Spreading dynamics of reactive surfactants driven by Marangoni convection
SSpreading dynamics of reactive surfactants driven by Marangoni convection
Thomas Bickel ∗ Univ. Bordeaux, CNRS, Laboratoire Ondes et Mati`ere d’Aquitaine (UMR 5798), F-33400 Talence, France
We consider the spreading dynamics of some insoluble surface-active species along an aqueousinterface. The model includes both diffusion, Marangoni convection and first-order reactionkinetics. An exact solution of the nonlinear transport equations is derived in the regime of largeSchmidt number, where viscous effects are dominant. We demonstrate that the variance of thesurfactant distribution increases linearly with time, providing an unambiguous definition for theenhanced diffusion coefficient observed in the experiments. The model thus presents new insightregarding the actuation of camphor grains at the water-air interface.
Keywords : enhanced diffusion, Marangoni convection, camphor boats.
Active particles are artificial systems that have theability to harness energy from their environment in orderto achieve self-propulsion. Although this issue has be-come increasingly popular over the last decade [1–3], thespontaneous motion of camphor grains at the water-airinterface has been documented for more than two cen-turies [4, 5]. The actuation mechanism of camphor boatsrelies on the surfactant properties of its constituants oncedissolved in water [6–9]. Motion then arises from the im-balance of interfacial tension along the contact line [10–12]. Other realizations of Marangoni surfers make use ofchemical reactions [13] or thermal energy [14–16] in orderto create and maintain surface tension gradients. This ef-fect is also of relevance in nature for the locomotion ofinsects [17].Self-propulsion of Marangoni surfers is accompanied bya flow in the aqueous phase. The flow contributes to theinteractions between particles [18–21] or with the bound-aries of the system [22–25]. Both the individual and col-lective dynamics that emerge are extremely rich and havenot been fully elucidated yet [26–28]. Still, several exper-iments have been performed recently in order to providequantitative information regarding the physico-chemicalparameters of the camphor-water system [29–31]. Theinterpretation of experimental data is delicate, however,since it requires a fine understanding of the relation to themodels’ parameters. For instance, the estimate for thediffusion coefficient of camphor molecules is particularlyintriguing: the value deduced from the experiments is in-deed 6 orders of magnitude larger than that under equi-librium condition [29–31]. This discrepancy is attributedto Marangoni-driven transport, though undeniable evi-dence is still missing. Our objective is thus to clarify theeffect of Marangoni convection on molecular transport,with particular emphasis on the camphor-water system.In this article, we focus on the spreading dynamics ofsome insoluble surface-active species along the free in-terface of a deep liquid layer. Axisymmetric spreadingdriven by Marangoni forces has been extensively stud-ied in various configurations — see for instance [32] for arecent overview. The dynamics typically exhibits power-law behaviors for the spreading radius vs. time, ξ ( t ) ∼ t a . A variety of spreading exponents a have been identi-fied [33, 34]. The latter are usually obtained through scal-ing analysis or numerical simulations. Here, we intend toderive an exact solution of the transport equations in-cluding reaction kinetics. This encompasses first-orderchemical reactions, or exchanges with the gaz phase ( e.g. camphor sublimation). The transport of surfactants isthus controlled by both advection, diffusion and reaction.The system under consideration is schematically drawnin Fig. 1. The interface coincides with the horizontalplane z = 0, the unit vector e z pointing upward. Define r (cid:107) = ( x, y ), the surfactant concentration Γ( r (cid:107) , t ) thenobeys the advection-reaction-diffusion