Alberto Vailati

Giant fluctuations in a free diffusion process

Macroscopic concentration gradients in physical systems relax towards equilibrium by diffusion, in the absence of bulk motion. This is normally regarded as a spatially homogeneous mixing process. Here, however, we show that unexpectedly large spatial fluctuations in concentration can occur during a free diffusion process. We set up an initially sharp interface between two miscible fluids by letting a mixture phase-separate below the critical consolution temperature and then raising the temperature quickly to the single-phase region. Shadowgraph images and low-angle light scattering show evidence for large fluctuations in composition, orders of magnitude larger in amplitude than those seen in the equilibrium state. We show that these pronounced inhomogeneities are due to a coupling between velocity and concentration fluctuations in the non-equilibrium state. Gravity cuts off the fluctuations above a certain wavelength, and the amplitude of the fluctuations at longer wavelengths does not depend on any relevant thermodynamic property of the fluid. As a consequence, these giant fluctuations should be observable in any mixture undergoing mixing by diffusion.

Fractal fronts of diffusion in microgravity

Spatial scale invariance represents a remarkable feature of natural phenomena. A ubiquitous example is represented by miscible liquid phases undergoing diffusion. Theory and simulations predict that in the absence of gravity diffusion is characterized by long-ranged algebraic correlations. Experimental evidence of scale invariance generated by diffusion has been limited, because on Earth the development of long-range correlations is suppressed by gravity. Here we report experimental results obtained in microgravity during the flight of the FOTON M3 satellite. We find that during a diffusion process a dilute polymer solution exhibits scale-invariant concentration fluctuations with sizes ranging up to millimetres, and relaxation times as large as 1,000 s. The scale invariance is limited only by the finite size of the sample, in agreement with recent theoretical predictions. The presence of such fluctuations could possibly impact the growth of materials in microgravity.

Use of dynamic schlieren interferometry to study fluctuations during free diffusion

We used a form of schlieren interferometry to measure the mean-squared amplitude and temporal autocorrelation function of concentration fluctuations driven by the presence of a gradient during the free diffusion of a urea solution into water. By taking and processing sequences of images separated in time by less than the shortest correlation time of interest, we were able to simultaneously measure dynamics at a number of different wave vectors. The technique is conceptually similar to the shadowgraph method, which has been used to make similar measurements, but the schlieren method has the advantage that the transfer function is wave-vector independent rather than oscillatory.

Heterodyne near-field scattering

We describe an optical technique based on the statistical analysis of the random intensity distribution due to the interference of the near-field scattered light with the strong transmitted beam. It is shown that, from the study of the two-dimensional power spectrum of the intensity, one derives the scattered intensity as a function of the scattering wave vector. Near-field conditions are specified and discussed. The substantial advantages over traditional scattering technique are pointed out, and is indicated that the technique could be of interest for wavelengths other than visible light.

Diffusive mass transfer by nonequilibrium fluctuations: Fick's law revisited.

Recent experimental and theoretical works have shown that giant fluctuations are present during diffusion in liquid systems. We use linearized fluctuating hydrodynamics to calculate the net mass transfer due to these nonequilibrium fluctuations. Remarkably, the mass flow turns out to coincide with the usual Ficks one. The renormalization of the hydrodynamic equations allows us to quantify the gravitational modifications of the diffusion coefficient induced by the gravitational stabilization of long wavelength fluctuations.

Near-field intensity correlations of scattered light

We show that the two-point correlation function in the near field of scattered light is simply related to the scattered intensity distribution. We present a new, to our knowledge, optical scheme to measure the correlation function in the near field, and we describe a processing technique that permits the subtraction of stray light on a statistical basis. We present experimental data for solutions of latex spheres, and we show that this novel technique is a powerful alternative to static light scattering.

