Abstract
The Leibniz–Reynolds transport theorem yields an omnimetric interface energy balance, i.e., one valid over all continuum length scales. The transport theorem, moreover, indicates that solid–liquid interfaces support capillary-mediated redistributions of energy capable of modulating an interface’s motion—a thermodynamic phenomenon not captured by Stefan balances that exclude capillarity. Capillary energy fields affect interfacial dynamics on scales from about 10 nm to several mm. These mesoscopic fields were studied using entropy density multiphase-field simulations. Energy rate distributions were exposed and measured by simulating equilibrated solid–liquid interfaces configured as stationary grain boundary grooves (GBGs). Negative rates of energy distributed over GBGs were measured as residuals, by subtracting the linear potential distribution contributed by applied thermal gradients constraining the GBGs from the nonlinear distributions actually developed along their solid–liquid interface. Rates of interfacial cooling revealed numerically confirm independent predictions based on sharp-interface thermodynamics, variational calculus, and field theory. This study helps answer a long-standing question: What creates patterns for diffusion-limited transformations in nature and in material microstructures?
Highlights
Grain boundary grooves (GBGs) are commonly occurring features found along polycrystalline solid– liquid interfaces
A century beyond Poincare’s quoted remark, and six decades after Turing’s explanation of instability in chemically reacting systems, major issues remain unresolved concerning pattern formation dynamics in diffusion-limited systems. These issues entail the following questions: (1) How do specific patterns initiate during crystal growth, solidification, and other diffusion-limited phase transformations? (2) Is there an agent that provides a template, or guide, for pattern development, especially where neither prior structures nor preferred directionality is present in the metastable melt, solution, or vapor phase? This paper explores the origin of ‘‘very small causes,’’ or perturbations, that appear spontaneously in diffusion-limited systems and guide pattern formation
Wang et al concluded from their experiments on grain boundary grooves (GBGs) on succinonitrile interfaces [31]: ‘‘. . . the interface instability occurring first at the grain boundary groove probably becomes the origin of the entire planar interface instability.’’
Summary
‘‘A very small cause that escapes our notice determines a considerable effect that we cannot fail to see, and we say that the effect is due to chance.’’—Henri Poincare, Science et methode, 1908. We will combine insights based on experimental observation involving quantitative shape and thermal analysis [13] with interface energy conservation, via Leibniz–Reynolds transport theory, which identifies the higher-order energy terms responsible for interfacial shape changes during melting and guides pattern formation during solidification [16]. These higher-order energy terms are identified, their functions in pattern dynamics in transforming systems are not They include in particular capillary energy stored or released as an interface changes its area and crystallographic orientation to the melt, plus energy redistributed over the interface via divergence of capillary-mediated fluxes arising from interfacial gradients of the Gibbs–Thomson thermo-potential. The basis for proportionality between shifts in interface thermo-potential and field energy rate is discussed in detail in ‘‘Detecting interfacial energy fields’’ section
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