Axisymmetric Surface Diffusion: Dynamics and Stability of Self-Similar Pinchoff

Bernoff, Andrew J.; Bertozzi, Andrea L.; Witelski, Thomas P.

doi:10.1023/b:joss.0000033251.81126.af

Cited by 120 publications

(154 citation statements)

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“…Further solutions are expected at cone angles smaller than 30 • . However, as in the surface-diffusion-limited case 12 , we expect that only the solution with the largest cone angle is stable and thus observable experimentally.…”

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confidence: 85%

Universality and self-similarity in pinch-off of rods by bulk diffusion

et al. 2010

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As rodlike domains pinch off owing to Rayleigh instabilities, a finite-time singularity occurs as the interfacial curvature at the point of pinch-off becomes infinite. The dynamics controlling the interface become independent of initial conditions and, in some cases, the interface attains a universal shape 1 . Such behaviour occurs in the pinching of liquid jets and bridges 2-9 and when pinching occurs by surface diffusion 10-12 . Here we examine an unexplored class of topological singularities where interface motion is controlled by the diffusion of mass through a bulk phase. We show theoretically that the dynamics are determined by a universal solution to the interface shape (which depends only on whether the highdiffusivity phase is the rod or the matrix) and materials parameters. We find good agreement between theory and experimental observations of pinching liquid rods in an Al-Cu alloy. The universal solution applies to any physical system in which interfacial motion is controlled by bulk diffusion, from the break-up of rodlike reinforcing phases in eutectic composites 13-16 to topological singularities that occur during coarsening of interconnected bicontinuous structures 17-20 , thus enabling the rate of topological change to be determined in a broad variety of multiphase systems.As pinch-off is approached, the length and timescales near the singularity become much smaller than the scale of the initial conditions that caused the singularity to form. In this regime the interfacial morphology can be written in scale-independent self-similar coordinates 2 . Experimental investigations of self-similar evolution that determine both the interfacial morphology and dynamics near topological singularities have largely been confined to multiphase fluids. In some cases, the interface asymptotically approaches a symmetric cone shape as the time of pinch-off is approached. For pinching by surface diffusion, a countable set of solutions can be identified, each with a different cone angle. A stable solution is found for only the largest cone angle, and thus the geometry of the system close to pinching is universal 12 . This universality has not been observed experimentally. There is no corresponding analysis for pinching through mass diffusion in the bulk phases.We examine the evolution of interfacial morphology of liquid rods in a solid matrix. An Al-15 wt.% Cu alloy is employed along with 4D synchrotron radiation-based tomographic microscopy. The interfaces evolve by bulk diffusion of solute through the high-diffusivity liquid 21 . The rods were created by holding a 1-mmdiameter sample isothermally 5 • C above the eutectic temperature to form a solid-liquid mixture. The interfacial morphology of liquid rods was observed in situ using tomographic microscopy at the TOMCAT beamline of the Swiss Light Source over a 12 h period 22,23 (see Methods for details). Hundreds of pinching events were observed during this time. Figure 1 shows one such pinching event.To determine whether a self-similar solution exists near pinchof...

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Universality and self-similarity in pinch-off of rods by bulk diffusion

et al. 2010

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“…Analysis of finite-time blow-up and rupture in many higher order PDEs starts along similar lines [35,11,76,77,79,34,22,25].…”

Section: Self-similar Finite-time Blow-up In a Higher-order Pdementioning

confidence: 99%

“…The first few of a discrete, countably infinite family of self-similar solutions,Ū ℓ (η), for ℓ = 1, 2, 3, · · ·, were computed using an under-relaxed Newton-Raphson method [1,11,73] and are shown in Figure 7. These solutions can be indexed by the number of maxima.…”

Section: Similarity Variablesmentioning

confidence: 99%

“…In the pinch-off of a thin liquid filament, as in a lava lamp or the break-up of a fluid jet, the profile of the pinching neck assumes a universal shape as detachment is approached [18,32]. Other examples include rupture of a solid filament due to surface diffusion [11], rupture of thin films [79,69,70,75,76], and the ignition of a combustible substance [6].…”

Section: Introductionmentioning

confidence: 99%

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Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

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Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

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“…The applications of singularity formation are diverse, including fourth-order lubrication theory models from fluids (cf. [21][22][23]), chemotactic aggregation governed by systems of nonlinear PDEs (cf. [24,25]), and quenching behaviour in a nonlinear PDE model of a MEMS capacitor (cf.…”

Section: Similaritymentioning

confidence: 99%