Velocity fluctuations during particle sedimentation probed by multispeckle XPCS

20-06-2017

Settling of suspensions under the influence of gravity is an everyday experience, commonly observed in a coffee cup or a beverage bottle. It is widely exploited in industrial purification and separation processes as well as being the origin of geological formations such as sedimentary rocks.

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Sedimentation at the colloidal scale has been extensively investigated for more than a century [1] and it has been identified as a powerful tool for probing the interactions and dynamics in particulate suspensions [2]. A sedimenting suspension is an out-of-equilibrium system and the physics of which has been at the forefront of modern statistical physics.  Despite over a century of research, many puzzles in sedimentation dynamics remain unresolved [1]. A particular case is when the thermally-driven Brownian motion is comparable to the gravity-induced flow, which is characterised by the Peclet number being similar to 1. The Peclet number (Pe) is the ratio of gravitational energy to thermal energy. If there are only a few independent particles in the suspension, their sedimentation velocity is given by the well-known Stokes law.  However, in a suspension with finite concentration, the hydrodynamic back flow causes a reduction of the sedimentation velocity from the Stokes velocity and fluctuations in the local particle number density lead to variance in the sedimentation speed [3]. Plausible divergence of these velocity fluctuations [3] up to the smallest container dimension is an unsettled issue in the physics of sedimentation [1].  

Nevertheless,  experimental conditions may preclude observation of the divergence of the velocity fluctuations, one such influence frequently encountered is the rapid onset of stratification (gravity induced concentration gradient) [1,4].  The stratification effects can be delayed to time scales of several hours by using Brownian particles (Pe < 1). But then it becomes experimentally challenging to separate contributions of gravity induced advective and thermal diffusive motions. Here we used multispeckle ultra-small-angle X-ray photon correlation spectroscopy (USA-XPCS) at beamline ID02 to probe the gradual transition of dynamics in sedimenting suspensions of charge-stabilised Brownian particles prior to the onset of the macroscopic sedimentation front. A key advantage of multispeckle XPCS is that it allows a direction dependent analysis of the dynamics (e.g., the gravity-induced advective motions along the vertical direction). The ultra-small-angle range provides access to the pertinent length scales. Figure 1 shows typical intensity autocorrelation functions measured along the vertical direction parallel to the sedimentation. The periodic modulation in the autocorrelation function is a signature of the well-defined advective motions that dominate the colloid dynamics during the early stages of sedimentation. With elapsing time, these periodic modulations become indistinct corresponding to a decay of advective currents and eventually diffusive motions become the dominating contribution in the dynamics.

Representative intensity autocorrelation functions as a function of elapsed time parallel to the sedimentation direction

Figure 1. (a) Representative intensity autocorrelation functions as a function of elapsed time parallel to the sedimentation direction. (b) Corresponding autocorrelation functions calculated from a model assuming a distribution of velocities around the mean value. (c) The ratio of vertical and horizontal functions displaying the purely advective contribution and the corresponding temporal evolution.

The measured XPCS correlation functions can be modelled by taking into account the Doppler shifts caused by all particle pairs in the scattering volume in terms of a relatively simple velocity distribution (Figure 1b). In the specific XPCS scheme, the decay of the correlation function is particularly sensitive to relative velocity fluctuations and the different scattering vector (q) dependence permits separation of the advective contribution (linear in q) from the conventional diffusive part (quadratic in q) as shown in Figure 1c.  These velocity fluctuations decay over the first hour of the experiment, and then the measured correlation functions along the vertical and horizontal directions become identical. In the picture of turbulence, this corresponds to velocity fluctuations not only slowing down but also decreasing in their overall extent, i.e. the breakup of flow structures over time.  This change of the dynamic behaviour occurs on a much shorter timescale (minutes) than the appearance of the macroscopic sedimentation front (several hours). Figure 2a displays the sensitivity of the initial condition on the decay of the velocity fluctuations (Vfluc) along the height of the capillary prior to the onset of the macroscopic sedimentation front. Nevertheless, once normalised by the initial Vfluc and the characteristic decay time (τ), all data superimpose on an exponential master curve as depicted in Figure 2b. A similar exponential time decay of velocity fluctuations has been observed in non-Brownian systems over larger size scales which have been attributed to the concentration stratification effect mentioned above [4]. However, a different mechanism operates in the case of Brownian particles which is likely to be the stronger coupling between advective and diffusive motions. 

Temporal decay of velocity fluctuations along the height of the sedimentation column

Figure 2. (a) Temporal decay of velocity fluctuations (Vfluc) along the height of the sedimentation column (z). (b) Normalised Vfluc (to its initial value, V0) as a function of the reduced time with t the characteristic decay time of velocity fluctuations for different z and colloid concentrations.

In summary, multispeckle USA-XPCS enabled velocity fluctuations to be studied over micrometre scales in colloidal sedimentation at low Pe regime (< 1). Well-defined advective motions are manifested as oscillations in the measured intensity autocorrelation functions that can be quantitatively described by a simple model involving velocity fluctuations around a mean sedimentation velocity.  When compared to non-Brownian particles, the timescales of macroscopic sedimentation and the exponential decay of the microscopic velocity fluctuations are well separated at low Pe range. Furthermore, when the diffusion and sedimentation persist on similar timescales, the two contributions are usually difficult to separate experimentally. This study illustrates the advanced experimental capabilities enabled by multi-speckle USA-XPCS to study faster out-of-equilibrium processes in colloidal systems. Applications include self-driven active colloids and particles in a strong external flow.

 

Principal publication and authors
Velocity fluctuations in sedimenting Brownian particles, J. Möller and T. Narayanan, Phys. Rev. Lett. 118, 198001 (2017); doi: 10.1103/PhysRevLett.118.198001.
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References
[1] S. Ramaswamy, Adv. Phys. 50, 297 (2001).
[2] R. Piazza, Rep. Prog. Phys. 77, 056602 (2014).
[3] R.E. Caflisch and J.H. Luke, Phys. Fluids 28, 759 (1985).
[4] S.-Y. Tee et al., Phys. Rev. Lett. 89, 054501 (2002).