Resolving a paradox of anomalous scalings in the diffusion of granular materials

Christov, Ivan; Stone, Howard A.

doi:10.1073/pnas.1211110109

Cited by 34 publications

(41 citation statements)

References 50 publications

(80 reference statements)

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“…The result is shown in inset of the same figure. All the curves collapse onto a single curve, demonstrating the self-similarity of the long-term evolution. However, this intermediate asymptotics appears to be reached only at large times, as in [42]. During the short-term evolution, the oscillations are observed to grow with time before saturating.…”

Section: Resultsmentioning

confidence: 93%

Numerical solutions of thin-film equations for polymer flows

et al. 2012

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We report on the numerical implementation of thin-film equations that describe the capillary-driven evolution of viscous films, in two-dimensional configurations. After recalling the general forms and features of these equations, we focus on two particular cases inspired by experiments: the leveling of a step at the free surface of a polymer film, and the leveling of a polymer droplet over an identical film. In each case, we first discuss the long-term self-similar regime reached by the numerical solution before comparing it to the experimental profile. The agreement between theory and experiment is excellent, thus providing a versatile probe for nanorheology of viscous liquids in thin-film geometries.

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Section: Resultsmentioning

confidence: 93%

Numerical solutions of thin-film equations for polymer flows

et al. 2012

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“…The emergence of the core is well understood as the consequence of a kinetic sieving effect. However, there is still an open scientific debate [25,28,37,41,[44][45][46] triggered by Khan and Morris [24], whether the transport of the beads along the core is subdiffusive or diffusive.…”

Section: Introductionmentioning

confidence: 99%

The mechanism of long-term coarsening of granular mixtures in rotating drums

2015

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Three fundamental segregation and pattern formation processes are known in granular mixtures in a rotating cylindrical drum: radial segregation, axial banding, and coarsening of the band pattern. While the mechanism for the first effect is well understood and for the second effect, several models have been proposed, the long-term coarsening mechanism remained unexplained so far. We demonstrate that the unidirectional flow between the bands in an axially segregated pattern is driven by small differences in size of the small beads at the band edges. Due to a process of microsegregation inside each band of small particles, which was so far unrecognized, this difference in diameter will be effective in all experiments with polydisperse beads. In consequence the stability of individual bands can be easily controlled by minor alterations of their composition. Our results make evident that a new mechanism as the driving force behind the axial particle flow has to be sought. We suggest possible hypotheses for such a mechanism.

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“…In particular, we show that the regular part of this essential solution converges in time towards the Green's function of the purely viscous linear thin film equation [34]. Using these results, we then analyse the temporal relaxation of a canonical Gaussian height perturbation on a thin viscoelastic film, and study its evolution in terms of a transient levelling exponent [46]. Finally, we discuss the different observed regimes.…”

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confidence: 99%

Viscoelastic effects and anomalous transient levelling exponents in thin films

Benzaquen¹,

Salez²,

Raphaël³

2014

EPL

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-We study theoretically the profile evolution of a thin viscoelastic film supported onto a no-slip flat substrate. Due to the nonconstant initial curvature at the free surface, there is a flow driven by Laplace pressure and mediated by viscoelasticity. In the framework of lubrication theory, we derive a thin film equation that contains local viscoelastic stress through the Maxwell model. Then, considering a sufficiently regular small perturbation of the free surface, we linearise the equation and derive its general solution. We analyse and discuss in details the behaviour of this function. We then use it to study the viscoelastic evolution of a Gaussian initial perturbation through its transient levelling exponent. For initial widths of the profile that are smaller than a characteristic length scale involving both the film thickness and the elastocapillary length, this exponent is shown to reach anomalously high values at the elastic-to-viscous transition. This prediction should in particular be observed at sufficiently short times in experiments on thin polymer films.Over the past decades, the study of soft materials in confined geometries and thin films [1][2][3] has widely attracted the interest of physicists, biophysicists, chemists and engineers. In particular, thin polymer films are nowadays of major importance in several industrial applications such as biocompatible coatings, organic microelectronics and polymeric data storage devices. In order to gain insights into the behaviour of these films and their constitutive macromolecules, a wide class of experiments, including dewetting [4,5], nanoindentation with gold particles [6], and levelling of stepped films [7][8][9], have been performed. Enhanced mobility effects in ultra-thin polymer films have been predicted [10,11], and observed [12][13][14]. Film preparation by spincoating has also been widely studied [15][16][17][18], and is known to govern surface instabilities and pattern formation [19][20][21].The evolution of the free surface of a thin Newtonian liquid film with nonconstant curvature is driven by the Laplace pressure and mediated by viscosity. This is well understood from the theoretical point of view through the so-called capillary-driven thin film equation [1][2][3]22]. Extensive analytical work on the thin film equation [23][24][25][26][27] and numerous accurate numerical schemes [28][29][30][31] have been performed in the past decades, and have allowed for a deeper understanding of its mathematical features. Long-term traveling-wave solutions have been discussed [32]. Convergence of the solutions to intermediate asymptotic regimes [33] has also been revealed. In particular, it was shown that the vertically-rescaled solution for any summable initial profile uniformly converges in time towards a universal self-similar attractor that is precisely given by the Green's function of the capillary-driven linear thin film equation multiplied by the initial algebraic volume of the perturbation [34].The aforementioned capillary-driven thin fi...

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Resolving a paradox of anomalous scalings in the diffusion of granular materials

Cited by 34 publications

References 50 publications

Numerical solutions of thin-film equations for polymer flows

Numerical solutions of thin-film equations for polymer flows

The mechanism of long-term coarsening of granular mixtures in rotating drums

Viscoelastic effects and anomalous transient levelling exponents in thin films

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