Measuring solid–liquid interfacial energy fields: diffusionlimited patterns
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 capillarymediated 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 multiphasefield 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 sharpinterface thermodynamics, variational calculus, and field theory. This study helps answer a longstanding question: What creates patterns for diffusionlimited transformations in nature and in material microstructures?
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 longstanding scientific interest and practical engineering importance [1, 2]. Alan Turing’s paper, “The Chemical Basis of Morphogenesis” [3], is credited as explaining that diffusionlimited 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.
A century beyond Poincaré’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 diffusionlimited systems. These issues entail the following questions: (1) How do specific patterns initiate during crystal growth, solidification, and other diffusionlimited 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 diffusionlimited systems and guide pattern formation.
Most fluid phases (gases and liquids) lack discernible internal structures, order, or directionality, excepting ephemeral correlations associated with their localized molecular configurations. For example, shortrange order in supersaturated melts seldom extends beyond a few nanometers [4]. It is unlikely, therefore, that information residing at the atomic/molecular level guides patterns emerging at scales three to six ordersofmagnitude larger. Diffusionlimited patterns of interest falling within this category include wellorganized cellular and dendritic forms, invaginated and highly ramified interfaces, including “seaweed” and other irregularly branched microstructures. Indeed, increasingly complex patterns might even be extensible to those found in biological systems.
In other words, to understand the origin of many natural patterns, and, ultimately, control microstructures derived from materials processes involving solidification, welding, and crystal growth, one must determine: (1) whether patternforming “signals” or “instructions” exist, and, if so, (2) do they fundamentally devolve from stochastic processes, or from higherorder deterministic sources.
This paper addresses both issues for crystal–melt interfaces in unary systems, by exploring the presence of interfacial energy fields that provide patternforming stimuli in 2D. The presence of such stimuli on solid–liquid interfaces described in earlier publications [5, 6] is finally revealed and measured here through novel measurements extracted from phasefield simulations. Capillary fields in the form of interfacial energy distributions are exposed and measured on simulated microstructures in the form of equilibrated solid–liquid grain boundary grooves (GBGs). Simulated interfacial data also allow quantifiable comparison with analytic predictions of interfacial energy fields derived from sharpinterface thermodynamics. Simulations and measurements reported in this study also confirm that equivalent patternforming fields arise within standard phasefield physics that manifest themselves as deterministic perturbations.
Numerical simulations are compared with predictions based on interface energy conservation and classic field theory. The comparison reveals the existence of persistent capillarymediated energy fields that influence the dynamics of interfacial shape changes during phase transformation. Such fields stimulate complex pattern formation on unstable interfaces with, or without, benefit of noise.
Experimental observations: capillaryinduced shape change
Prior microgravity studies
The idea that innate capillary phenomena, rather than “selectively amplified” noise [7, 8, 9, 10], control pattern formation in diffusionlimited phase transformations was initially prompted by experimental observations of unusual solid–liquid interfacial shape changes. Needlelike crystallites progressively melted selfsimilarly under convectionfree conditions in orbital microgravity, but suddenly, and surprisingly, spheroidized [11].
The first gravityfree studies yielding quantitative kinetic data on diffusionlimited crystal growth, melting, and pattern formation were the Isothermal Dendritic Growth Experiments (IDGE). Three IDGE experiments were flown by NASA during the midtolate 1990s on Space Shuttle Columbia using the US Microgravity Payload (USMP) platform. Video and precision thermal data, streamed directly from the IDGE3/USMP4 orbital experiments, indicated that capillary effects might be responsible for such unusual changes. These experiments recorded the evolving shapes of melting crystallites of ultrapurified pivalic anhydride (PVA) (2,2dimethylpropionic anhydride) [11]. PVA is a transparent facecentered cubic crystal that melts at approximately 36 °C. Video (shape) data and precision thermal data for quasistatically melting PVA crystallites were analyzed kinetically and reported through encouragement received from both NASA and the German Aerospace Center (DLR) at KölnPorz, Germany [12, 13, 14].
