Nonequilibrium Concentration Fluctuations Research Articles

Dynamics of non-equilibrium concentration fluctuations during free-diffusion in highly stratified solutions of glycerol and water.

We investigate the non-equilibrium fluctuations occurring during free diffusion between two solutions of glycerol and water with various concentration differences. The non-linearity of the system, determined by the strong stratification of the sample, requires introducing an interpretation model able to characterize the dependence of the correlation properties of the non-equilibrium fluctuations on the local thermophysical variables of the system. The proposed model allows us to characterize the dynamics of non-equilibrium fluctuations in the presence of a wide range of relaxation times determined by the strong stratification of the sample, at variance with the cumulant methods commonly used in dynamic light scattering experiments, which work well in the presence of a moderate dispersion of relaxation times.

The Kraichnan Model and Non-equilibrium Statistical Physics of Diffusive Mixing

We discuss application of methods from the Kraichnan model of turbulent advection to the study of non-equilibrium concentration fluctuations arising during diffusion in liquid mixtures at high Schmidt numbers. This approach treats nonlinear advection of concentration fluctuations exactly, without linearization. Remarkably, we find that static and dynamic structure functions obtained by this method reproduce precisely the predictions of linearized fluctuating hydrodynamics. It is argued that this agreement is an analogue of anomaly non-renormalization which does not, however, protect higher-order multi-point correlations. The latter should thus yield non-vanishing cumulants, unlike those for the Gaussian concentration fluctuations predicted by linearized theory.

Giant Fluctuations Induced by Thermal Diffusion in Complex Liquids

The GRADFLEX experiment of ESA has shown that under microgravity conditions a stationary thermodiffusion process is accompanied by giant non-equilibrium fluctuations with size as large as the size of the sample. In the presence of small concentration gradients, the features of the non-equilibrium fluctuations can be described by means of linearized hydrodynamics. However, the linear models are not suitable to describe most cases of applicative interest, such as fluctuations induced by large gradients and under non-stationary conditions. Moreover, presently the investigation of non-equilibrium fluctuations has mainly involved single component fluids and binary mixtures, but recently transport processes in ternary mixtures have attracted increasing interest due to the experiments performed on the International Space Station in the framework of the DCMIX project of ESA. The Giant Fluctuations (NEUF-DIX) project of ESA will investigate non-equilibrium fluctuations during diffusive processes occurring in complex multi-component mixtures, where one of the components is a macromolecule, such as a polymer, a colloid or a protein. Important objectives will be the exploration of the features of the fluctuations under non-ideal conditions, such as large gradients, transient processes, and concentrated samples, and the understanding of how the fluctuations affect the interactions between macromolecules. The project involves the development of a dedicated facility, consisting of an array of shadowgraph optical instruments working in parallel, each one equipped with a thermal gradient cell. Here we outline the design concept of the facility and the results of performance tests performed on a breadboard to evaluate the suitability of the designed instrument to carry out scientific measurements of non-equilibrium concentration fluctuations in space.

Asymmetric time-cross-correlation of nonequilibrium concentration fluctuations in a ternary liquid mixture.

Equilibrium phenomena are characterized by time symmetry. Thermodynamic fluctuations are also time-symmetric at equilibrium. Conversely, diffusion of a solute in a liquid in the presence of a gradient is a nonequilibrium phenomenon, which gives rise to long-range fluctuations with amplitude much larger than the equilibrium one for small enough wave number. In the case of diffusion in binary mixtures such fluctuations are time-symmetric, notwithstanding the fact that they are generated by a nonequilibrium condition. In this paper, we investigate diffusion of two solutes in a ternary liquid mixture by means of fluctuating hydrodynamics theory. We show that the time-cross-correlation function of the concentrations is not time-symmetric, hence showing that time symmetry is violated for such nonequilibrium fluctuations. We discuss the feasibility of experiments aimed at the detection of the asymmetry of the cross-correlation function of nonequilibrium concentration fluctuations in ternary mixtures, as envisaged in the Giant Fluctuations (NEUF-DIX) microgravity project of the European Space Agency.

Kinetics of growth of non-equilibrium fluctuations during thermodiffusion in a polymer solution.

