Slowing-down of non-equilibrium concentration fluctuations in confinement

Transcription

1 Slowing-down of non-equilibrium concentration fluctuation in confinement Cédric Giraudet, Henri Bataller, Yifei Sun 2, Alekandar Donev 2, Joé María Ortiz de Zárate 3 and Fabrizio Croccolo Laboratoire de Fluide Complexe et leur Réervoir, Univerité de Pau et de Pay de l Adour, Anglet, France 2 Courant Intitute of Mathematical Science, New York Univerity, New York, NY 002, USA and 3 Departamento de Fíica Aplicada I, Univeridad Complutene, Madrid, Spain (Dated: June 2, 205) Fluctuation in a fluid are trongly affected by the preence of a macrocopic gradient making them long-ranged and enhancing their amplitude. While mall-cale fluctuation exhibit diffuive lifetime, moderate-cale fluctuation live horter becaue of gravity. In thi letter we explore fluctuation of even larger ize, comparable to the extent of the ytem in the direction of the gradient, and find experimental evidence of a dramatic lowing-down of their dynamic. We recover diffuive behavior for thee trongly confined fluctuation, but with a diffuion coefficient that depend on the olutal Rayleigh number. Reult from dynamic hadowgraph experiment are complemented by theoretical calculation and numerical imulation baed on fluctuating hydrodynamic, and excellent agreement i found. Hence, the tudy of the dynamic of non-equilibrium fluctuation allow to probe and meaure the competition of phyical procee uch a diffuion, buoyancy and confinement; i.e. the ingredient included in the Rayleigh number, which i the control parameter of our ytem. PACS number: a, Ln, 47..-j, Va The phyic of ytem out of thermodynamic equilibrium i intrumental in everal reearch area uch a phyic of fluid, oft matter, atrophyic, tatitical phyic, biology, metallurgy and many other [, 2]. A an example, we conider in thi letter a binary liquid mixture ubjected to a teady temperature gradient parallel to gravity. Two component liquid experience eparation in the preence of temperature difference due to different affinitie of molecule to heat [3]. A phenomenon referred to a thermodiffuion or Soret effect that will induce a teady concentration gradient in the ytem. Thi o-called olutal Rayleigh-Bénard etting, allow for a very refined control of denity profile within the ytem and the ability to invetigate intimate propertie of fluid like molecular interaction [4 6]. Any full decription of non-equilibrium ytem mut include pontaneou fluctuation, whoe nature i quite different from fluctuation around equilibrium tate mainly due to the former long-ranged nature [7 9], not retricted to the proximity of critical point [0]. Non-equilibrium fluctuation are thu a baic problem in undertanding tranport phenomena like ma diffuion [], a well a fluctuation-induced, or Caimir, force [2 4]. In our non-equilibrium problem, the coupling between pontaneou velocity fluctuation and the macrocopic gradient reult in giant non-equilibrium concentration fluctuation (c-nef) in the quiecent tate [9, 5]. Gravity quenche the intenity (tatic) of fluctuation with length cale larger than a characteritic (horizontal) ize 2π/q related to the dimenionle olutal Rayleigh number Ra of the ytem [5, 6]: Ra = β g c νd L4 ; Ra =(q L) 4, () where β = ρ ( ρ/ c) i the olutal expanion coefficient, ρ the fluid denity, g the gravity acceleration, c the concentration (ma fraction) of the dener component of the fluid, c the concentration gradient, D the ma diffuion coefficient, ν the kinematic vicoity, and q a characteritic olutal wave vector. Thi number i the equivalent for the concentration of the Rayleigh number for the temperature and decribe the competition of oppoite force like buoyancy, diffuion and boundarie. It i alo known that, in addition to gravity, the preence of boundarie further uppre the intenity of c-nef with length cale larger than the confinement length L in the direction of the gradient [9, 7]. The role played by the different phyical mechanim (diffuion, buoyancy, confinement) on the dynamic of the fluctuation ha received comparatively little attention. It i known that gravity accelerate the dynamic of c-nef with (horizontal) length cale larger than 2π/q [8, 9]. However, thi mean that fluctuation of a larger ize decay fater, which i a rather non-intuitive behavior [20] and cannot be extrapolated to zero wave number. To invetigate thee open iue further, we have developed a new hadowgraph machine, with a tate-ofthe-art CCD detector able to meaure at wave vector down to q min = 8.9 cm. Hence, we explored an entire new range of wave number, maller than ever before, and dicovered a dramatic lowing-down in the dynamic of c-nef. We interpret thi lowing-down a caued by confinement, whoe role on the dynamic of c-nef ha not been invetigated o far. Our work demontrate that the tudy of the dynamic, rather than the intenity, of non-equilibrium fluctuation give deep inight into the competition of phyical procee uch a diffuion, buoyancy, and confinement. Typically, the dynamic of fluctuation i characterized in term of an Intermediate Scattering Function (ISF

