Stephen J. Watson

Coarsening dynamics of the convective Cahn-Hilliard equation

Abstract We characterize the coarsening dynamics associated with a convective Cahn-Hilliard equation (cCH) in one space dimension. First, we derive a sharp-interface theory through a matched asymptotic analysis. Two types of phase boundaries (kink and anti-kink) arise, due to the presence of convection, and their motions are governed to leading order by a nearest-neighbors interaction coarsening dynamical system ( CDS ). Theoretical predictions on CDS include: • The characteristic length L M for coarsening exhibits the temporal power law scaling t1/2; provided L M is appropriately small with respect to the Peclet length scale L P . • Binary coalescence of phase boundaries is impossible. • Ternary coalescence only occurs through the kink-ternary interaction; two kinks meet an anti-kink resulting in a kink. Direct numerical simulations performed on both CDS and cCH confirm each of these predictions. A linear stability analysis of CDS identifies a pinching mechanism as the dominant instability, which in turn leads to kink-ternaries. We propose a self-similar period-doubling pinch ansatz as a model for the coarsening process, from which an analytical coarsening law for the characteristic length scale L M emerges. It predicts both the scaling constant c of the t1/2 regime, i.e. L M =ct 1/2 , as well as the crossover to logarithmically slow coarsening as L M crosses L P . Our analytical coarsening law stands in good qualitative agreement with large-scale numerical simulations that have been performed on cCH.

Fluid flow and heat transfer in a dual-wet micro heat pipe

Micro heat pipes have been used to cool micro electronic devices, but their heat transfer coefficients are low compared with those of conventional heat pipes. In this work, a dual-wet pipe is proposed as a model to study heat transfer in micro heat pipes. The dual-wet pipe has a long and narrow cavity of rectangular cross-section. The bottom-half of the horizontal pipe is made of a wetting material, and the top-half of a non-wetting material. A wetting liquid fills the bottom half of the cavity, while its vapour fills the rest. This configuration ensures that the liquid–vapour interface is pinned at the contact line. As one end of the pipe is heated, the liquid evaporates and increases the vapour pressure. The higher pressure drives the vapour to the cold end where the vapour condenses and releases the latent heat. The condensate moves along the bottom half of the pipe back to the hot end to complete the cycle. We solve the steady-flow problem assuming a small imposed temperature difference between the two ends of the pipe. This leads to skew-symmetric fluid flow and temperature distribution along the pipe so that we only need to focus on the evaporative half of the pipe. Since the pipe is slender, the axial flow gradients are much smaller than the cross-stream gradients. Thus, we can treat the evaporative flow in a cross-sectional plane as two-dimensional. This evaporative motion is governed by two dimensionless parameters: an evaporation number E defined as the ratio of the evaporative heat flux at the interface to the conductive heat flux in the liquid, and a Marangoni number M . The motion is solved in the limit E →∞ and M →∞. It is found that evaporation occurs mainly near the contact line in a small region of size E −1 W , where W is the half-width of the pipe. The non-dimensional evaporation rate Q * ~ E −1 ln E as determined by matched asymptotic expansions. We use this result to derive analytical solutions for the temperature distribution T p and vapour and liquid flows along the pipe. The solutions depend on three dimensionless parameters: the heat-pipe number H , which is the ratio of heat transfer by vapour flow to that by conduction in the pipe wall and liquid, the ratio R of viscous resistance of vapour flow to interfacial evaporation resistance, and the aspect ratio S . If HR ≫1, a thermal boundary layer appears near the pipe end, the width of which scales as ( HR ) −1/2 L , where L is the half-length of the pipe. A similar boundary layer exists at the cold end. Outside the boundary layers, T p varies linearly with a gradual slope. Thus, these regions correspond to the evaporative, adiabatic and condensing regions commonly observed in conventional heat pipes. This is the first time that the distinct regions have been captured by a single solution, without prior assumptions of their existence. If HR ~ 1 or less, then T p is linear almost everywhere. This is the case found in most micro-heat-pipe experiments. Our analysis of the dual-wet pipe provides an explanation for the comparatively low effective thermal conductivity in micro heat pipes, and points to ways of improving their heat transfer capabilities.