A thermal diffusion process occurring in a binary liquid mixture is accompanied by long ranged non-equilibrium concentration fluctuations. The amplitude of these fluctuations at large length scales can be orders of magnitude larger than that of equilibrium ones. So far non-equilibrium fluctuations have been mainly investigated under stationary or quasi-stationary conditions, a situation that allows to achieve a detailed statistical characterization of their static and dynamic properties. In this work we investigate the kinetics of growth of non-equilibrium concentration fluctuations during a transient thermodiffusion process, starting from a configuration where the concentration of the sample is uniform. The use of a large molecular weight polymer solution allows to attain a slow dynamics of growth of the macroscopic concentration profile. We focus on the development of fluctuations at small wave vectors, where their amplitude is strongly limited by the presence of gravity. We show that the growth rate of non-equilibrium fluctuations follows a power law [Formula: see text] as a function of time, without any typical time scale and independently of the wave vector. We formulate a phenomenological model that allows to relate the rate of growth of non-equilibrium fluctuations to the growth of the macroscopic concentration profile in the absence of arbitrary parameters.

Propagating modes in a binary liquid mixture under thermal stress.

Nonequilibrium temperature and concentration fluctuations inside a binary liquid mixture under the action of a temperature gradient relax back to equilibrium either due to conduction and diffusion at large wave numbers, or due to the quenching determined by gravity at small wave numbers. We investigate the dynamics of nonequilibrium fluctuations in a binary liquid mixture of polystyrene and toluene heated from above under stationary conditions in a thermodiffusion experiment. We show that the strong gravitational stabilization at small wave numbers determines the appearance of propagating modes of nonequilibrium fluctuations as detected through the structure function of shadowgraph images. The propagating modes are the combined effect of temperature and velocity nonequilibrium fluctuations induced by the buoyancy force. The experimental results are in good agreement with a fluctuating hydrodynamics theroretical model including the coupling of fluctuations of velocity, temperature and concentration.

Nonequilibrium fluctuations during diffusion in liquid layers.

Theoretical analysis and experiments have provided compelling evidence of the presence of long-range nonequilibrium concentration fluctuations during diffusion processes in fluids. In this paper, we investigate the dependence of the features of the fluctuations from the dimensionality of the system. In three-dimensional fluids the amplitude of nonequilibrium fluctuations can become several orders of magnitude larger than that of equilibrium fluctuations. Notwithstanding that, the amplitude of nonequilibrium fluctuations remains small with respect to the concentration difference driving the diffusion process. By extending the theory to two-dimensional systems, such as liquid monolayers and bilayers, we show that the amplitude of the fluctuations becomes much stronger than in three-dimensional systems. We investigate the properties of the fronts of diffusion and show that they have a self-affine structure characterized by a Hurst exponent H=1. We discuss the implications of these results for diffusion in liquid crystals and in cellular membranes of living organisms.

Soret coefficient of the n-dodecane-n-hexane binary mixture under high pressure.

In the present work, the Soret coefficient has been determined at high pressure for a binary hydrocarbon mixture by combining the thermogravitational column and the dynamic near-field imaging techniques. The analyzed mixture is an iso-massic n -dodecane-n -hexane mixture at 298.15K. The molecular diffusion coefficient has been measured up to 20MPa by means of the dynamic analysis of the light scattered by non-equilibrium concentration fluctuations. With a cylindrical thermogravitational column the thermodiffusion coefficient was determined from 0.1MPa to 10MPa. Density, as well as, mass expansion and thermal expansion have been measured with a high pressure densimeter. Dynamic viscosity at up to 20MPa has been determined with a high pressure viscometer. This work shows the decreasing tendency of both the molecular diffusion and the thermodiffusion coefficient with increasing pressure.

Non-equilibrium concentration fluctuations in superparamagnetic nanocolloids.

We investigate non-equilibrium concentration fluctuations during the free diffusion of a colloidal suspension against pure water. We investigate Fe2O3 superparamagnetic nanocolloids with sizes between 1 and 10 nm by means of a shadowgraph apparatus to determine the mixture mass diffusion coefficient and kinematic viscosity. The experiments were performed in three distinct conditions: Experiment 1 is without any magnetic field; Experiment 2 with a vertical magnetic field; Experiment 3 after turning off the magnetic field. We found no correlation between the kinematic viscosity coefficient and the external magnetic field. Conversely, we found that the mass diffusion coefficient decreases in the presence of the external magnetic field and slowly rebounds after the magnetic field was turned off.

