Spontaneous motion of an oil droplet driven by chemical nonequilibricity is
reported. It is shown that the droplet undergoes regular rhythmic motion under
appropriately designed boundary conditions, whereas it exhibits random motion
in an isotropic environment. This study is a novel manifestation on the direct
energy transformation of chemical energy into regular spatial-motion under
isothermal conditions. A simple mathematical equation including noise
reproduces the essential feature of the transition from irregularity into
periodic regular motion. Our results will inspire the theoretical study on the
mechanism of molecular motors in living matter, working under significant
influence of thermal fluctuation.Comment: 4 pages, 4 figure

C. D. Bain

F. Brochard

G. Nicolis

H. Mori

H. P. Greenspan

J. D. Murray

Kenichi Yoshikawa

M. K. Chaudhury

Nobuyuki Magome

P. G. de Gennes

T. Takahashi

Tsutomu Hamada

Y. Inoue

Yutaka Sumino

English

arXiv

Sumino, Y

Magome, N

Hamada, T

Yoshikawa, K

Kyoto University Research Information Repository

Title Self-running droplet: Emergence of regular motion fromnonequilibrium noiseAuthor(s)Sumino, Y; Magome, N; Hamada, T; Yoshikawa, KCitationPHYSICAL REVIEW LETTERS (2005), 94(6)Issue Date2005-02-18URL http://hdl.handle.net/2433/49863RightCopyright 2005 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversitySelf-Running Droplet: Emergence of Regular Motion from Nonequilibrium NoiseYutaka Sumino, Nobuyuki Magome,* Tsutomu Hamada, and Kenichi Yoshikawa†Depertment of Physics, Graduate School of Science, Kyoto University and CREST, Kyoto 606-8502, Japan(Received 30 June 2004; published 14 February 2005)Spontaneous motion of an oil droplet driven by nonequilibrium chemical conditions is reported. It isshown that the droplet undergoes regular rhythmic motion under appropriately designed boundaryconditions, whereas it exhibits random motion in an isotropic environment. This study is a novelmanifestation on the direct energy transformation of chemical energy into regular spatial motion underisothermal conditions. A simple mathematical equation including noise reproduces the essential feature ofthe transition from irregularity into periodic regular motion. Our results will inspire the theoretical studyon the mechanism of molecular motors in living matter, working under significant influence of thermalfluctuation.DOI: 10.1103/PhysRevLett.94.068301 PACS numbers: 83.80.Jx, 47.70.FwThe second law of thermodynamics prohibits specificspontaneous motion in macroscopic systems under equi-librium; nature obeys the law of equipartition. In contrast,living organisms exhibit a specific spatiotemporal structurein a self-organized manner under thermodynamically openconditions, where chemical processes play the major role.Over the past several decades, reaction-diffusion systemshave attracted considerable attention as a simple model forthe formation of a spatiotemporal structure in living organ-isms. It has been shown both experimentally and theoreti-cally that various dissipative structures can be generated,such as traveling waves and a Turing pattern [1–3]. How-ever, in a reaction-diffusion system there is essentially nomass flow, or geometric motion, in contrast to actual ob-servations in living things. In the present study, we foundregular spatial movement could be induced in a reactivedroplet by adopting an appropriate boundary conditionunder chemically nonequilibrium conditions. We will dis-cuss a novel scenario for the generation of macroscopicdirected and/or periodic motion under a large fluctuation.Since the 19th century, it has been known that an oil-water system exhibits spontaneous mechanical agitation,i.e., the Marangoni effect [4–9]. Two different kinds ofself-agitation, thermal and chemical Marangoni convec-tion, have been reported. Chemical Marangoni convectionis described as the time-dependent fluctuation of interfacialtension under isothermal conditions, where the drivingforce is the transfer of solute through the interface. Onetype of chemical Marangoni convection is the spontaneousmechanical motion of a reactive liquid droplet on