined by eye and with a video cam-
∗ [email protected] era. We made observations on several types of hot wa-
2
the droplets are relatively monodispersed at each tem-
perature; that there is a clear boundary between the
dense and the sparse regions of the droplets; and that
thedropletstendtoformatriangularlatticeinthedense
LED light guide
regions.
Figure 4 showsthe size distributions andaveragesizes
of the droplets at several temperatures. The radius dis-
tributions show narrowdispersion,whose standarddevi-
Inner & outer
chambers ations are typically less than half of their average value.
The average radius of the droplet gradually decreases
◦
when the temperature decreases. The data at T <60 C
may containsystematic errorbecause the averageradius
becomescomparabletotheresolutionlimitoftheimages,
i.e., 4.74 µm for 1 pixel.
Hot water bath
Observations with high-speed camera.
Careful examination of the video images reveals that
the droplets occasionally fall from above to settle above
Lens & Camera
the surface,andthey occasionallyfall downto the water
surfacetoindividuallydisappear. Inadditiontosuchin-
dividual disappearances,weobservedoccasionalcollective
FIG. 2. Schematic diagram of theexperimental setup.
disappearances of hundreds of droplets within a single-
frame interval of the 30 fps video camera; this collective
event corresponds to the aforementioned splitting event.
ter, such as coffee, tea, water with detergent, and pure
Tostudythisfastprocessduringthecollectivedisappear-
water filtered using a Milli-Q system, and we did not
ance, we made observations using the high-speed video
find significant differences in the white membranes for
camera with frame speeds of 1000 and 8000 fps and a
all cases. The microscope observations showed that the
shutter speed of 1/16000 sec.
white membranes consist of micro droplets of the order
of 10 µm. The droplets were also found on the surface Figure 5 shows three consecutive frames of 1000 fps
of moderately hot water with a temperature of approxi- video, which captured a collective vanishing event. A
mately 50◦C, although they were notably sparse. millimetre-sizeddroplet cluster vanishedwithin a couple
of frame intervals. The streaks appeared in the middle
Afterthepreliminaryobservations,wemadecloserob-
framebecauseofthedropletmotionduringasingle-frame
servationsofhottapwaterwithahigh-speedvideocam-
exposure. It should be noted that not all droplets in the
erausinga setupdesignedfor this phenomenon(Fig. 2).
region vanished, and there were surviving droplets, i.e.,
The experimental setup had a water container, a liquid
some isolated droplets and/or clusters of droplets sur-
light-guideilluminationsystem,andamicroscopewitha
vivedintheregionthathadacollectiveevent. Thevideos
highspeed-videocamera(KEYENCEVW-9000,Japan).
alsoshowacontinuousfallingofdropletsfromaboveonto
To avoid water condensation on the lens, the water sur-
the surface, some of which settled in the clusters and
face wasobservedfrom below under bright- ordark-field
caused rearrangementof the surrounding droplets (Sup-
illuminations from above. The dual wall chamber was
plementary video 1).
used for the water container to reduce external distur-
bances. The container was filled with hot water of 60 ∼ Figure6showsnine consecutiveframesofthe 8000fps
90◦Cintheroomtemperatureenvironmentof20∼25◦C; video, which reveal that the vanishing event propagated
the room temperature and humidity were not controlled as a wave front at speed of approximately 1m/s. It is
during the experiments. The recordedimages wereanal- also observed that the wave front was accompanied by
ysed using image-processingsoftware, mainly ImageJ. a surface wave of the water, which propagatedat an ap-
proximatelyidenticalspeed. Thefourconsecutiveframes
Snapshots. Figure 3 shows typical microscope pictures in Fig. 7 capture the initial stage of a collective event.
at three different temperatures. Droplets are scattered The collective event was triggered by the disappearance
on the surface and drifting together at approximately a ofasingledroplet,fromwhichthedisturbancewavefront
few mm/s. They do not stay on the surface but levitate propagatedoutwardandcausedamassivedisappearance
above the surface, which is confirmed with the indepen- event that involved thousands of droplets. We also ob-
dence of their drift motions on those of dust particles servedconcurrentevents,wheretwocollectiveand/orin-
thatfloatonthe surface. The distance abovethe surface dividual events occurred within a 10 ms time interval,
is estimated to be 10 ∼ 100 µm from the focus depth of although they did not seem causally related through the
the microscope. surface wave because they were spatially separated from
Figure 3 shows the following: that the droplets are each other.
more populous and larger at higher temperature; that Finally, Fig. 8 shows a vanishing event in the system
3
FIG. 3. Microscope images of thehot-water surface at thetemperatures T =62.5, 73, and 78◦C.
withthesurfactantTritonX-100. Theconcentrationwas immediately above the water surface; 3) the typical size
0.3mM,whichisabovethecriticalmicelleconcentration of the droplets is on the orderof 10 µm, the averagesize
atroomtemperature,i.e.,0.24mM.Onecanobservethat decreases as the water temperature decreases, and the
the event is qualitatively similar to the one in the tap levitation height is 10∼100 µm; 4) the splitting events
water. The surfactant does not appear to significantly in the membranes are the propagation of the collective
change the droplet size and the propagation speed. disappearance of droplets, which is triggered by a single
droplet; and 5) their propagation fronts are accompa-
nied by the surfacewave,both ofwhich propagateat ca.