equation ∂ t Γ + ∇ (cid:107) · (cid:0) v (cid:107) Γ (cid:1) = − k Γ + D ∇ (cid:107) Γ , (1)with D the diffusion coefficient and k the reaction rate.Here we use the notation A (cid:107) = ( − e z e z ) · A for thehorizontal projection of the vector field A on the freeinterface. Note that, in general, the interfacial velocity v (cid:107) is not divergence-free since ∇ (cid:107) · v (cid:107) = − ∂ z v z (cid:54) = 0. x z 0 FIG. 1. Schematic illustration of the spreading dynamicsof surface-active molecules along the interface. Transport isdriven by both diffusion, Marangoni convection and reactionkinetics. a r X i v : . [ c ond - m a t . s o f t ] A p r Marangoni self-convection involves a nonlinear cou-pling between the flow field v and the concentration Γ.For the sake of simplicity, we neglect fluid inertia andfocus on the incompressible Stokes regime η ∇ v = ∇ p , and ∇ · v = 0 , (2)with p the pressure and η the viscosity of the liquid.The Stokes Eq.s (2) have to be solved together with theboundary conditions at the free interface v z (cid:12)(cid:12) z =0 = 0 , (3a) η (cid:0) ∂ z v (cid:107) + ∇ (cid:107) v z (cid:1) (cid:12)(cid:12)(cid:12) z =0 = ∇ (cid:107) γ . (3b)The first condition (3a) implicitly assumes that the inter-face remains flat: although a tiny depression is expectednear the origin, the effect is negligible for the values of theparameters considered in this work [35]. The Marangonicondition (3b) states that an inhomogeneity of surfacetension induces a shear stress at the interface, thereforeleading to a flow in the aqueous phase [36].A key ingredient in the analysis is then provided bythe equation of state γ (Γ). At low surface coverage, onecan assume a linear relationship [37, 38] γ (Γ) = γ − γ ΓΓ , (4)with γ the surface tension of the clean interface, andΓ the concentration scale. We also define the positiveconstant γ = Γ | ∂γ/∂ Γ | that controls the strength ofthe Marangoni flow.The lack of analytical solution for the transport Eq. (1)makes it difficult to interpret experimental data. Toachieve an exact solution, it can first be noticed thatEqs. (2)–(4) provide a closed set of linear equations forthe velocity field. One thus expects a linear relation-ship between the effect — the flow in the bulk — andthe cause — concentration inhomogeneities along theinterface [39, 40]. In the derivation that follows, weshall make a further simplification and assume that thesystem is two-dimensional. The flow field then reads v ( x, z ) = v x ( x, z ) e x + v z ( x, z ) e z . Solving the Stokesequations in the deep water limit, one obtains [41] v x ( x,
0) = γ η Γ H [Γ]( x ) , (5)with the Hilbert transform operator H defined as [42, 43] H [ f ]( x ) = 1 π − (cid:90) ∞−∞ f ( y ) x − y y. . (6)Here, the dashed integral refers to the Cauchy principalvalue. These relations show that the flow depends onthe distribution of surfactants over the entire interface.This nonlocality is inherent to the long-range nature ofhydrodynamic interactions. To proceed further, it is convenient to consider rescaledvariables. We introduce ξ and Γ as the characteristiclength and concentration, respectively. One also definesthe Marangoni speed U = γ / (2 η ), as well as the as-sociated time scale τ = ξ /U . We then set ˜ x = x/ξ ,˜ t = t/τ , ˜Γ = Γ / Γ , and ˜ v = v/U . For notational conve-nience, the tilde mark is dropped where there is no chanceof confusion, keeping in mind that all physical quantitiesare non-dimensional. The set of Eqs. (1)–(5) can then bere-expressed as a single nonlinear equation ∂ t Γ + ∂ x (Γ H [Γ]) = − α Γ + β∂ x Γ . (7)The spreading dynamics is thus controlled by two dimen-sionless parameters, α = kτ and β = 2 ηD/ ( γ ξ ). Thelatter is just the inverse of the P´eclet number, β = Pe − ,with Pe = ξ U /D . For experimentally relevant values ξ ∼ − m, U ∼ − m · s − and D ∼ − m · s − ,one gets Pe ∼ . Diffusive transport is thus completelynegligible compared to advection, so that we are led toset β = 0 in Eq. (7). Regarding the