2 2 or, equivalently, a normalized time correlation function) f(q, t), with f(q, 0) =. Thi function decribe how an pontaneou fluctuation of a thermodynamic variable decay in time. In firt approximation, the ISF can be modeled by a ingle exponential, with decay time τ(q) depending on the fluctuation wave number q, or length cale in the horizontal direction (perpendicular to gravity and gradient). Available theorie accounting for the imultaneou preence of diffuion (d) and gravity (g) [8, 9], but not for confinement, predict for a table configuration (Ra < 0): τ( q) τ = τ( q) d+g = ( ), d+g q 2 Ra (2) q 4 where the wave vector i expreed in dimenionle form q = ql, and τ = L 2 /D i the typical olutal time it take diffuion to travere the thickne of the ample. Equation (2) implie different behavior for the decay time of mall-cale and large-cale fluctuation, namely, τ( q) d = / q 2 for q q, and τ( q) g = q 2 /Ra for q q. Hence, mall length-cale fluctuation decay diffuively and evolve lower the larger the cale. Buoyancy effect, for wave number maller than q, lead to a non-diffuive decay of fluctuation [20]. Separating thee two regime, the decay time of c-nef exhibit a maximum at q, a clearly hown by Eq. (2). The exitence of thi maximum, which identifie the mot peritent fluctuation in the ytem if confinement i neglected, ha been experimentally demontrated in a number of experiment on c-nef both with a pure concentration gradient (iothermal ma diffuion) [20, 2] or with a concentration gradient induced by the Soret effect [, 22, 23]. Our purpoe here i to go beyond thee previou invetigation, into a q-range where confinement effect are to be expected. To oberve c-nef we ued the thermal-gradient cell ketched in Fig. : Two apphire window kept at fixed vertical ditance contain the fluid ample while being thermally controlled by two Peltier element with a central hole. The entire ytem allow a quai-monochromatic parallel light beam pa through in the direction of the temperature gradient. Further detail of the thermal gradient cell can be found elewhere [, 24]. A tabilizing temperature difference of T = 20 K (with an average temperature of T 0 =298 K) i applied to a horizontal layer of tetralin and n-dodecane at 50% weight fraction. The ample thickne can be varied by uing different platic pacer and ealing O-ring, and for thi work three thicknee L = 0.7,.3 and 5.0 mm (and a contant lateral extent of R = 3.0 mm) were ued. The olutal Rayleigh number are: Ra = 4 0 4, and 0 7, repectively [25, 26]. To apply a temperature difference by heating the fluid mixture from above reult in a linear temperature profile FIG.. Experimental cell: two apphire window are kept at different temperature T 0 + T/2 (the top, red one) and T 0 T/2 (the bottom, blue one) while the ample fluid (colored pattern) i contained by an O-ring (black circle) at a thickne L preciely defined by three platic pacer (gray rectangle). acro the ample in a thermal time τ T = L 2 /κ, where κ i the fluid thermal diffuivity. Due to the maller value of the ma diffuion coefficient, a nearly linear concentration profile i generated by the Soret effect [, 27] in a much larger olutal diffuion time τ = L 2 /D. Since our mixture ha a poitive eparation ratio, for negative Ra both the temperature and the concentration profile reult in a tabilizing denity profile [28]. Shadowgraphy [29 32] allow recording image whoe intenitie I(x, t) contain a mapping of the ample refractive-index fluctuation, over pace and time, averaged along the direction of the gradient. An example of one of thee image i hown in Fig. 2(a). Thee intenity pattern are generated at the enor plane by the heterodyne uperpoition of the light cattered by the ample refractive-index fluctuation and the much more intene tranmitted beam ( local ocillator ). Quantitative image analyi i performed by the Differential Dynamic Algorithm [, 20, 2, 33]. One firt compute difference of normalized image i m (x, t) a hown in Fig. 2(b). Thee difference image are then 2D-pace-Fourier tranformed in ilico, Fig. 2(c), to eparate the contribution of light cattered at different wave vector. Thi procedure provide reult imilar to conventional light cattering, but with a hadowgraph one can acce much maller wave vector. Quantitative image analyi yield the ocalled tructure function: C(q, t) = i m (q, t) 2 t, q =q = = i(q, t) i(q, t + t) 2 t, q =q, (3) with i(q, t) = F[I(x, t)/ I(x, t) x ] the 2D-Fourier tranform of a normalized image I(x, t) and t the time delay between the pair of analyzed image. Example of experimental C(q, t) are hown in Fig. 2(d-e). The phyical optic theory of hadowgraphy relate the tructure function to the ISF a [, 20, 2, 33]: C(q, t) = 2A{T (q)s(q) f(q, t) +B(q)}, (4)

3 3 where T (q) i the optical tranfer function of the intrument (a ocillating function for a hadowgraph, ee [30, 3]), S(q) the tatic tructure factor of c-nef, A an intenity pre-factor, and B(q) a background including all the phenomena with time-correlation function decaying fater than the CCD frame rate, uch a contribution due to hot noie and temperature fluctuation. Hence, experimental ISF f(q, t) can be evaluated via Eq. (3)- (4). Reult for three different wave vector are hown in Fig. 2(f). Eentially, for all the wave vector acceible in the experiment the ISF can be modeled by a ingle exponential function. For direct comparion with theory and imulation we extract effective experimental decay time a the time needed to f(q, t) to decay to /e. Figure 3 preent thee experimental decay time for the three Ra invetigated, the raw data in panel (a), and in dimenionle form in panel (b). Note that in Fig. 3(b) for almot all wave vector maller than q = 4 Ra, the effective decay time depart from the theoretical decription of Eq. (2), hown a a dahed line. A the wave vector decreae the decay time preent a minimum for a dimenionle wave vector q b = 5, while for maller wave vector the decay time recover a diffuive behavior τ q 2. Thee concluion are clear in Fig. 3 except for the larger Ra = 0 7, with no experimental point available at low enough q. To interpret thee experimental finding and undertand the phyical origin of the dicovered lowing-down of large-cale c-nef, we ue a Fluctuating Hydrodynamic (FHD) model [7] that incorporate gravity and confinement. FHD, baed on original idea by Onager and Landau, upplement diipative fluxe with random contribution o a to derive in a conitent way a fluctuating or tochatic verion of any thermodynamic or hydrodynamic problem [9]. Previou FHD invetigation of our problem [7] focued on the intenity (tatic) of the c-nef. Here we invetigate the dynamic and were able to expre the theoretical dynamic tructure factor a a erie of exponential: S(q)f(q, t) = N= [ A N (q) exp t ], (5) τ N (q) ee [34] for further detail. The decay time in Eq. (5) are the invere of the eigenvalue Γ N (q) = /τ N (q) dicued in Ref. [7]. The amplitude A N are analytically related to Γ N and q. The tatic tructure factor dicued in [7] i then S(q) = A N (q). In general, the Γ N can only be computed numerically, however, in the limit q 0, a full analytical invetigation i poible by mean of power expanion in q, and a clear hierarchy of well-eparated Γ N can be identified [7]. In that limit, the firt amplitude in Eq. (5) dominate, and f(q 0, t) become a ingle-exponential in practice, with decay time due to FIG. 2. (a) Shadowgraph image I(x, t); (b) difference of normalized image i m(x, t); (c) power pectrum of (b) i m(q, t) 2 ; (d) tructure function C(q, t) for three different time delay, vertical line tand for wave vector ued in (e); (e) tructure function C(q, t) for three different wave vector, vertical line tand for delay time ued in (d); (f) ISF for three different wave vector f(q, t): ymbol are for experimental data while line how ingle-exponential modeling. All data are for Ra = confinement (c): τ( q 0) c = ( q 2 Ra ) = ( q Ra 2 Ra ), (6),c 720 where Ra,c = 720 i the critical olutal Rayleigh number at which the convective intability appear in thi ytem [28]. Prediction from the aymptotic Eq. (6) are hown in Fig. 3(b) a dotted line. Hence, the theory how a croover from Eq. (2) (not-including confinement) at large and intermediate q, to the confinement behaviour of Eq. (6) at mall q, preciely the kind of behaviour experimentally oberved. We etimate the wave number q b correponding to the minimum decay time by equating Eq. (2) and (6). Thi give q b = 4 Ra,c = = 5.2 independent of Ra, in further agreement with the experimental obervation in Fig. 3(b). For arbitrary value of q, the decay time τ N (q) and correponding amplitude can only be evaluated numerically. We have performed a numerical invetigation for the experimental Ra [34], yielding imilar reult in the three cae. For very large q 50, all decay time collape to the bulk value, τ N q 2, and the ISF i approximately a ingle exponential. A already commented,