Effect of Gravity on the Dynamics of Nonequilibrium Fluctuations in a Free-Diffusion Experiment

Abstract:  Diffusion is commonly believed to be a homogeneous process at the mesoscopic scale, being driven only by the random walk of fluid molecules. On the contrary, very large amplitude, long wavelength fluctuations always accompany diffusive processes. 1–4 In the presence of gravity, fluctuations in a fluid containing a stabilizing gradient are affected by two different processes: diffusion, which relaxes them, and the buoyancy force, which quenches them. These phenomena affect both the overall amplitude of fluctuations and their time dependence. For the case of free diffusion, the time‐correlation function of the concentration fluctuations is predicted to exhibit an exponential decay with correlation time depending on the wave vector q. For large wave vector fluctuations, diffusion dominates, and the correlation time is predicted to be 1 / (Dq2). For small wave vector fluctuations, gravitational forces have time to play a significant role, and the correlation time is predicted to be proportional to q2. The effects of gravity and diffusion are comparable for a critical wave vector qc determined by fluid properties and gravity. We have utilized a quantitative dynamic shadowgraph technique to obtain the temporal correlation function of a mixture of LUDOX® TMA and water undergoing free diffusion. This technique allows one to simultaneously measure correlation functions achieving good statistics for a number of different wave vectors in a single measurement. Wave vectors as small as 70 cm−1 have been investigated, which is very difficult to achieve with ordinary dynamic light‐scattering techniques. We present results on the transition from the diffusive decay of fluctuations to the regime in which gravity is dominant.

Gradient-driven fluctuations experiment: fluid fluctuations in microgravity

We describe an experimental breadboard developed for the investigation of nonequilibrium fluctuations induced by macroscopic temperature and concentration gradients under microgravity conditions. Under these conditions the amplitude of the fluctuations diverges strongly for long wavelengths. The setup was developed at the University of Milan and at the University of California at Santa Barbara within the gradient-driven fluctuations experiment (GRADFLEX) project of the European Space Agency, in collaboration with the National Aeronautics and Space Administration. The apparatus uses a quantitative shadowgraph technique for characterization of the static power spectrum of the fluctuations S(q) and the measurement of their dynamics. We present preliminary experimental results for S(q) obtained in the presence of gravity for gradient-driven fluctuations for two cases, those induced in a liquid mixture with a concentration gradient produced by the Soret effect and those induced in a single-component fluid by a temperature gradient.

A schlieren method for ultra-low–angle light scattering measurements

We describe a self-calibrating optical technique that allows to perform absolute measurements of scattering cross-sections for the light scattered at extremely small angles. Very good performances are obtained by using a very simple optical layout similar to that used for the schlieren method, a technique traditionally used for mapping local refraction index changes. The scattered intensity distribution is recovered by a statistical analysis of the random interference of the light scattered in a half-plane of the scattering wave vectors and the main transmitted beam. High-quality data can be obtained by proper statistical accumulation of scattered intensity frames, and the static stray light contributions can be eliminated rigorously. The potentialities of the method are tested in a scattering experiment from non-equilibrium fluctuations during a free-diffusion experiment. Contributions of light scattered from length scales as long as Λ = 1 mm can be accurately determined.

Giant thermophoresis of poly(N-isopropylacrylamide) microgel particles

Thermophoresis is the rectification of Brownian motion induced by the presence of a thermal gradient ∇T, yielding a net drift of colloidal particles along or against the direction of ∇T. The effect is known to depend on the specific interactions between solute and solvent, and quantitative theoretical models are lacking except in a few simple experimental cases. Both the order of magnitude and the temperature dependence of the thermophoretic mobility DT are known to be very similar for a wide class of aqueous colloidal systems, ranging from latex colloids to polymers, surfactant micelles, proteins, and DNA. Here we show that thermoresponsive microgel particles made of poly(N-isopropylacrylamide) (PNIPAM) do not share, in the temperature range around the ϑ-point, these common features. Instead, DT displays an unusually strong temperature dependence, maintaining a linear growth across the collapse transition. This behaviour is not shared by linear PNIPAM chains, for which existing data show DT falling at the transition, with similar values between the expanded coil and collapsed globule states away from the transition point. A possible connection of the observed giant temperature dependence of DT to microgel hydration is suggested.