IDGE melting data from USMP4 showed that unexpected crystal shape changes occurred in microgravity during diffusionlimited melting. Slender, melting ellipsoidal crystallites initially experienced slowly increasing majortominor axial ratios. Then, at a further reduced size, these crystallites suddenly underwent dramatic decreases of more than an orderofmagnitude from their typical high axial ratios of 15–20, down to almost unity. The onset of large interfacial shape changes always occurred when the crystallites during slow melting entered a narrow range of sizes: viz., when their major axes were reduced to about 5 mm, and their minor axes simultaneously decreased to about 500 microns. These dramatic shape changes—viz., needles suddenly contracting into spheroids—were consistently recorded as diffusionlimited melting progressed toward extinction of PVA crystallites under quiescent microgravity conditions.
During terrestrial melting, by contrast, where gravity is present, buoyancy convection and rapid sedimentation totally obscure the onset of shape changes. Several aspects of these experimental observations in microgravity were later reproduced in a phasefield simulation of melting crystallites reported by Mullis [15]. Mullis’s numerical results provided the first inkling that conventional phasefield simulations, without modification, might also produce interface fields responsible for transformationinduced shape changes. Indeed, this paper advances that initial suggestion toward near certainty.
Our explanation of deterministic—versus stochastic—pattern dynamics is based on two independent insights: (1) heuristic observations in microgravity, as summarized above, of dynamic shape changes for crystallites melting under diffusion control, and, (2) to be described next in “Interface energy balances” section, the application of sharpinterface thermodynamics that imposes interfacial energy conservation over all relevant length scales. 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 higherorder energy terms responsible for interfacial shape changes during melting and guides pattern formation during solidification [16].
Interface energy balances
Conservation principles applied in the form of the Leibniz–Reynolds transport theorem [17, 18] identify an overlooked higherorder energy term. We shall demonstrate that this term represents a thermodynamic field that exists ubiquitously on heterophase interfaces exhibiting capillarity and curvature gradients. Moreover, general formulas published recently for calculating energy redistribution on interfaces are also capable of predicting the initial geometric character of patterns evolved in both isotropic and anisotropic solid–liquid systems [5]. These same energy fields are, in a sense, nature’s patternforming “information,” which is provided autonomously by thermodynamics.
In this paper, we expose and measure the active presence of capillarymediated energy fields on stationary interfaces, chosen as equilibrated grain boundary grooves simulated using a multiphasefield model. We then compare them to theoretical predictions from thermodynamics and field theory.
Stefan conditions
The Stefan condition conserves energy and/or species by balancing their release or intake at moving phase boundaries [2]. Indeed, Stefan conditions provide the accepted method, through control volume requirements, to connect the speed of a moving interface—related to its rate of phase transformation energy and species release—to the transport rates of these quantities. The rates of generation and transport must balance. Although usually credited to Josef Stefan’s lecture notes and journal publications from the late nineteenth century, for example, [19, 20], an energy balance similar to Stefan’s eponymous condition was introduced much earlier, appearing in 1831, by Lamé and Clapeyron [21]. So, interface energy and mass balances have been under discussion and use for nearly two centuries. Today, phase change kinetics, freeboundary problems, and most theories of diffusionlimited pattern dynamics invariably employ “Stefan balances” to impose conservation laws at moving heterophase interfaces.
The fundamental issue raised here regarding limitations in Stefan balances is to reexamine closely the length scales over which an interface must precisely balance its energy/mass budget. Our view, to be examined in detail, is that (1) energy and matter at interfaces, of course, are conserved, and (2) sources and fluxes of these quantities must remain balanced in control volumes of arbitrary spatial extent. In short, energy rates on, to, and from, an interface must balance “omnimetrically”, i.e., over all continuum length scales. As will be shown next in “Omnimetric energy balances” section, the Stefan condition, per se, does not satisfy these basic requirements if capillarity is present. As humorist Mark Twain once famously remarked, “. . . this [finding] will gratify some people, and surprise the rest.”
Omnimetric energy balances
Interface energy balances in the presence of capillarity were recently analyzed by applying the Leibniz–Reynolds transport theorem [17]. This theorem imposes Leibniz differentiation on both volume and surface integrals, taken in this case over a timedependent 3D domain of pure solid and liquid undergoing freezing or melting. The soughtafter omnimetric energy balance is captured by adding two steps to the Leibniz–Reynolds transport analysis. These include: (1) applying contraction mapping, to focus “bookkeeping” energy rates in a 3D domain to those exchanges occurring over the solid–liquid interface and (2) applying the 2D divergence theorem to a line integral tracking energy crossing the space curve formed by the intersection of that interface with the domain’s outer boundary [5, 22].