DISCUSSION 1m/s.
These observations provide more questions than an-
With these observations, we found the following: 1) swers: 1) What is the levitation force? 2) Why are the
The phenomenon is not sensitive to the type of water, droplets so monodispersed? 3) What triggers the collec-
i.e., coffee, tea, water with detergent, and Milli-Q water; tive vanishing event? 4) How does the vanishing wave
2) the white membranes consist of the fairly monodis- front propagate? 5) Why do some droplets survive after
persed microdroplets that fall from above and levitate a vanishing event?
Non-coalescentdrops. Beforediscussingsomeofthese
a issues, let us briefly mention an apparently similar phe-
84°C
nomenon, i.e., non-coalescent drops, which you may ob-
servewhenyoumakedripcoffee. Somedropsstayonthe
76.3°C
coffee surface for several seconds and roll over it before
y
nc they coalesce into the coffee in the pot. The droplets
que 64.1°C float above the surface also in this case, but the droplet
e
Fr size is notably different: 1 cm for the dripping droplets
59°C
and10µmforthedropletsinthewhitemembranes. This
type of non-coalescent liquid drops has been studied in
56°C
various situations[6, 7], and it has been shown that two
surfaces of the same liquid that are pressed against each
0 5 10 15 20
Radius (µm) othercanbestabilisedwithoutcoalescingforquitealong
20 time because of a thin air layer, which is maintained
b
m) by various mechanisms, such as the air drag by liquid
µ 15 viscosity[8], externally driven air flow[9], air flow driven
us ( by the thermal Marangoni effect[10–13], surfactant sur-
di
a 10 face elasticity[14], or vibration[8], etc. However, the rel-
R
e evance to the present phenomenon is not direct because
g
a
er 5 of the size difference.
v
A
0 Levitation force. The required force to levitate a
40 50 60 70 80 90
Temperature (°C) 10µm diameter droplet against gravitation is approxi-
mately 5pN. The levitation force may depend on the
FIG.4. Sizedistribution(a)andaveragesizeofthedroplets water temperature because the average droplet size in-
(b) at various temperatures. The error bars for the average creases with the water temperature. The evaporation
radii represent the standard deviation of the distributions. flow from the hot water surface is a candidate as Schae-
The data set for each temperature contains several hundreds fer previously conjectured in his essay [1]. Recently, this
of data points of droplet radii. mechanismwasexaminedinmoredetailbyFedoretsand
4
FIG. 5. Collective vanishingevent of a droplet cluster. Three consecutive frames of the1000 fps video are shown. Levitating
water droplets are observed as the bright spots underdark-field illumination. (Supplementary video2)
co-workers[3,5]. Theyalsoexploredanothermechanism; If there is no particular size of approximately 10µm for
the droplets may be levitated by the air flow induced by the droplets to be thermodynamically stable, their size
the spinning motion because of the Marangoni effect[3]. maybeselectedbasedontheforcebalancebetweengrav-
Theyestimatedthatthe dropletscouldspinasfastas50 itation and the levitation force.
rotationspersecond(rps)byapossibletemperaturevari- In a slightly different situation from the present case,
ation in the droplets. Both mechanisms should be sensi- ArinshteinandFedoretsmade aninterestingobservation
tivetothewatertemperatureandenvironmentalhumid- onthisconnection[4]: thereappeareddropletsaslargeas
ity. Another possibility is an electrostatic force because 100µm in diameter over the water layer on the ebonite
smalldropletscanbeeasilycharged. Inthepresentcase, substrate when the water surface was heated at a local-
the droplets are likely charged because they form a tri- ized region of 1 mm in size.
angular lattice in densely populated regions. With this
mechanism,thelevitationforceshouldbesensitivetothe Vanishing events. Droplets individually and collec-
electricstatusofthesurface,whichshouldbeaffectedby tively vanish. In a collective event, the vanishing front
salt or other impurities in the water. It is puzzling that propagates at approximately 1m/s, which is close to the
the phenomenon appears insensitive to variations in the capillary surface wave speed of the water with a wave-
water temperature, humidity, and water purity. length of 0.1∼1 mm. Thus, it is natural to assume that
the droplets are swamped by the surface wave, which is
Size distribution of the droplets. The droplets are
relativelymonodispersed. Thestandarddeviationsofthe
linearsizedistributionsarelessthan50%oftheaverages.