Confinement effect on the dynamics of non-equilibrium concentration fluctuations far from the onset of convection.

In a recent letter (C. Giraudet et al., EPL 111, 60013 (2015)) we reported preliminary data showing evidence of a slowing-down of non-equilibrium fluctuations of the concentration in thermodiffusion experiments on a binary mixture of miscible fluids. The reason for this slowing-down was attributed to the effect of confinement. Such tentative explanation is here experimentally corroborated by new measurements and theoretically substantiated by studying analytically and numerically the relevant fluctuating hydrodynamics equations. In the new experiments presented here, the magnitude of the temperature gradient is changed, confirming that the system is controlled solely by the solutal Rayleigh number, and that the slowing-down is dominated by a combined effect of the driving force of buoyancy, the dissipating force of diffusion and the confinement provided by the vertical extension of the sample cell. Moreover, a compact phenomenological interpolating formula is proposed for easy analysis of experimental results.

Equilibrium and non-equilibrium concentration fluctuations in a critical binary mixture.

When a macroscopic concentration gradient is present across a binary mixture, long-ranged non-equilibrium concentration fluctuations (NCF) appear as a consequence of the coupling between the gradient and spontaneous equilibrium velocity fluctuations. Long-ranged equilibrium concentration fluctuations (ECF) may be also observed when the mixture is close to a critical point. Here we study the interplay between NCF and critical ECF in a near-critical mixture aniline/cyclohexane in the presence of a vertical concentration gradient. To this aim, we exploit a commercial optical microscope and a simple, custom-made, temperature-controlled cell to obtain simultaneous static and dynamic scattering information on the fluctuations. We first characterise the critical ECF at fixed temperature T above the upper critical solution temperature Tc, in the wide temperature range [Formula: see text] °C. In this range, we observe the expected critical scaling behaviour for both the scattering intensity and the mass diffusion coefficient and we determine the critical exponents [Formula: see text], [Formula: see text] and [Formula: see text], which are found in agreement with the 3D Ising values. We then study the system in the two-phase region (T < T c). In particular, we characterise the interplay between ECF and NCF when the mixture, initially at a temperature Ti, is rapidly brought to a temperature T f > T i. During the transient, a vertical diffusive mass flux is present that causes the onset of NCF, whose amplitude vanishes with time, as the flux goes to zero. We also study the time dependence of the equilibrium scattering intensity I eq, of the crossover wave vector q co and of the diffusion coefficient D during diffusion and find that all these quantities exhibit an exponential relaxation enslaved to the diffusive kinetics.

Correlations and scaling properties of nonequilibrium fluctuations in liquid mixtures.

Diffusion in liquids is accompanied by nonequilibrium concentration fluctuations spanning all the length scales comprised between the microscopic scale a and the macroscopic size of the system, L. Up to now, theoretical and experimental investigations of nonequilibrium fluctuations have focused mostly on determining their mean-square amplitude as a function of the wave vector. In this work, we investigate the local properties of nonequilibrium fluctuations arising during a stationary diffusion process occurring in a binary liquid mixture in the presence of a uniform concentration gradient, ∇c_{0}. We characterize the fluctuations by evaluating statistical features of the system, including the mean-square amplitude of fluctuations and the corrugation of the isoconcentration surfaces; we show that they depend on a single mesoscopic length scale l=sqrt[aL] representing the geometric average between the microscopic and macroscopic length scales. We find that the amplitude of the fluctuations is very small in practical cases and vanishes when the macroscopic length scale increases. The isoconcentration surfaces, or fronts of diffusion, have a self-affine structure with corrugation exponent H=1/2. Ideally, the local fractal dimension of the fronts of diffusion would be D_{l}=d-H, where d is the dimensionality of the space, while the global fractal dimension would be D_{g}=d-1. The transition between the local and global regimes occurs at a crossover length scale of the order of the microscopic length scale a. Therefore, notwithstanding the fact that the fronts of diffusion are corrugated, they appear flat at all the length scales probed by experiments, and they do not exhibit a fractal structure.

