04.05.2018 | Computation | Ausgabe 15/2018 Open Access

# Measuring solid–liquid interfacial energy fields: diffusion-limited patterns

- Zeitschrift:
- Journal of Materials Science > Ausgabe 15/2018

## Introduction

Patterns encountered in nature, such as those exhibited by snow flakes and many crystallized mineral forms, and those found in the microstructures of cast alloys and fusion weldments, remain subjects of long-standing scientific interest and practical engineering importance [1, 2]. Alan Turing’s paper, “The Chemical Basis of Morphogenesis” [3], is credited as explaining that diffusion-limited processes can drive thermodynamically “open” systems to instability. Patterns then evolve spontaneously in response to Poincaré’s “very small cause(s).” But what, in fact, are the nature and function of such very small causes?“A very small cause that escapes our notice determines a considerable effect that we cannot fail to see, and then we say that the effect is due to chance.”—Henri Poincaré, Science et méthode, 1908.

## Experimental observations: capillary-induced shape change

### Prior microgravity studies

## Interface energy balances

### Stefan conditions

### Omnimetric energy balances

#### Moving interfaces

^{1}—is essential in achieving omnimetric balance. The Stefan condition, which excludes capillary, does not yield the desired multi-scale energy balance, particularly at smaller mesoscopic scales. For example, a positive flux divergence withdraws energy at points on an advancing freezing interface and slightly reduces the net energy rate released at those points. This reduction in energy rate would locally imbalance the nearly constant energy rate entering and required by the surrounding phases’ slowly changing macro-gradients. Long-range macroscopic thermal gradients act quasi-statically and change relatively slowly over time, compared to either latent heat or capillary-mediated energy sources that arise from fast-changing (microscopic) molecular scale events. The energy rate reduction resulting from an onset of positive flux divergence must, therefore, be “cancelled” by a prompt compensatory increase in local interface speed, from \({v}_{St}({\mathbf {r}})\rightarrow {v}_{LR}({\mathbf {r}})\), a modulation that boosts the local latent heat rate slightly and restores the local balance of interfacial energy. Conversely, a negative flux divergence that adds energy would raise the interface’s energy release rate by a small amount. A locally increased energy rate requires a compensatory decrease in interface speed that reduces the local latent heat rate and restores balance to that region’s energy budget. The capillary bias field, in this context, allows omnimetric energy balances down to the smallest mesoscopic scales affected by capillarity and pattern formation.

#### Stationary isotropic interfaces

## Grain boundary grooves

### Background

Wang et al.’s observation and conclusion, quoted above, indicates that GBGs provide the “trigger” mechanism for actually inducing morphological instabilities on polycrystalline solid–liquid interfaces.“\(\dots \) the interface instability occurring first at the grain boundary groove probably becomes the origin of the entire planar interface instability.”

### Variational grooves

^{2}The free energy of a variational solid–liquid GBG consists of capillary energy stored along its solid–liquid interface, which is increased by the “pull” of an intersecting grain boundary that curves and lengthens that interface. The grain boundary itself decreases its total area and energy as it contracts. Also included in a GBG’s variational energy functional is the volume-free energy within the slightly undercooled melt confined within the groove cusp, relative to its adjacent curved solid phase at the same temperature and pressure. The sequence of variational grooves plotted in Fig. 2 illustrates how crystal boundaries with different energy density, or surface tension, “pull” against a solid–liquid interface to form, under the same thermal gradient, unique variational profiles, each with its specified dihedral angle, \({{\varPsi }}\), and equilibrium groove depth, \(\eta ^{\star }({{\varPsi }})\).

### Equilibrated GBGs

### Isotropic grain boundary grooves

^{3}variational GBGs have been analyzed in detail in earlier investigations: (1) isotropic grooves separating phases with equal thermal conductivity, the profiles, and linear thermo-potentials for which were solved by variational methods as used in this study [42]; and (2) the more general situation of isotropic grooves separating phases with unequal thermal conductivities. For unequal thermal conductivities, nonlinear potentials develop, which were analyzed theoretically, calculated, and then confirmed using an analog potentiometric device. The potentiometer consisted of alloy sheets, with dissimilar electrical conductivities, shaped and joined as a meter-sized variational “GBG.” As steady-state temperature and voltage distributions both obey Laplace’s equation, the nonlinear voltage distribution measured along this analog GBG profile for a large (7:1) conductivity ratio served to reflect the thermo-potential distribution of its equivalent GBG [53]. GBG profiles for other unequal liquid–solid thermal conductivities were computed by identical numerical methods, from which the theoretical profile for a 4:1 thermal conductivity profile was calculated, and later employed by Hardy [50], to determine the solid–liquid interface energy surrounding the rhombohedral c-axis of pure water ice.

### Variational GBGs

## Thermodynamic properties of variational GBGs

### Thermo-potential and interface curvature

### Tangential interface gradients

^{4}Dimensionless arc length is scaled similarly with the groove’s thermo-capillary length, \(2{\varLambda }\). The tangential, or arc-length, gradient may be found by applying the vector sequence,

^{5}that appear in definition (13) reflects the derivatives, \({\mathrm{d}}\eta /{\mathrm{d}}\hat{s}\), taken along the groove’s left and right profiles, per equations (5), which reverse sign when crossing the GBG’s triple junction,

### Capillary-mediated energy fluxes

#### Fourier’s law

### Capillary flux divergence

### Bias fields on equilibrated GBGs

### Bias-field distributions

## Phase-field simulations

### Diffuse and sharp interfaces

## Detecting interfacial energy fields

### Potential change and energy rate

^{6}between small changes in temperature and the associated energy rate. Thus, in an open steady-state system at constant pressure, \(\delta \dot{Q}\propto \delta {T}\), indicating that capillary-mediated interface energy rates are precisely proportional to the changes they induce in the local interface temperature, or thermo-potential.