a solidsubstrate. The motion of a reactive droplet has been pre-dicted theoretically [10–14] and verified experimentally[15–21]. However, in such a system the trajectory ofmotion cannot overlap itself because the substrate surfaceis changed in an irreversible manner, which would pre-clude periodic or circular motion. In the present study, wefound that irregular agitation induced by a chemicalMarangoni effect [4,7,8] can be transduced into character-istic repetitive, rhythmic motion, simply by choosing ap-propriate boundary conditions.The aqueous phase, 1 mM stearyl trimethyl ammoniumchloride (STAC), and the organic phase, 5 mM iodinesolution of nitrobenzene saturated with potassium iodide,were used. STAC was prepared through recrystallizationfrom acetone. For glass substrates, we used microslideglass (Matsunami, Osaka; S9111) without any pretreat-ment. As for a measurement 20–60 l of the oil was placedin an aqueous phase on a glass substrate. The movement ofthe oil droplet was recorded by high speed video camera(RedLake MASD, Inc. Motion Scope PCI) in 60 frames persecond. All the measurements were carried out at the roomtemperature.Figure 1 shows the motion of an oil droplet on a glasssubstrate in an aqueous phase. We put a small amount(30 l) of oil on the glass substrate through a pipette.The oil droplet spontaneously began to move immediatelyafter it was transferred to the glass substrate. As shown inFig. 1(a), the oil droplet exhibited random agitation withinthe circular glass vessel. Note that the trajectory of the oil-droplet motion crosses itself. In contrast to the irregularmotion shown in Fig. 1(a), a regular rhythmic motion canbe generated in the same chemical system simply bychanging the spatial geometry, as in Fig. 1(b). Corre-sponding movies are available online [22]. Such periodicmotion along the glass substrate continues over several tensof seconds and then stops. Figure 1 also shows time tracesof the x component of velocity, together with their Fouriertransformation, indicating the appearance of periodicityf  0:5 Hz for motion on the glass stripe (b) in contrastto the noisy behavior in the circular vessel (a). Carefulobservation of Fig. 1(b) reveals that the advancing contactangle of the oil droplet was smaller than its recedingcontact angle. Figure 1(c) shows the enlarged image ofan oil droplet moving towards the right, where the advanc-ing contact angle is greater than its receding angle. Thislarge difference in contact angles suggests that the drivingforce can be attributed to the difference in surface tensionbetween the front and the rear of the oil droplet.Next, we conducted an experiment to check for differ-ences in the surface of the glass plate by mimicking thePRL 94, 068301 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 20050031-9007=05=94(6)=068301(4)$23.00 068301-1  2005 The American Physical Societyconditions before and after the oil droplet had passed. Weexamined the contact angle of a droplet with pure water onthe glass plates A and B under air. The glass plate A waspretreated with the procedure of immersion in an aqueoussolution for 5 min, rinsed with distilled water, and thendried in the air. The other substrate B was pretreated in thesame procedure as plate A, then dipped into an organicphase for 5 min, carefully rinsed with distilled water, anddried in the air. A large difference in the contact angle wasobserved [A  68, B  43] [Fig. 2(a)], indicating thatplate B is less hydrophobic than plate A. Thus, it isapparent that a difference in surface tension  betweenthe front and the rear of an oil droplet is generated accom-panied by its translational motion. Once  becomes non-zero due to some spontaneous small motion driven by theMarangoni effect, an oil droplet tends to continue thedirected motion by avoiding the backward movement dueto the memory effect of the chemical condition [14,19].The generation of a difference in surface tension be-tween the front and the rear can be explained as schemati-cally shown in Fig. 2(b). Since the surface of the glasssubstrate has a negative charge, STA (Stearyl TrimethylAmmonium Ion) is arranged on the