Slowing-down of non-equilibrium concentration fluctuations in confinement

Abstract

Fluctuations in a fluid are strongly affected by the presence of a macroscopic gradient making them long-ranged and enhancing their amplitude. While small-scale fluctuations exhibit diffusive lifetimes, larger-scale fluctuations live shorter because of gravity, as theoretically and experimentally well-known. We explore here fluctuations of even larger size, comparable to the extent of the system in the direction of the gradient, and find experimental evidence of a dramatic slowing-down in their dynamics. We recover diffusive behaviour for these strongly-confined fluctuations, but with a diffusion coefficient that depends on the solutal Rayleigh number. Results from dynamic shadowgraph experiments are complemented by theoretical calculations and numerical simulations based on fluctuating hydrodynamics, and excellent agreement is found. The study of the dynamics of non-equilibrium fluctuations allows to probe and measure the competition of physical processes such as diffusion, buoyancy and confinement.PACS numbers: 05.70.Ln, 42.30.Va It is well established that fluctuations are long-ranged in systems out-of-equilibrium [1-3], even far from critical points where the long-range behaviour is observed also in equilibrium conditions [4]. In a binary fluid mixture subject to a stabilizing (vertical) temperature or concentration gradient, the coupling between the spontaneous velocity fluctuations and the macroscopic gradient results in giant concentration fluctuations in the quiescent state [3, 5]. Gravity quenches the intensity of fluctuations with length scales larger than a characteristic (horizontal) size 2π/q s related to the dimensionless solutal Rayleigh number Ra s of the system [5, 6]:where β s = ρ −1 (∂ρ/∂c) is the solutal expansion coefficient, ρ the fluid density, g the gravity acceleration, c the concentration (mass fraction) of the denser component of the fluid, ∇c the modulus of the concentration gradient, D the mass diffusion coefficient, ν the kinematic viscosity, and q s a characteristic solutal wave vector. Vertical boundaries suppress fluctuations larger than the confinement length L in the direction of the gradient [3, 7]. Gravity also accelerates the dynamics of the fluctuations for wavenumbers smaller than q s via buoyancy effects, leading to non-diffusive decay of large-scale fluctuations [8].The dynamics of concentration non-equilibrium fluctuations (c-NEFs) in the presence of a vertical concentration gradient in a binary liquid mixture can be characterized in terms of the Intermediate Scattering Function (ISF or, equivalently, normalized time correlation function) f (q, t), with f (q, 0) = 1. At first approximation the ISF can be modeled by a single exponential with decay time τ (q) depending on the analysed wave vector q.Available theories accounting for the simultaneous presence of diffusion (d) and gravity (g) [9, 10], but not for confinement, predict for a stable configuration (Ra s < 0):where the wave vector is expressed in its dimensionless formq = qL and τ s = L 2 /D is the typical so...

Correlations and scaling properties of nonequilibrium fluctuations in liquid mixtures

“…A confirmation of this result has been obtained in a recent study on the effect of the confinement on the dynamics of nonequilibrium concentration fluctuations [17]. This study showed that the confinement effect starts to be felt at the reduced wave number qL = 5.18 = Ra 1/4 sc , where Ra sc = 720 is the critical solutal Rayleigh number.…”

“…An additional mechanism breaking the scale invariance of the fluctuations at small wave vectors was predicted theoretically to be the finite size of the sample [13], a finding confirmed experimentally during the GRADFLEX experiment by the European Space Agency [14][15][16]. Recent experiments showed that finite-size effects also affect the dynamics of the fluctuations in the presence of gravity [17]. The current understanding of nonequilibrium fluctuations is that they do not represent a mere perturbation of macroscopic nonequilibrium.…”

“…In this case, both theoretical models [13] and experiments performed under microgravity conditions [14,16] have shown that the quenching of the fluctuations occurs at wave vectors smaller than Q ≈ π/L. Recently, the effect of confinement has been shown to affect the dynamics of the fluctuations on Earth, too [17]. A second cutoff mechanism is determined on Earth by the gravity force [7,8].…”

Gravity effects on Soret-induced non-equilibrium fluctuations in ternary mixtures

“…However, for the rest of this paper we shall neglect the presence of boundaries. Previous studies show that confinement affects both the amplitude [25,26] and the dynamics [27] of NE fluctuations at lateral wave numbers q L −1 . Hence, our current results are expected to be valid only for fluctuations with q large enough, while we leave for future publications a thorough investigation of confinement effects on NE fluctuations in ternary mixtures.…”

“…One other notable example of FHD success is the prediction of the influence of gravity on the fluctuations [23], an effect initially considered to be not accessible to experiments, and later confirmed by novel optical techniques [24]. Similarly, detailed FHD predictions about finite size effects on non-equilibrium fluctuations [25][26][27] have been later experimentally verified, by Gradflex [28] in microgravity and by ground-based measurements [27,29] in the presence of buoyancy force. Hence, as a preliminary step in developing the theory of thermodynamic fluctuations in NE ternary mixtures, it was necessary to re-derive [30] the equilibrium results for ternary mixtures on the basis of FHD, for which the simplifications of Bardow [4] were adopted.…”