The Leibniz–Reynolds energy balance, when modified as indicated above, includes the energy terms found in Stefan’s balance, plus additional rates linked to capillarity, all of which have been recognized and discussed in theoretical surface thermodynamics [23]. Although these higherorder 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 capillarymediated fluxes arising from interfacial gradients of the Gibbs–Thomson thermopotential.
Moving interfaces
The Leibniz–Reynolds transport theorem specifically identifies six independent sources (excluding thermal radiation) that contribute to the interfacial energy balance for a moving anisotropic solid–liquid interface [5], whereas the Stefan balance, which excludes capillarity, includes only three. One such interfacial energy source, easily eliminated for our present purposes, is to consider isotropic solid–liquid interfaces, so that crystallographic orientation, per se, does not influence an interface’s energy density, i.e., \(\gamma _{s\ell }={\hbox {const}}.\ ({\hbox {J}}\,\hbox {m}^{2})\).
Another capillarymediated energy source identified by Leibniz–Reynolds analysis is the rate of interfacial deformation, or “stretching.” Area changes on evolving interfaces require some energy storage or release as a moving interface advances or retreats during phase transformation. The rate of area change for moving interfaces is proportional to the product of their local mean curvature and speed. Insofar as interfacial stretching and latent heat production both occur at rates proportional to an interface’s normal velocity, energetic effects from stretching or shrinkage can be combined with the local rate of latent heat production. This is accomplished by inserting a small correction to the volumetric latent heat, \({{\Delta }}{H}_f/\varOmega \, ({\hbox {J}}\,{\hbox {m}}^{3})\), where \(\varOmega \,({\hbox {m}}^3\cdot\,{\hbox {mol}}^{1})\) is the system’s molar volume, and \({{\Delta }}{H}_f\ ({\hbox {J}}\,{\hbox {mol}}^{1})\) is the molar enthalpy change upon melting. Corrections to \({{\Delta }}{H}_f\) added to account for any stretching may be safely ignored provided that the average mean curvature of the microstructure, \({\mathcal {H}}\ll {\Delta }{H}_f/\gamma _{s\ell }\varOmega \approx 10\ ({\hbox {nm}}^{1})\). Mean curvatures of mesoscopic solid–liquid microstructures would seldom exceed this level. This explains why Stefan’s balance, which excludes capillarity, predicts net (overall) transformation rates correctly, irrespective of the detailed intermediate solid–liquid microstructure, but fails to address properly energy exchanges occurring at smaller length scales.
The fourth energy rate, appearing in Eq. (1), is responsible for local capillarybased energetic addition and subtraction. It represents the interfacial energy rate associated with the surface divergence of the capillarymediated tangential flux vector, \({{\phi }}_{\tau }({\mathbf {r}})\cdot {\varvec{\tau }}\). This flux, itself a conservative vector field, arises in response to gradients of the Gibbs–Thomson thermopotential. Much more will be discussed later about the capillary flux, \(\phi _{\tau }\cdot {\varvec{\tau }}\), and its scalar divergence.
The Leibniz–Reynolds theorem tracks the fourth interfacial energy exchange rate as a line integral taken round the intersection of the solid–liquid interface with the exterior boundary of the 3D solid–liquid domain. This line integral sums any interfacial energy losses or gains that exit or enter this closed space curve. The line integral transforms to a standard area integral over the solid–liquid interface by applying the 2D divergence theorem [22], yielding the last term in Eq. (1). See again “Omnimetric energy balances” section.
Despite its technical origin, the fourth term appearing in the Leibniz–Reynolds interfacial energy budget—termed the “bias field”^{1}—is essential in achieving omnimetric balance. The Stefan condition, which excludes capillary, does not yield the desired multiscale 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 macrogradients. Longrange macroscopic thermal gradients act quasistatically and change relatively slowly over time, compared to either latent heat or capillarymediated energy sources that arise from fastchanging (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.