glass plate with thehead group attached to the surface and the hydrophobic tailtoward the aqueous phase; as a result, the glass surface thatfaces the aqueous solution is hydrophobic. When an oildroplet exhibits a directed motion due to the Marangonieffect, STA attached to the surface tends to dissolve intothe organic phase. Consequently, the glass surface becomesless hydrophobic, and the difference in surface tension isgenerated between the front and the rear of the droplet.This imbalance in the surface tension promotes directedmotion against viscous damping. The existence of I3 madefrom I and I2 in an oil droplet accelerates the transfer ofSTA from the surface into the organic phase [8] bycreating a hydrophobic ion pair with STA. Sig-nificantly, the surface of the glass substrate spontaneouslyreturns to the hydrophobic state soon after an oil dropletpasses by, since STA in the aqueous phase tends to beabsorbed onto the glass substrate. This mechanism allowsthe trajectory of an oil droplet to cross itself; thus, rhythmicand repetitive motion is generated in our system.An oil droplet ceases its motion accompanied with thedecrease of the nonequilibrium chemical conditions. Thisis explained as follows. As I3 concentration decreases inthe oil droplet because of pairing with STA, the oildroplet cannot afford enough driving force, then the oildroplet stops. We have also found that with the decrease ofthe size of droplet, its translational speed tends to be largerand the lifetime of self-motion to be shorter. This may bedue to the enhanced surface effect on smaller systems.Let us now discuss the mechanism of the appearance ofregular, rhythmic motion in the self-agitating system. Thecharacteristic feature of the motion of an oil droplet can bedescribed asmdvdt v  vjvj  t: (1)The first term represents the resistance force and the sec-3.0s70 mmOil droplet10 mm10 mm(a)xyGlass substrate(b)26 mmWidth 3.4mm20 mmt5mm(c)recθ advθvx [mm/sec]f [Hz]1-40040v x [mm/sec]t [s]0 42 6 8 0 420x7.0s9.0s 1.0s5.0s0 42 6 8-400400 4210f  [Hz]t [s]FIG. 1 (color). According to the configuration of the glass substrate, an oil droplet showed a distinct mode of motion. (a) Upper left:Schematic diagram of the experimental setup. The volume of the oil droplet is 30 l. Upper right: Random motion appears underisotropic conditions. Also see movies in [22]. Lower panel: Trace of the x component of oil-droplet velocity and its Fouriertransformation, where there is no distinct peak. The y-component velocity shows similar behavior. (b) Upper left: Schematic diagramof the experimental setup. The volume of the oil droplet is 30 l. Upper right: Periodic forward-and-backward motion appears on anarrow and straight glass substrate. The time scale in the picture on the right is 1=6 s per frame. The rhythmic motion continues for10 s. Also see movies in [22]. Lower panel: Trace of the x component of oil-droplet velocity and its Fourier transformation, indicatinga peak at 0.5 [Hz]. (c) Schematic representation of an oil droplet, moving toward the red arrow. adv was smaller than rec. On average,adv  105, rec  116.PRL 94, 068301 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005068301-2ond term represents the driving force caused by the differ-ence in surface tension, where we incorporate the charac-teristics of self-motion to prohibit backward motion as inFig. 2(b). The third term  represents the Gaussian randomforce whose standard deviation is  resulting from theinhomogeneity of the glass surface and the oil-water inter-face. Figure 3 shows schematic examples of simulated oil-droplet motion based on Eq. (1) [23]. We consider that thedriving force is comparable to the Gaussian random force(	 ). As for the boundary condition corresponding tothe edge of a glass substrate, an oil droplet is assumed toexhibit inelastic rebound at the boundary since there existsdissipation caused by inversing convective flow inside theoil droplet. In the absence of edge effect on a vessel,random motion is generated [Fig. 3(a)]. In contrast, whenthe width of the glass substrate d becomes narrower, i.e.,when a quasi-one-dimensional boundary