4 4 for very mall q 0.3 the firt mode dominate in amplitude and a ingle-exponential decay i again recovered, with decay time given by Eq. (6). For intermediate 0.6 q 30, the econd mode lead in amplitude but having a maller decay time mean that the two lower mode play a ignificant role and the theoretical ISF how ignature of a double exponential decay. Indeed, data from imulation how uch ignature in the predicted wave-vector range. However uch ignature were not detected in the experiment due to limited range, frame acquiition rate, and inufficient ignal to noie ratio. In Fig. 2(e) we reported three example of experimental ISF for different wave vector, with ingleexponential modeling. Regardle of the multiple exponential character, a ingle effective theoretical decay time τ eff (q) can be defined by f(q, τ eff ) = /e. In Fig. 3(b) we how reult for theoretical τ eff (q), computed via Eq. (5) from the numerical decay rate and amplitude. All the feature een in the experimental data are well reproduced by the theory. Noticeably the lowing-down oberved for mall wave number i clearly related to confinement, ince thi i the only ingredient added to the bulk theory of Eq. (2). The FHD theory [7] aume that vicou diipation dominate, and neglect the effect of fluid inertia; thi i jutified by the fact that in all liquid momentum diffuion i much fater than ma diffuion, i.e., the Schmidt number Sc = ν/d i very large. While neglecting inertial effect i a good approximation at mot wavenumber of interet, it i known that, depending on Ra, it fail at ufficiently mall wavenumber due to the appearance of propagative mode [35] (cloely related to gravity wave) driven by buoyancy. In order to confirm that the oberved lowing down i due to confinement and not to inertia we have performed FHD numerical imulation [36, 37] that account for inertial effect and confinement, ee [34] for further detail. Data point from a numerical imulation with fluid parameter matching the experimental one are alo diplayed in Fig. 3. The excellent agreement of thi dataet with experimental and theoretical reult, how that inertia effect are not relevant in our experiment. We note, however, that for thicknee L 5 mm the imulation do how ocillatory time-correlation function (propagative mode) at the mallet wavenumber [37], but thi range i not acceible in the experiment reported here. We conclude that, although confinement ha a moderate damping impact on the intenitie of large-cale nonequilibrium concentration fluctuation [7], in the preence of gravity, it trongly affect their dynamic. Our current finding are in contrat to the cae of diffuion in microgravity where the coupling to velocity fluctuation greatly enhance the intenity of the c-nef but doe not alter their Fickian diffuive dynamic [38]. Although the focu of thi letter i on the dynamic and we leave for future publication a full dicuion of FIG. 3. Effective decay time: (a) Log-log plot of the experimental decay time τ a a function of wave vector q for different Rayleigh number. (b) Same in term of dimenionle variable. In panel (b), filled red ymbol are experimental data, open blue are for calculation baed on the FHD model, and open-dotted black are from numerical imulation. Dahed curve repreent Eq. (2) for q > q b, which account for gravity and diffuion only. Dotted line repreent Eq. (6) for q < q b, which account for confinement. the tatic, we note that the minimum q b in τ eff correpond to a minimum in the intenity of fluctuation S(q). Hence, the current reult might be interpreted a a kind of de Genne narrowing [39]. In analogy to diffuion in colloid, where a competition between interparticle interaction and hydrodynamic effect exit, here we have competition between gravity and confinement. Interetingly, we find that the dimenionle wave number where confinement coexit with gravity i related to the critical olutal Rayleigh number Ra,c = 720 where the convective intability firt appear [28]. Thi i a ignature of the Onager regreion hypothei tating that the dynamic of the fluctuation contain all of the ignature een in the determinitic dynamic, which i known to be controlled by the Rayleigh number. We acknowledge fruitful dicuion with Alberto Vailati, Doriano Brogioli, Roberto Cerbino and Jan Senger. J.O.Z. acknowledge upport from the UCM- Santander Reearch Grant PR6/ during a