Rheology modulated non-equilibrium fluctuations in time-dependent diffusion processes

The effect of non-Newtonian rheology, manifested through a viscoelastic linearized Maxwell model, on the time-dependent non-equilibrium concentration fluctuations due to free diffusion as well as thermal diffusion of a species is analyzed theoretically. The non-equilibrium process is quantified through the concentration fluctuation auto-correlation function, also known as the dynamic structure factor. The analysis reveals that the effect of rheology is prominent for both the cases of free diffusion and thermal diffusion at long times where the rheology dictates not only the location of the peaks in concentration auto-correlations, but also the magnitudes; such peaks in the auto-correlation function are absent in the case of a Newtonian fluid. At smaller times, for the case of free diffusion presence of time-dependent peak(s) are observed, which are weakly dependent on the influence of rheology, a phenomenon which is absent in the case of thermal diffusion. Lastly, different regimes of the frequency dependent overall dynamic structure factor depending on the interplay of the fluid relaxation time and momentum diffusivity are evaluated.

Nonequilibrium fluctuation-induced Casimir pressures in liquid mixtures

In this article we derive expressions for Casimir-like pressures induced by nonequilibrium concentration fluctuations in liquid mixtures. The results are then applied to liquid mixtures in which the concentration gradient results from a temperature gradient through the Soret effect. A comparison is made between the pressures induced by nonequilibrium concentration fluctuations in liquid mixtures and those induced by nonequilibrium temperature fluctuations in one-component fluids. Some suggestions for experimental verification procedures are also presented.

Variance-corrected Michaelis-Menten equation predicts transient rates of single-enzyme reactions and response times in bacterial gene-regulation.

Many chemical reactions in biological cells occur at very low concentrations of constituent molecules. Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E.coli lac repressor with few tens of copies. Here we study the effects of inherent concentration fluctuations of substrate-molecules on the seminal Michaelis-Menten scheme of biochemical reactions. We present a universal correction to the Michaelis-Menten equation for the reaction-rates. The relevance and validity of this correction for enzymatic reactions and intracellular gene-regulation is demonstrated. Our analytical theory and simulation results confirm that the proposed variance-corrected Michaelis-Menten equation predicts the rate of reactions with remarkable accuracy even in the presence of large non-equilibrium concentration fluctuations. The major advantage of our approach is that it involves only the mean and variance of the substrate-molecule concentration. Our theory is therefore accessible to experiments and not specific to the exact source of the concentration fluctuations.

Dynamic scaling for the growth of non-equilibrium fluctuations during thermophoretic diffusion in microgravity.

Diffusion processes are widespread in biological and chemical systems, where they play a fundamental role in the exchange of substances at the cellular level and in determining the rate of chemical reactions. Recently, the classical picture that portrays diffusion as random uncorrelated motion of molecules has been revised, when it was shown that giant non-equilibrium fluctuations develop during diffusion processes. Under microgravity conditions and at steady-state, non-equilibrium fluctuations exhibit scale invariance and their size is only limited by the boundaries of the system. In this work, we investigate the onset of non-equilibrium concentration fluctuations induced by thermophoretic diffusion in microgravity, a regime not accessible to analytical calculations but of great relevance for the understanding of several natural and technological processes. A combination of state of the art simulations and experiments allows us to attain a fully quantitative description of the development of fluctuations during transient diffusion in microgravity. Both experiments and simulations show that during the onset the fluctuations exhibit scale invariance at large wave vectors. In a broader range of wave vectors simulations predict a spinodal-like growth of fluctuations, where the amplitude and length-scale of the dominant mode are determined by the thickness of the diffuse layer.

Non-equilibrium concentration fluctuations in binary liquids with realistic boundary conditions.

Because of the spatially long-ranged nature of spontaneous fluctuations in thermal non-equilibrium systems, they are affected by boundary conditions for the fluctuating hydrodynamic variables. In this paper we consider a liquid mixture between two rigid and impervious plates with a stationary concentration gradient resulting from a temperature gradient through the Soret effect. For liquid mixtures with large Lewis and Schmidt numbers, we are able to obtain explicit analytical expressions for the intensity of the non-equilibrium concentration fluctuations as a function of the frequency ω and the wave number q of the fluctuations. In addition we elucidate the spatial dependence of the intensity of the non-equilibrium fluctuations responsible for a non-equilibrium Casimir effect.