condition is im-posed, periodic regular motion is generated [Fig. 3(b)][24]. This result indicates that the reduction of the effectivefreedom of motion with an appropriate boundary conditioncan cause a transition from irregular behavior to regularbehavior for such a droplet under nonequilibrium chemicalconditions.An oil droplet shows various types of regular motiondepending on the shape of the glass substrate. Figure 4(a)shows an oil droplet moving around a vertical circle, like aroller coaster [25]. The droplet can move up the circleagainst gravity, where the density of the oil droplet oil 1:2 and the density of the aqueous phase water  1:0.Figure 4(b) shows the stepping-up motion of an oil droplet.Based on observations in multiple experiments, we havefound that the oil droplet tends to climb up the stairs ratherthan to fall down them. Corresponding movies are avail-able online [22].In the present study, we have realized regular motionunder nonequilibrium noise by setting suitable boundaryconditions, where the broken spatial symmetry in the con-tainer of the oil-water system reduces the freedom ofmotion, inducing regular motion of the droplet. Recentstudies on single motor proteins [26] have indicated thatthe mode of such mechanical motion can be switched bychanging the internal and boundary conditions [27–29].Although the spatial scale of molecular motors in livingorganisms is much smaller than that of the present experi-mental system, the generation of regular motion undernonequilibrium in a real experiment may inspire the theo-retical consideration of the working mechanism of biologi-cal molecular motors.This work is supported by the Grant-in-Aid for the 21stCentury COE ‘‘Center for Diversity and Universality inPhysics’’ from the Ministry of Education, Culture, Sports,Science and Technology (MEXT) of Japan and by theJapan Space Forum.xy(a) (b)-a axtxydFIG. 3 (color). Numerical simulation on the spontaneous mo-tion of an oil droplet at different boundary condition [23].(a) Random motion of an oil droplet under boundary-free iso-tropic environment, which is shown as the two-dimensional traceof the motion. (b) Repetitive motion of an oil droplet on a narrowstripe that restricts the motion to a < x< a and d=2< y<d=2, where the upper panel shows the trace of the motion in realspace and the lower panel shows spatiotemporal representationof the motion. Oil-droplet motion described by Eq. (1) variesdistinctively depending on the effective freedom of motion thatis restricted by a boundary condition. In (b), an oil droplet showsforward and backward motion when d is small enough.rearγ frontγPlate A Plate BAirWaterθ Α θ BAirWater(a)(b)rearγFIG. 2 (color). (a) A water droplet on a glass substrate in air.We estimate the difference in water-glass surface tension from   pure watercosB  cosA. Since A  68, B 43, and pure water  72 mN=m, the difference in the interfa-cial tension is   14 mN=m. (b) Schematic diagram of themechanism of the difference in surface tension. Red arrows showthe direction of oil-droplet motion. STA is represented as ahydrophobic bar with a polar head. Since STA tends toassemble with I3 in the droplet, STA dissolves into the oildroplet from the glass surface. Thus, a difference surface tensionis generated as front > rear. The glass surface returns to hydro-phobic shortly after an oil droplet passes by, since STA in theaqueous phase again adheres to the glass surface.PRL 94, 068301 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005068301-3*Present address: Department of Food and Nutrition,Nagoya Bunri College, Nagoya 451-0077, Japan.†To whom correspondence should be addressed.Fax: 011-81-75-753-3779Electronic address: yoshikaw@scphys.kyoto-u.ac.jp[1] G. Nicolis and I. Prigogine, Self Orgainization inNonequilibrium Systems (Wiley, New York, 1977).[2] J. D. Murray, Mathematical Biology (Springer-Verlag,Berlin, 1990).[3] H. Mori and Y. Kuramoto, Dissipative Structures andChaos (Springer-Verlag, Berlin, 1997).[4] M. Dupeyrat and E. Nakache, Bioelectrochem. Bioenerg.5, 134 (1978).[5] T. Takahashi, H. Yui, and T. Sawada, J. Phys. Chem. B106, 2314 (2002).