0/8000s 1/8000s 2/8000s
3/8000s 4/8000s 5/8000s
6/8000s 7/8000s 8/8000s FIG. 7. Initial stage of a collective vanishing event. The
enlarged images from the high-speed video frames show that
FIG. 6. Wave front propagation of a collective vanishing the collective vanishing event is triggered by a disturbance,
event. Nine consecutive video frames of 8000 fps are shown. which is caused by the disappearance of a single droplet, as
(Supplementaryvideo 3) indicated bythe arrow.
5
FIG.8. Vanishingeventon thehot waterwith detergent. Triton X-100was dispersed with aconcentration of 0.3 mM,which
is above thecritical micelle concentration at room temperature, i.e., 0.24 mM. The water temperature was 78◦C.
sustainedbythesurfaceenergyoftheswampeddroplets. perature are important.
A simple estimate of the energy balance shows that the
energy input from the swamped droplets can maintain In conclusion, a cup of hot tea continues to provide
the surface wave with a sufficiently large amplitude to us with interesting phenomena that deserve scientific re-
swamp the droplets in the regions of high density of search. Thephenomenonthatwestudiedherecanbeob-
droplets (see Appendix section). served everyday and should have been noticed by many
We observed that a single-droplet disappearance scientists, yet very few people appear to have imagined
caused a collective event, although there are also many such fascinating phenomena happening in a teacup.
isolatedeventsthatdonottriggeracollectiveevent. The
single-droplet events tend to occur in less populous re-
gions, which is consistent with the above mechanism of APPENDIX
collective vanishing because in a sparse region, the dis-
turbance decays before it reaches neighbouring droplets;
In this section, the amplitude of the surface wave is
otherwise, it should trigger a collective vanishing event.
estimated based on the described swamping mechanism.
Whether a vanishing event triggers a collective event
In the steady propagation, there should be an energy
or not, it is not clear what causes a droplet to vanish
balance between the energy input from the swamped
in the first place; the reason may be fluctuations in the
droplets and the energy dissipation in the surface wave
evaporationfloworanelectricdisturbancethatiscaused
becauseofthe viscosity. Assumingthatthe surfacewave
by a cosmic ray if the droplets are levitated by the elec-
is localised around the wave front region with the width
trostaticforce. Theobservedconcurrenteventsarelikely
and depth of the wave-length λ, the energy dissipation
to be triggered by a common disturbance in the levita-
per unit time per unit length along the wave front is es-
tionforcebecauseitishighlyimprobablethatmorethan
timated as
oneeventoccursinlessthan10ms inasmallareaofthe
microscope field of view. h 2
If the levitation force is provided by the temperature, ediss ∼η ×λ2, (1)
(cid:18)τλ(cid:19)
the spatial pattern of the droplet disappearance events
may reflect spatial variations in the surface tempera-
whereηisthewaterviscosity,τ isthewaveperiod,andh
ture of the water, which would explain the observation
istheheightoramplitudeofthe wave. Theenergyinput
of Terada and Schaefer[1] that the splits in the white
is estimated as the product of the number of swamped
membranesappeartodelineatetheRayleigh-B´enardcon-
droplets and the surface energy of a droplet,
vection pattern in the cup. The collective disappear-
anceeventsshouldinthelow-surface-temperatureregion e ∼vn×σd2, (2)
input
of descending convection, where the levitation force is
weaker and thus the droplets stay closer to the surface. wherev ≡λ/τ, n,σ,anddarethewavespeed,areaden-
sity of the droplets, surface tension of the water, and
Surviving droplets. If the swamp mechanism of the
droplet diameter, respectively. By equating e and
collective droplet disappearance is true, some droplets diss
e , we obtain
can survive if they are levitated higher. However, the input
survivingdropletsdonotnecessarilyappearsmallerthan
1
the vanished droplets, which does not appear consistent h∼ τ2vnσ ×d. (3)
with the idea that the levitationheight is determined by (cid:18)rη (cid:19)
the force balance because a smaller droplet should stay
If we use
higher in this mechanism. To determine the mechanism,
experimental observations of the levitation heights and
λ∼100µm, v ∼1m/s, n∼1/(30µm)2 (4)
their correlation with the droplet size and surface tem-
6
with the following viscosity and the surface tension for
water,
η ∼10−3Pa·s, σ ∼7×10−2N/m, (5)
Eq.(3) is estimated as
h∼30×d. (6)
This estimate suggests that the sufficiently large ampli-
FIG. 9. Cluster of droplets being swamped by surface wave tudeofthesurfacewavecanbemaintainedbythesurface
front. The shaded area is the wave front region of the wave energy of the swamped droplets for the observeddroplet
length width. density in densely populated regions.
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a fluid bath. Phys. Rev. Lett. 94, 177801 (2005). The authors thank Clive Ellegaards, Namiko Mitarai,
Takahiro Sakaue, and Yoko Ishii for stimulating discus-
sions. This work was supported by JSPS KAKENHI
Grant Number 25610124.