In general, we argue, capillarymediated divergences occurring along an interface are automatically buffered by small countervailing speed adjustments (modulations) to the interface. These modulations insure “spectral” compliance of local energy conservation at every continuum spatial scale. Thus, it is multiscale energy (and/or mass) conservation that is exposed here as fundamental dynamic mechanisms influencing pattern initiation on moving interfaces. Random noise might well be present during solidification and melting; however, it is the spectral, or omnimetric, balancing of the interface’s energy budget that thermodynamics demand. This intrinsic control mechanism, apparently, has not appeared in prior dynamic pattern analyses. As shown later, moreover, the multiscale balances just described are also captured by phasefield theories and their numerical models.
Connections between pattern formation, interfacial speed modulations, and the presence of fourthorder capillary fields were overlooked in early mathematical formulations of dendritic growth [26], in subsequent patternformation models for diffusionlimited systems [2], including a major monograph on pattern formation in solidification by Xu [27].
The authors also suggest, without formal proof at this time, that use of the Leibniz–Reynolds multiscale energy balance, rather than the Stefan balance, might avoid mathematical singularities introduced by capillarity in sharpinterface models of patternforming dynamics. Another unintended consequence of relying on the Stefan condition to satisfy interfacial energy and/or mass conservation is that it limits use of the interface’s Gibbs–Thomson potential as a scalar boundary condition to match chemical potentials along curved heterophase interfaces. As shown later in “Thermodynamic properties of variational GBGs” section, applying the Gibbs–Thomson scalar potential, but ignoring its vector gradient and flux, overlooks a critical function of allowing omnimetric energy conservation to hold, especially at interfacial scales where patterns form.
Lastly, we demonstrate in “Phasefield model and results” section that thermodynamically consistent diffuse interface simulations, as used in this study, do not suffer these limitations, as their physics admit formation of equivalent interface energy fields.
Stationary isotropic interfaces
Stationary solid–liquid interfaces provide a simple, nontrivial setting on which to measure capillarymediated potential gradients, fluxes, and their vector divergences. In particular, static solid–liquid interfaces neither generate nor absorb latent heat, nor do they change shape, undergo interface “stretching” or change orientation over time. Such interfaces, moreover, remain strain free, are neither subject to morphological instability nor stress relaxation, and, perhaps most importantly, may be probed with great precision to measure the distribution of their thermodynamic potentials and evaluate their active energy fields.
Equation (4), moreover, also suggests a basis for detecting bias fields that manifest their presence as small nonlinear shifts in the interface’s thermopotential. The change in thermopotential along static solid–liquid interfaces is also proportional to the local energy rate of the bias field [the lefthand side of Eq. (4)], and proportional to the jump developed in the gradient across the interface [the righthand side of Eq. (4)]. The basis for proportionality between shifts in interface thermopotential and field energy rate is discussed in detail in “Detecting interfacial energy fields” section.
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.”
Dynamic interactions of grain boundaries with stationary and moving solid–liquid interfaces were described in the 1960s by investigators using hotstage optical and electron microscopy [32, 33, 34, 35]. In situ studies of moving solid–liquid interfaces demonstrated that persistent defects, such as grain boundaries and screw dislocations, were often responsible for initiating morphological instabilities that led to increasingly complex patterns in solidifying dilute alloys [36, 37, 38]. Isolated GBGs on solid–liquid interfaces, moreover, provide wellstudied examples of microstructures analyzed for both energy measurement and stability [39]. Dynamic effects induced by GBGs on solid–liquid interfaces were examined in greater detail by Yeh et al., who applied a phasefield model to simulate these instabilities and subsequent pattern formation [40].
Variational grooves
The mathematical term “variational” applied to analytically derived GBG profiles denotes interface forms obtained as solutions to the Euler–Lagrange differential equation, using methods of the calculus of variations [41]. Figure 1 is a variational profile derived for an isolated GBG, embedded, and held stationary, by a vertical thermal gradient. The configuration of solid and liquid represents a static thermodynamically “open” system, insofar as heat is uniformly conducted downward through the gradient from the hotter liquid through the cooler solid. Both phases have identical thermal conductivities, so that their steadystate heatflow fields exhibit horizontal isotherms, vertical lines of heat flow, and a spatially uniform rate of entropy production in 2D.