[6] N. M. Kovalchuk and D. Vollhardt, Phys. Rev. E 69,016307 (2004).[7] N. Magome and K. Yoshikawa, J. Phys. Chem. 100,19 102 (1996).[8] A. Shioi, K. Katano, and Y. Onodera, J. Colloid InterfaceSci. 266, 415 (2003).[9] K. Yoshikawa, T. Omochi, Y. Matsubara, and H. Kourai,Biophys. Chem. 24, 111 (1986).[10] H. P. Greenspan, J. Fluid Mech. 84, 125 (1978).[11] P. G. de Gennes, Rev. Mod. Phys. 57, 827 (1985).[12] F. Brochard, Langmuir 5, 432 (1989).[13] P. G. de Gennes, F. Brochard, and D. Que´re´, Capillarityand Wetting Phenomena Drops, Bubbles, Pearls, Waves(Springer-Verlag, Berlin, 2004).[14] P. G. de Gennes, Physica (Amsterdam) 249A, 196(1998).[15] C. D. Bain and G. M. Whitesides, Langmuir 5, 1370(1989).[16] M. K. Chaudhury and G. M. Whitesides, Science 256,1539 (1992).[17] C. D. Bain, G. D. Burnett-hall, and R. R. Montgomerie,Nature (London) 372, 414 (1994).[18] F. D. D. Santos and T. Ondarc¸uhu, Phys. Rev. Lett. 75,2972 (1995).[19] S. Lee and P. E. Laibinis, J. Am. Chem. Soc. 122, 5395(2000).[20] J. Bico and D. Que´re´, Europhys. Lett. 51, 546 (2000).[21] S. W. Lee, D. Y. Kwok, and P. E. Laibinis, Phys. Rev. E 65,051602 (2002).[22] See EPAPS Document No. E-PRLTAO-94-066507for movies of irregular motion (movie1.mpg), shuttlingmotion (movie2.mpg), rotating motion (movie3.mpg),and climbing motion on stairlike substrates (movie4.mpg).Every movie shown is in half speed of its real motion.Detailed setup is explained in suminomovietext.doc,which is also in EPAPS Document. A direct link tothis document may be found in the online article’sHTML reference section. The document may also bereached via the EPAPS homepage (http://www.aip.org/pubserves/epaps.html) or from ftp.aip.org in the direc-tory /epaps/. See the EPAPS homepage for more informa-tion.[23] Simulation was performed through the fourth-order Runge-Kutta method with the time step of 0.0001.Each parameter was taken to be as follows: =m  24,=m  0:9, =m  10, d  0:001, a  0:013, andcoefficient of rebound  0:5.[24] The occurrence of the quasi-one-dimensional motion inthe simulation is understood as follows. From Eq. (1), thecharacteristic velocity vc and the relaxation time r aregiven as vc  = and r  m=. The characteristic timespan c between each rebound is c  d=vc  d=,where d is the free width of glass substrate. We haveconcluded that, when c  r, i.e., when d is smallenough, the velocity that is perpendicular to long axis ofsubstrate is suppressed as long as there is loss of velocityat each rebound.[25] We can estimate the difference in surface tension from theresult of Fig. 4(a), where an oil droplet moves up verticallyon a glass substrate. In this case, the driving force gen-erated by the difference in surface tension, , betweenthe rear and front should be greater than the gravitationalforce, fgrav, acting on the oil droplet. For the experiment inFig. 4(a), w  fgrav  0:04 mN, where w is thewidth of the oil droplet. Since w  3:4 mm,  shouldbe larger than 11 mN=m, which is the same order as theacquired value in Fig. 2(a).[26] M. Schliwa and G. Woehlke, Nature (London) 422, 759(2003).[27] S. A. Endow and H. Higuchi, Nature (London) 406, 913(2000).[28] Y. Okuda and N. Hirokawa, Proc. Natl. Acad. Sci. U.S.A.97, 640 (2000).[29] Y. Inoue, A. H. Iwane, T. Miyai, E. Muto, and T. Yanagida,Biophys. J. 81, 2838 (2001).3.4mm1.5s(b)5mm(a)5.2mm3.4mm25mm5mm25 mm20 mmt1mmUpDown1.0s2.0s 2.5s3.0s3.5sFIG. 4 (color). (a) Rotational motion within a circle normal tothe horizontal plane. In the experiment shown in this figure, thedroplet underwent rotation up to 3 times and then stops. Thevolume of the oil droplet is 20 l. Also see movies in [22].(b) Climbing motion on a stairlike substrate. The oil droplettends to move up the stairs, implying that the stairlike substratecan regulate the direction of motion. The time interval in thepictures on the right is 1=6 s per frame. The volume of the oildroplet is 60 l. Also see movies in [22].PRL 94, 068301 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending18 FEBRUARY 2005068301-4

Self Running Droplet: Emergence of Regular Motion from Nonequilibrium Noise

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