Nonequilibrium Casimir-like Forces in Liquid Mixtures.

In this Letter, we consider a liquid mixture confined between two thermally conducting walls subjected to a stationary temperature gradient. While in a one-component liquid nonequilibrium fluctuation forces appear inside the liquid layer, nonequilibrium fluctuations in a mixture induce a Casimir-like force on the walls. The physical reason is that the temperature gradient induces large concentration fluctuations through the Soret effect. Unlike temperature fluctuations, nonequilibrium concentration fluctuations are also present near a perfectly thermally conducting wall. The magnitude of the fluctuation-induced Casimir force is proportional to the square of the Soret coefficient and is related to the concentration dependence of the heat and volume of mixing.

Low Mach number fluctuating hydrodynamics of multispecies liquid mixtures

We develop a low Mach number formulation of the hydrodynamic equations describing transport of mass and momentum in a multispecies mixture of incompressible miscible liquids at specified temperature and pressure, which generalizes our prior work on ideal mixtures of ideal gases [Balakrishnan et al., “Fluctuating hydrodynamics of multispecies nonreactive mixtures,” Phys. Rev. E 89 013017 (2014)] and binary liquid mixtures [Donev et al., “Low mach number fluctuating hydrodynamics of diffusively mixing fluids,” Commun. Appl. Math. Comput. Sci. 9(1), 47-105 (2014)]. In this formulation, we combine and extend a number of existing descriptions of multispecies transport available in the literature. The formulation applies to non-ideal mixtures of arbitrary number of species, without the need to single out a “solvent” species, and includes contributions to the diffusive mass flux due to gradients of composition, temperature, and pressure. Momentum transport and advective mass transport are handled using a low Mach number approach that eliminates fast sound waves (pressure fluctuations) from the full compressible system of equations and leads to a quasi-incompressible formulation. Thermal fluctuations are included in our fluctuating hydrodynamics description following the principles of nonequilibrium thermodynamics. We extend the semi-implicit staggered-grid finite-volume numerical method developed in our prior work on binary liquid mixtures [Nonaka et al., “Low mach number fluctuating hydrodynamics of binary liquid mixtures,” arXiv:1410.2300 (2015)] and use it to study the development of giant nonequilibrium concentration fluctuations in a ternary mixture subjected to a steady concentration gradient. We also numerically study the development of diffusion-driven gravitational instabilities in a ternary mixture and compare our numerical results to recent experimental measurements [Carballido-Landeira et al., “Mixed-mode instability of a miscible interface due to coupling between Rayleigh–Taylor and double-diffusive convective modes,” Phys. Fluids 25, 024107 (2013)] in a Hele-Shaw cell. We find that giant nonequilibrium fluctuations can trigger the instability but are eventually dominated by the deterministic growth of the unstable mode, in both quasi-two-dimensional (Hele-Shaw) and fully three-dimensional geometries used in typical shadowgraph experiments.

Multiscale temporal integrators for fluctuating hydrodynamics

Following on our previous work [S. Delong, B. E. Griffith, E. Vanden-Eijnden, and A. Donev, Phys. Rev. E 87, 033302 (2013)], we develop temporal integrators for solving Langevin stochastic differential equations that arise in fluctuating hydrodynamics. Our simple predictor-corrector schemes add fluctuations to standard second-order deterministic solvers in a way that maintains second-order weak accuracy for linearized fluctuating hydrodynamics. We construct a general class of schemes and recommend two specific schemes: an explicit midpoint method and an implicit trapezoidal method. We also construct predictor-corrector methods for integrating the overdamped limit of systems of equations with a fast and slow variable in the limit of infinite separation of the fast and slow time scales. We propose using random finite differences to approximate some of the stochastic drift terms that arise because of the kinetic multiplicative noise in the limiting dynamics. We illustrate our integrators on two applications involving the development of giant nonequilibrium concentration fluctuations in diffusively mixing fluids. We first study the development of giant fluctuations in recent experiments performed in microgravity using an overdamped integrator. We then include the effects of gravity and find that we also need to include the effects of fluid inertia, which affects the dynamics of the concentration fluctuations greatly at small wave numbers.