Importantly, variational GBGs do not accommodate any additional interfacial energy fields, including the bias field. Although variational GBGs are theoretical constructs, their mathematical profiles prove useful for our purposes, as they closely approximate real equilibrated GBGs. Specifically, the analytic shapes of variational GBGs may be used to calculate accurate estimates of capillarymediated bias fields that, according to thermodynamics, should be present on real, or simulated, GBGs with the same geometric profile.
Equilibrated GBGs
Stationary GBGs equilibrate in thermal gradients aligned with their grain boundaries. The photomicrograph of an equilibrated GBG, shown in Fig. 3, exhibits a stationary microstructure similar in shape with its variational profile, having the same dihedral angle and groove depth. (Cf. Fig. 2, \({{\varPsi }}=0\)). Note also how the variational profile coordinates plotted as points on the micrograph in Fig. 3 fit along the crystal–melt interface.
There is, as mentioned, a critical distinction between variational GBGs, with linear potential and curvature distributions, and what we term here as equilibrated GBGs, with nonlinear potential and curvature distributions and groove depths that are altered slightly by the presence of capillarymediated energy fields. These differences, linear versus nonlinear potential and curvature distributions, allow us to evaluate the resident bias fields supported on equilibrated GBGs.
In the example offered in Fig. 3, the dihedral angle is small, \({{\varPsi }}\approx 0\), and the groove depth is large (\(\approx 170\ \upmu {\hbox {m}}\)). Equilibrated solid–liquid GBGs develop on spatial scales typically between a few tens of nanometers and several hundred microns, depending primarily on the magnitude of their thermal gradient. Similar GBGs have been equilibrated over a range of thermal gradients in experiments that yield values for the solid–liquid interface energy density of a number of transparent crystalline substances [48, 49, 50, 51, 52].
The plot in Fig. 4 correlates the norm, or magnitude, of applied thermal gradients, \({{\mathbf {\nabla }}}T\), with experimentally observed steadystate sizes (cusp depths) of an equilibrated GBG in pure succinonitrile, similar to the one displayed in Fig. 3. The scaling relationship established with these data for equilibrated groove size and the applied thermal gradient agrees with that predicted from Eq. (6), expressing an inverse squareroot relationship among thermal gradient, thermocapillary length, \({{\varLambda }}\), and equilibrium groove depth.
Isotropic grain boundary grooves
Two categories of isotropic^{3} variational GBGs have been analyzed in detail in earlier investigations: (1) isotropic grooves separating phases with equal thermal conductivity, the profiles, and linear thermopotentials 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 metersized variational “GBG.” As steadystate 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 thermopotential 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 caxis of pure water ice.
Additional features attending strong anisotropy of the solid–liquid interfacial energy density on GBGs were analyzed by Voorhees et al. [54]. More recently, the influence of weak anisotropy on the dynamic stability of GBGs was studied experimentally by Wang et al. [39]. The capillarymediated fields calculated later in “Capillary flux divergence” section, and then confirmed numerically using phasefield simulations in “Thermopotentials on analytic profiles and phasefield isolines” section, also explain the observations of Wang et al. of how GBGs when set into motion affect interface stability.
The goals in the present study, however, remain: (1) to calculate capillarymediated bias fields expected on equilibrated GBGs with variational profiles, and (2) establish the existence of interface energy fields independently as thermodynamic entities on simulated GBGs. Both tasks are accomplished more easily by studying isotropic GBGs: the first with variational profiles, and the second with simulated phasefield equilibrated GBGs.
Variational GBGs
Bolling and Tiller [42] solved the Euler–Lagrange equation [41] for the energy functional of a stationary GBG subject to flat (zero curvature) endpoint conditions as \(\mu \rightarrow \pm\, \infty \), and \(\eta \rightarrow 0\). Details for constructing this functional may be found in [43].
