Steven Roper

Steady growth of nanowires via the vapor-liquid-solid method

Understanding the dynamics of the growth of nanowires by the vapor-liquid-solid (VLS) process is essential in order to relate the properties of the wire to their processing conditions. A theory for VLS growth is developed that incorporates the surface energy of the solid-liquid, liquid-vapor, and solid-vapor interfaces, allows for supersaturation of growth material in the droplet, and employs contact-line conditions. We predict the profile of catalyst concentration in the droplet, the degree of supersaturation, and the modification to the shape of the solid-liquid interface due to growth, as functions of the material properties and process parameters. Under typical experimental conditions the interface deflection due to growth is predicted to be practically zero. We also find that the growth rate of the wire inherits the same dependence on diameter as the flux of growth material at the liquid-vapor interface; thus, if we assume that the flux is independent of radius, we obtain a growth rate that is also independent of radius. To make a prediction about the actual variation with diameter requires a detailed knowledge of the decomposition kinetics at the liquid-vapor interface.

Buoyancy-driven crack propagation : the limit of large fracture toughness

We study steady vertical propagation of a crack filled with buoyant viscous fluid through an elastic solid with large effective fracture toughness. For a crack fed by a constant flux Q, a non-dimensional fracture toughness K=Kc /(3μQm 3/2)1/4 describes the relative magnitudes of resistance to fracture and resistance to viscous flow, where Kc is the dimensional fracture toughness, μ the fluid viscosity and m the elastic modulus. Even in the limit K ≫ 1, the rate of propagation is determined by viscous effects. In this limit the large fracture toughness requires the fluid behind the crack tip to form a large teardrop-shaped head of length O(K 2/3) and width O(K 4/3), which is fed by a much narrower tail. In the head, buoyancy is balanced by a hydrostatic pressure gradient with the viscous pressure gradient negligible except at the tip; in the tail, buoyancy is balanced by viscosity with elasticity also playing a role in a region within O(K 2/3) of the head. A narrow matching region of length O(K −2/5) and width O(K −4/15), termed the neck, connects the head and the tail. Scalings and asymptotic solutions for the three regions are derived and compared with full numerical solutions for K ≤ 3600 by analysing the integro-differential equation that couples lubrication flow in the crack to the elastic pressure gradient. Time-dependent numerical solutions for buoyancy-driven propagation of a constant-volume crack show a quasi-steady head and neck structure with a propagation rate that decreases like t −2/3 due to the dynamics of viscous flow in the draining tail.

Buoyancy-driven crack propagation from an over-pressured source

The propagation of a liquid-filled crack from an over-pressured source into a semi-infinite uniform elastic solid is studied. The fluid is lighter than the solid and propagates due to its buoyancy and to the source over-pressure. The role of this over-pressure at early and late times is considered and it is found that the combination of buoyancy and over-pressure leads to significantly different behaviour from buoyancy or over-pressure alone. Lubrication theory is used to describe the flow, where the pressure in the fluid is determined by the elastic deformation of the solid due to the presence of the crack. Numerical results for the evolution of the crack shape and speed are obtained. The crack grows exponentially at early times, but at later times, when buoyancy becomes important, the crack growth accelerates towards a finite-time blow-up. These results are explained by asymptotic similarity solutions for early and late times. The predictions of these solutions are in close agreement with the full numerical results. A different case of crack geometry is also considered in order to highlight connections with previous work. The geological application to magma-filled cracks in the Earths crust, or dykes, is discussed.