Crystal Growth, Coarsening and the Convective Cahn—Hilliard Equation

The coarsening dynamics of a faceted vicinal crystalline surface growing into its melt by attachment kinetics is considered. The convective Cahn-Hilliard equation (CCH) is derived as a small amplitude expansion of such surface evolutions restricted to 1-D morphologies, with the local surface slope serving as the order parameter. A summary of the sharp interfacetheory for CCH that follows from a matched asymptotic analysis, is also presented [26]. It takes the form of a nearest neighbor interaction between two non-symmetrically related phase boundaries (kinkand anti-kink).The resulting coarsening dynamical system CDS for the phase boundaries exhibits novel coarsening mechanisms. In particular, binary coalescence of phase boundaries is impossible. Also, ternary coalescence occurs only through two kinks meeting an anti-kink resulting in a kink (kink-ternary);the alternative of two anti-kinks meeting a kink is impossible. This behavior stands in marked contrast to the Cahn-Hilliard equation CH where binary coalescence of phase boundaries is generic. Numerical simulations of CCH are presented to validate the predictions of the sharp interface theory CDS.

Mean-field theory for coarsening faceted surfaces.

A mean-field theory is developed for the scale-invariant length distributions observed during the coarsening of generic one-dimensional faceted surfaces. This theory closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in two-phase systems, but the mechanism of coarsening in faceted surfaces requires the addition of convolution terms recalling work on particle coalescence, and induces an unexpected coupling between the convolution and the rate of facet loss. As a generic framework, the theory concisely illustrates how the universal processes of facet removal and neighbor merger are moderated by the system-specific mean-field velocity describing average rates of length change. For a simple, example facet dynamics associated with the directional solidification of a binary alloy, agreement between the predicted scaling state and that observed after direct numerical simulation of 40,000,000 facets is reasonable given the limiting assumption of noncorrelation between neighbors; relaxing this assumption is a clear path forward toward improved quantitative agreement with data in the future.

Simulating the kinematics of completely faceted surfaces

We fully generalize a previously-developed computational geometry tool [1] to perform large-scale simulations of arbitrary 2+1D faceted surfaces z=h(x,y). The method is an explicit front-tracking scheme that uses a compact, three-component facet/edge/junction storage mode. Because it naturally mirrors the intrinsic surface structure, this scheme allows both rapid simulation of large ensembles, and easy extraction of geometrical statistics. To do so, it must overcome the barrier of detecting and resolving a wide variety of topological changes that occur during surface evolution. However, while the variety of topological events is larger than in the case of 2D cellular networks, it is still limited, and our main result is a comprehensive algorithm performing these changes in the code.

Emergent parabolic scaling of nano-faceting crystal growth

Nano-faceted crystals answer the call for self-assembled, physico-chemically tailored materials, with those arising from a kinetically mediated response to free-energy disequilibria (thermokinetics) holding the greatest promise. The dynamics of slightly undercooled crystal–melt interfaces possessing strongly anisotropic and curvature-dependent surface energy and evolving under attachment–detachment limited kinetics offer a model system for the study of thermokinetic effects. The fundamental non-equilibrium feature of this dynamics is explicated through our discovery of one-dimensional convex and concave translating fronts (solitons) whose constant asymptotic angles provably deviate from the thermodynamically expected Wulff angles in direct proportion to the degree of undercooling. These thermokinetic solitons induce a novel emergent facet dynamics, which is exactly characterized via an original geometric matched-asymptotic analysis. We thereby discover an emergent parabolic symmetry of its coarsening facet ensembles, which naturally implies the universal scaling law L∼t1/2 for the growth in time t of the characteristic length L.

Lorentzian symmetry predicts universality beyond scaling laws

We present a covariant theory for the ageing characteristics of phase-ordering systems that possess dynamical symmetries beyond mere scalings. A chiral spin dynamics which conserves the spin-up (+) and spin-down (−) fractions, and , serves as the emblematic paradigm of our theory. Beyond a parabolic spatio-temporal scaling, we discover a hidden Lorentzian dynamical symmetry therein, and thereby prove that the characteristic length L of spin domains grows in time t according to , where (the invariant spin-excess ) and β is a universal constant. Furthermore, the normalised length distributions of the spin-up and the spin-down domains each provably adopt a coincident universal ( σ -independent) time-invariant form, and this supra-universal probability distribution is empirically verified to assume a form reminiscent of the Wigner surmise .