Thermodynamic properties of variational GBGs
Thermopotential and interface curvature
A convenient thermopotential, \(\vartheta (\eta (\mu );{\varPsi })\), may be defined along the curved solid–liquid interface of a GBG constrained by a uniform thermal gradient of magnitude G, pointing along the vertical \(\eta \)axis. The location of the system’s melting point isotherm, \(T_{\mathrm{int}}=T_{\mathrm{m}}\), occurs as \(\eta \rightarrow 0\), where its local dimensionless thermopotential is also set equal to zero, so \(\vartheta (\eta (\mu );{\varPsi })=0\) as \(\mu \rightarrow \pm \infty \).
Curvature of the solid–liquid interface, \(\kappa (y(x))\), at constant pressure, results in small shifts in the equilibrium thermopotential. These shifts allow the thermodynamic activities of curved solid and its contacting liquid phase to match along a GBG’s curving solid–liquid interface. Specifically, convex interfaces (\(\kappa >0\)), where the outward normal vector points toward the liquid phase, have equilibrium temperatures slightly below the melting point of a flat interface (\(<\,T_{\mathrm{m}}\)), whereas concave interfaces (\(\kappa <0\)) have equilibrium temperatures elevated slightly (\(>\,T_{\mathrm{m}}\)).
Tangential interface gradients
Capillarymediated energy fluxes
Difficulties encountered specifying temperatures on surfaces and interfaces are briefly included in [55], and higherorder heat conduction phenomena near fastmoving solid–liquid phase boundaries are discussed by Serdyukov [56]. Superficial thermal gradients and heat fluxes on heterophase interfaces, as considered in this study, are, however, rarely addressed, as these gradients involve extremely small temperature differences mediated by capillarity, which vary spatially over mesoscopic distances along curved interfaces. So, when capillarity is present, irrespective of whether an interface is considered sharp—i.e., having zero thickness, or, more realistically, slightly diffuse—field theory shows that thermopotential gradients develop as a necessary condition for the transport of thermal and mass fluxes from higher potential (flatter or concave areas) toward regions of lower potential (curved convex areas). The sufficient condition required for the appearance of capillarymediated fluxes is that the interface have nonzero transport numbers for heat and mass flow.
Interfacial transport of capillarymediated energy is analogous to another wellstudied fourthorder phenomenon: viz., species diffusion along interfaces and free surfaces in response to superficial gradients of the chemical potential [57]. In fact, surface diffusivities and interfacial thermal conductances are always expected to be nonzero, especially at temperatures near \(T_{\mathrm{m}}\), appropriate to solid–liquid interfaces. Indeed, Mullis was able to compute interfacial fluxes near the tip region of 2D dendritic crystals with a noisesuppressed fourthorder accurate phasefield model [58]. Mullis’s flux data closely matched the capillarymediated fluxes calculated for an appropriately scaled parabolic solid–liquid interface [5].
A phenomenon studied in reasonable depth and closely connected with capillarymediated interfacial heat conduction is Bénard–Marangoni hydrodynamic flow. This thermomechanical phenomenon is associated with free fluid surfaces subject to tangential temperature gradients [59]. Bénard–Marangoni flows are usually driven by large thermal gradients that directly affect a fluid’s surface tension, inducing stress gradients and surface flow. Interface heat conduction via capillarity in solid–liquid systems, as explained in “Capillary flux divergence” section, arises instead through an indirect effect of small capillaryinduced gradients of the Gibbs–Thomson potential.
Fourier’s law
Fourier’s law of heat conduction relates heat fluxes to their associated thermal gradients and conductivities, irrespective of dimensionality. Specifically, a thermal flux directed along a 2D interface by tangential thermal gradients, \({\varvec{\phi}}_{\tau }(x,y){\varvec{\tau }}\), bears units of (\({\hbox {W}}\,{\hbox {m}}^{1}\)). Its corresponding thermal conductivity, \(k_{\mathrm{int}}\), has physical units of (\({\hbox {W}}\,{\hbox {K}}^{1}\)). The energy flux along a solid–liquid GBG with a variational profile is found by applying Fourier’s law [55]. The superficial gradient responsible for this flux is the tangential gradient of the GBG’s GibbsThomson thermopotential, Eq. (14).