5 5 abbatical leave at Anglet. A.D. wa upported in part by the U.S. National Science Foundation under grant DMS-534 and the Office of Science of the U.S. Department of Energy through Early Career award number DE-SC Correpondence and requet for material hould be addreed to F.C. [] S. R. de Groot and P. Mazur, Nonequilibrium thermodynamic (North-Holland, Amterdam, 962). [2] S. Kjeltrup and D. Bedeaux, Non-Equilibrium Thermodynamic Of Heterogeneou Sytem (World Scientific, Singapore, 2008). [3] S. Hartmann, G. Wittko, W. Köhler, K.I. Morozov, K. Alber, and G. Sadowki, Phy. Rev. Lett. 09, (202). [4] S. Wiegand, H. Ning, and R. Kita, J. of Non-Equilibrium Thermodynamic 32, 93 (2007). [5] G. Galliero, and S. Volz, J. Chem. Phy. 28, (2008). [6] P.-A. Artola, and B. Roueau, Molecular Phyic,, 3394 (203). [7] T. R. Kirkpatrick, E. G. D. Cohen, and J. R. Dorfman, Phy. Rev. A 26, 995 (982). [8] J. R. Dorfman, T. R. Kirkpatrick, and J. V. Senger, Ann. Rev. Phy. Chem. 45, 23 (994). [9] J. M. Ortiz de Zárate and J. V. Senger, Hydrodynamic fluctuation in fluid and fluid mixture (Elevier, Amterdam, 2006). [0] J. V. Senger and J. M. H. L. Senger, Annu. Rev. Phy. Chem. 37, 89 (986). [] F. Croccolo, H. Bataller, and F. Scheffold, J. Chem. Phy. 37, (202). [2] T.R. Kirkpatrick, J.M. Ortiz de Zárate and J.V. Senger, Phy. Rev. Lett. 0, (203). [3] A. Najafi, and R. Goletanian, Europhy. Lett. 68, 776 (2004). [4] A. Hanke, PloS one 8, e53228 (203). [5] A. Vailati and M. Giglio, Nature 390, 262 (997). [6] A. Vailati and M. Giglio, Phy. Rev. E 58, 436 (998). [7] J. M. Ortiz de Zárate, J. A. Forné and J. V. Senger, Phy. Rev. E 74, (2006). [8] P. N. Segrè, R. Schmitz and J. V. Senger, Phyica A 95, 3 (993). [9] P. N. Segrè, and J. V. Senger, Phyica A 98, 46 (993). [20] F. Croccolo, D. Brogioli, A. Vailati, M. Giglio and D. S. Cannell, Phy. Rev. E 76, 042 (2007). [2] F. Croccolo, D. Brogioli, A. Vailati, M. Giglio and D. S. Cannell, App. Opt. 45, 266 (2006). [22] C. Giraudet, H. Bataller, and F. Croccolo, Eur. Phy. J. E, 37, 07 (204). [23] F. Croccolo, H. Bataller, and F. Scheffold, Eur. Phy. J. E, 37, 05 (204). [24] F. Croccolo, F. Scheffold and A. Vailati, Phy. Rev. Lett., (203). [25] ρ = g cm 3, D = cm 2, ν = cm 2, S T = K, β T = K, β = 0.27, ψ = c o( c o)s T β /β T = 0.695, κ = cm 2 from [26] and reference therein. [26] J. K. Platten, M. M. Bou-Ali, P. Coteèque, J. F. Dutrieux, W. Köhler, C. Leppla, S. Wiegand, and G. Wittko, Phil. Mag. 83, 965 (2003). [27] C. Soret, Arch. Sci. Phy. Nat. 3, 48 (879). [28] A. Rykin, H. W. Müller, and H. Pleiner, Phy. Rev. E 67, (2003). [29] G. S. Settle, Schlieren and Shadowgraph Technique (Springer, Berlin, 200). [30] S. Trainoff, and D. S. Cannell, Phy. Fluid 4, 340 (2002). [3] F. Croccolo, and D. Brogioli, App. Opt. 50, (20). [32] The probing beam i a plane parallel beam of quaimonochromatic light a in previou etup [, 24]. After the ample no collecting len i ued. A Charged Coupled Device enor (IDS, UI-6280SE-M-GL) with a reolution of pixel of µ m 2 i placed at a ditance of z = (00 ± 0) mm from the detector. Image are cropped to quare reolution of pixel. In thi arrangement the ize of the image i dictated by the real ize of the CCD enor, which i mm. Thi fixe the minimum wave vector to q min = 8.89cm. [33] R. Cerbino, and V. Trappe, Phy. Rev. Lett. 00, 8802 (2008). [34] C. Giraudet, H. Bataller, Y. Sun, A. Donev, J. M. Ortiz de Zárate and F. Croccolo, [35] C. J. Takac, G. Nikolaenko and D. S. Cannell, Phy. Fluid 00, (2008). [36] F. Balboa Uabiaga, J. B. Bell, R. Delgado-Bucalioni, A. Donev, T. G. Fai, B. E. Griffith, and C. S. Pekin, SIAM J. Multicale Model. Simul. 0, 369 (202). [37] S. Delong, Y. Sun, B.E. Griffith, E. Vanden-Eijnden and A. Donev, Phy. Rev. E, 90,, (204). [38] A. Vailati, R. Cerbino, S. Mazzoni, C. J. Takac, D. S. Cannell and M. Giglio, Nature Comm. 2, 290 (20). [39] P. G. de Genne, Phyica A 25, 825 (959).