first parameter, thereaction rate for camphor molecules is on the order of k ∼ − s − , so that α ∼ − . Camphor sublimationcan thus be considered as a disturbance with respect toadvection, even though it cannot be neglected on exper-imental time scales.The issue is then to solve Eq. (7) in the regime wherethe spreading dynamics is controlled by both advec-tion and reaction. Despite the simplification β = 0,Eq. (7) still involves a term that is nonlinear and nonlo-cal. This would in general render hopeless any attemptto tackle the problem analytically. We shall however by-pass this difficulty by considering a special initial distri-bution, characterized by its width and amplitude, andthen studying its time and space evolution [39]. Theconcentration is assumed to follow a semi-circle lawΓ( x, t ) = A ( t ) (cid:112) ξ ( t ) − x , (8)for | x | < ξ ( t ), and Γ( x, t ) = 0 otherwise. The two un-known functions A ( t ) and ξ ( t ) are positive, with initialvalues A (0) and ξ (0) that are left unspecified. The func-tional form Eq. (8) is motivated by two reasons. The firstis that Γ( x,
0) is a representation of the delta function inthe limit ξ (0) →
0, provided that A (0) = 2 / [ πξ (0) ]. Thesolution to be discussed below might therefore be thoughtas a fundamental solution of the advection-reaction equa-tion. The second argument lies in the properties of theHilbert transform that make this form of Γ( x, t ) partic-ularly well suited regarding the algebra [39, 42]. To pro-ceed, we first take the Hilbert transform of Eq. (7). In-serting the ansatz Eq. (8) then provides two conditionsfor | x | < ξ ( t ) and | x | > ξ ( t ), that both need to be satis-fied for all values of x . This eventually yields to a coupleof ordinary differential equations (cid:40) ˙ A + 2 A + α A = 0 , ˙ ξ = A ξ . (9) α = α = α = α = α = � / τ ( � ) FIG. 2. Time evolution of the amplitude A ( t ) of the distri-bution, for different values of α (rescaled units). The initialconditions are set to A (0) = ξ (0) = 1. Although nonlinear, this set of equations is now tractableanalytically. We thus obtain the amplitude of the distri-bution A ( t ) = α [2 + α A (0) − ] e αt − . (10)Regarding the width of the distribution, it is connectedto the amplitude through the simple relation A ( t ) ξ ( t ) = A (0) ξ (0) e − αt .The time evolution of A ( t ) and ξ ( t ) are shown in Figs. 2and 3, respectively. At short time t (cid:28)
1, the width ofthe distribution increases linearly ξ ( t ) = ξ (0) + 2 t , (11)while the amplitude is given by A ( t ) = 1 /ξ ( t ) . Wefurther note that the amplitude vanishes in the long-time limit, lim t →∞ A ( t ) = 0, whereas the width satu-rates to a finite value, lim t →∞ ξ ( t ) = ξ ∞ , with ξ ∞ = ξ (0) (2 A (0) + α ) /α . The transition between the twoasymptotic regimes occurs on time scales such that αt ∼ α (cid:28) α (cid:29)
1, the amplitudedecays as A ( t ) = A (0) e − αt . This is precisely the re-sult that would have been obtained if the advection termhad been discarded from the beginning. Indeed, Eq. (7)would simply come down to a first-order kinetics equa-tion ∂ t Γ = − α Γ, hence the exponential decay. It alsoappears that the initial spatial distribution remains un-altered when α (cid:29) ξ ( t ) = ξ (0) for all t .In order to relate our findings with experiments, it isnow appropriate to switch back to dimensional quanti-ties. We first emphasize that the prediction regarding thevariance σ ( t ) = ξ ( t ) − ξ (0) of the distribution is espe- α= α= α= α= α= - - - - t /τ σ ( t ) FIG. 3. Time evolution of the variance σ ( t ) = ξ ( t ) − ξ (0) in logarithmic scale, for different values of α (rescaled units).The initial conditions are set to A (0) = ξ (0) = 1. cially relevant. Indeed, Eq. (11) suggests that the spread-ing dynamics shares many similarities with a purely dif-fusive process, even though diffusion has been explicitlyignored. An apparent diffusion coefficient D can then bedefined according to σ ( t ) = 2 D t , yielding D = U ξ = γ ξ η . (12)Taking as previously ξ ∼ − m and U ∼ − m · s − ,we obtain D ∼ − m · s − . This value has to be com-pared to that measured recently for the camphor-watersystem. In the experiments, the expansion of the cam-phor layer is visualized using calcium sulfate powder dis-persed at the water-air interface [29–31]. The apparentdiffusion coefficient is then extracted from the times se-ries of σ , leading to D exp ∼ − m · s − . Our roughestimate is thus in fairly good agreement with the ex-perimental value. But more importantly, the theoreticalanalysis provides a definite interpretation to the conceptof effective transport coefficient. It also elucidates the re-lation between enhanced diffusion and Marangoni flow.At this point, it is worth mentioning that a complemen-tary approach has been proposed recently [44]. Startingfrom the well-known stationary solution of the reaction-diffusion equation (Pe = 0), the authors account for ad-vection by invoking a wave-number-dependent diffusioncoefficient. They obtain D = D (1 + κ Pe) in the large-scale limit, with κ a positive constant. Although theeffect of the Marangoni flow cannot be rigorously repre-sented as a diffusion process, it is interesting to note thatthe approximate law derived in [44] is consistent with ourexact result Eq. (12) in the limit Pe (cid:29) t (cid:38) k − , when sublimation takesover. The variance is then no longer a linear functionof time but saturates to the finite value ξ ∞ . One couldargue that this effect might as well be interpreted as theconsequence of confinement, as if diffusion were restrictedto a region of finite extension [45]. But the fundamen-tal solution of the reaction-diffusion equation does notexhibit such saturation [46]. The long-time behavior ofthe variance is thus a definite signature of the dominanttransport process — provided that ξ ∞ remains smallerthan the system size L . Otherwise, additional finite-sizeeffects have to be included in the theory.So far, we have restricted the discussion to the Stokesregime, even though the Reynolds number Re = U ξ /ν may be finite ( ν = η/ρ is the kinematic viscosity, with ρ the fluid density). One gets for instance Re ∼ for camphor-water systems considered in this work. Butsince the Schmidt number Sc = Pe / Re = ν/D is on theorder of Sc ∼ , the advection term in Eq. (1) is ex-pected to be the dominant nonlinearity and to control theoverall dynamics. Fluid inertia might nevertheless be rel-evant in the early stages of the spreading process. Whenthe dynamics is controlled by momentum transport, thestationary [47–49] as well as unsteady [33] structure ofthe flow is self-similar. The competition between bulkand surface stresses can then be expressed by scalingrelations [33]. In the horizontal direction, the relevantlength is ξ so that the fluid velocity scales as v ∼ ξ/t .The thickness δ of the boundary layer sets the lengthscale in the depthwise direction. The balance of viscousforces ηv/δ and fluid inertia ρv /ξ leads to δ ∼ νt .Equating the Marangoni and the shear stress at the in-terface gives γ Γ / ( ξ Γ ) ∼ ηv/δ . Finally, enforcing massconservation Γ ξ = Γ ξ at short times t (cid:28) k − yieldsanother diffusive behavior ξ ( t ) ∼ D t , with D = ν / ( U ξ ) / = (cid:18) γ ξ ηρ (cid:19) / . (13)One gets the numerical value D ∼ − m · s − for ξ ∼ − m and U ∼ − m · s − . The contributionthat arises from the nonlinearities of the Navier-Stokesequation is thus one order of magnitude smaller thanEq. (12).To summarize, we have derived an exact solution forthe nonlinear spreading dynamics of reactive surfactantmolecules at the water-air interface. It is shown that thevariance of an initial surfactant distribution increases lin-early with time, thus providing an unambiguous defini- tion of the apparent diffusion coefficient frequently in-voked in the literature. These conclusion have been ob-tained in the (experimentally relevant) regime of largeSchmidt number Sc (cid:29)
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