An analysis of convection in a mushy layer with a deformable permeable interface

We study the dynamics of a mushy layer in directional solidification for the case of a thin near-eutectic mush with a deformable and permeable mush–liquid interface. We examine the onset of convection using linear stability analysis, and the weakly nonlinear growth of liquid inclusions that signal the onset of chimneys. This analysis is compared to past analyses in which the mush–liquid interface is replaced by a rigid impermeable lid. We find qualitative agreement between the two models, but the rigid-lid approximation gives substantially different quantitative behaviour. In linear theory, the rigid-lid approximation leads to an over-estimate of the critical Rayleigh number and wavenumber of the instability. The condition for the onset of oscillatory instability is also changed by a factor of about 5 in composition number C. In the weakly nonlinear theory, the location of the onset of liquid inclusions is near the undisturbed front for the free-boundary analysis, whereas it lies at the centre of the mushy layer when the rigid-lid approximation is used. For hexagonal patterns, the boundary between regions of parameter space in which up and down hexagons are stable, shifts as a result of coupling between the liquid and mush regions.

Radius selection and droplet unpinning in vapor-liquid-solid-grown nanowires

The requirements for steady nanowire growth under near-equilibrium conditions in the vapor-liquid-solid (VLS) method is examined with particular emphasis on the configuration of the liquid droplet. It is found that the final radius of a cylindrical wire is selected by the fixed volume of liquid VL and the surface-energy ratio γsl/γlv but is independent of the solid-vapor energy γsv. Existing models for growth, based on a balance of configurational forces at the triple junction, are shown to be consistent with the principle of maximal release of free energy. Gibbs’s results on allowable contact angles at a sharp corner predict conditions on γsl/γlv and γsv/γlv for the existence of straight-wire growth. For parameter values that violate these conditions the droplet atop the wire is expected to unpin. A range of alternative configurations for the liquid exist and their relative energies are compared. In particular, it is found that for a certain region in parameter space—not extraordinary in VLS growth—a sph...

The thinning of lamellae in surfactant-free foams with non-Newtonian liquid phase

Thinning rates of liquid lamellae in surfactant-free non-Newtonian gas–liquid foams, appropriate for ceramic or polymer melts and also in metals near the melting point, are derived in two dimensions by matched asymptotic analysis valid at small capillary number. The liquid viscosity is modelled (i) as a power-law function of the shear rate and (ii) by the Ellis law. Equations governing gas–liquid interface dynamics and variations in liquid viscosity are derived within the lamellar, transition and plateau border regions of a corner of the liquid surrounding a gas bubble. The results show that the viscosity varies primarily in the very short transition region lying between the lamellar and the Plateau border regions where the shear rates can become very large. In contrast to a foam with Newtonian liquid, the matching condition which determines the rate of lamellar thinning is non-local. In all cases considered, calculated lamellar thinning rates exhibit an initial transient thinning regime, followed by a t −2 power-law thinning regime, similar to the behaviour seen in foams with Newtonian liquid phase. In semi-arid foam, in which the liquid fraction is O(1) in the small capillary number, results explicitly show that for both the power-law and Ellis-law model of viscosity, the thinning of lamella in non-Newtonian and Newtonian foams is governed by the same equation, from which scaling laws can be deduced. This result is consistent with recently published experimental results on forced foam drainage. However, in an arid foam, which has much smaller volume fraction of liquid resulting in an increase in the Plateau border radius of curvature as lamellar thinning progresses, the scaling law depends on the material and the thinning rate is not independent of the liquid viscosity model parameters. Calculations of thinning rates, viscosities, pressures, interface shapes and shear rates in the transition region are presented using data for real liquids from the literature. Although for shear-thinning fluids the power-law viscosity becomes infinite at the boundaries of the internal transition region where the shear rate is zero, the interface shape, the pressure and the internal shear rates calculated by both rheological models are indistinguishable.

Propagation of dissection in a residually-stressed artery model

This paper studies dissection propagation subject to internal pressure in a residually-stressed two-layer arterial model. The artery is assumed to be infinitely long, and the resultant plane strain problem is solved using the extended finite element method. The arterial layers are modelled using the anisotropic hyperelastic Holzapfel–Gasser–Ogden model, and the tissue damage due to tear propagation is described using a linear cohesive traction–separation law. Residual stress in the arterial wall is determined by an opening angle

HEXAGONAL PATTERNS IN A SIMPLIFIED MODEL FOR BLOCK COPOLYMERS

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