Although it might seem unusual that a stationary solid–liquid microstructure can spontaneously “absorb” a continuous stream of thermal energy and insert it into the perturbing grain boundary, one should recall that an equilibrated GBG is a steadystate microstructure that remains open to the passage of copious amounts of thermal energy moving down the applied thermal gradient, from the melt, through the interface, to its cooler solid. The thermal gradient constraining the stationary GBG, in a sense, also supports the much weaker tangential Gibbs–Thomson thermal gradients responsible for capillaryinduced interfacial heat fluxes.
As shown next, it is the more subtle influence of the capillary flux’s vector divergence—not the energy flux itself—that causes energy removal to occur along a stationary solid–liquid interface that actively alters the local energy budget. Although the effect of capillary flux divergence is concentrated primarily in the steeper parts of the GBG’s cusp, it will also be shown that a persistent distribution of cooling occurs over the entire GBG microstructure.
Capillary flux divergence
As discussed thus far, the persistent capillarymediated energy flux along the solid–liquid interface of a stationary GBG does not result in changes to its energy budget. Only indirect effects result through that flux’s vector divergence. The deeper explanation for this distinction lies in a fundamental difference between steadystate thermal fields developed within the bulk solid and liquid phases, and the capillarymediated fields on and along the GBG’s interface separating these phases. Specifically, steadystate heat flow obeys Laplace’s equation—the fluxes for which are harmonic and nondivergent—whereas that restriction applies neither to the tangential flux driven by capillary gradients along a sharp interface, nor to capillary fluxes passing between equilibrated bulk phases separated by a thicker diffuse interface. Both of these situations instead obey Poisson’s equation. The difference between nondivergent “Laplace fluxes” and divergent “Poisson fluxes” explains the fundamental origin of patternforming perturbations on moving interfaces, and shapealtering fields on stationary GBGs.
This distinction, moreover, accounts for the presence of autogenous Poisson sources—i.e., bias fields—the significance of which has not been addressed in theories of microstructure dynamics and pattern formation. Poisson sources enter as higherorder contributions to both interfacial energy and mass balances, especially at smaller length scales where patterns initiate. They represent, in fact are identical to, the higherorder capillarymediated energy rate that appears in the Leibniz–Reynolds energy balance, Eq. (1).
Bias fields on equilibrated GBGs
Stationary microstructures, such as GBGs interacting with their own interface fields, provide unique opportunities to generate and study persistent capillary fields. Upon equilibration with its applied thermal gradient, GBGs allow exposure and inspection of their equilibrated microstructure’s capillary fields and absent the complications that would normally accompany moving interfaces, such as timedependent shape change, morphological instability, latent heat release, or species redistribution in the case of alloys.
An example of what has been observed experimentally using an equilibrated GBG is captured in the photomicrograph, Fig. 3. This microstructure developed by maintaining a GBG for several hours in a uniform thermal gradient in a highprecision, temporally stable thermostat. The supercooling developed at its triple junction is extremely small, less than 1 mK, which reflects the large radii of curvature even deep within the GBG’s cusp region—a result of its relatively small constraining thermal gradient of only \(4\ {\hbox {K}}/{\hbox {m}}\) [48].
Theoretical profile points added to Fig. 3 were calculated from Eq. (5), noting its nearzero dihedral angle, and calculating its characteristic thermocapillary length, \({\varLambda }=1.18\times 10^{4}\) (m). The thermocapillary length, \({\varLambda }\), was determined from the GBG’s cusp depth, \(y^{\star }=1.67\times 10^{4}\) (m), which corresponds to the dimensionless location of its triple junction at \(\eta ^{\star }=\sqrt{2}/2\). At the scale and resolution of this micrograph, one is unable to detect discernible differences in the equilibrated interface shape from the one predicted using the variational profile formula.
Now, whether or not an active bias field is present on such a stationary solid–liquid interface after equilibration and whether or not higherorder GBG profile changes are induced by the bias field interacting with its own microstructure are the issues next addressed.
Biasfield distributions
The bias field, \({\mathfrak {B}}(\eta (\mu );{\varPsi })\), Eq. (20), predicts rates of capillarymediated energy withdrawn at the interface at any cusp depth \(\eta \) for different dihedral angles, \({\varPsi }\). The associated energy rate distributions, \({\mathfrak {B}}(\mu (\eta );{\varPsi })\), are easily calculated from \({\mathfrak {B}}(\eta (\mu );{\varPsi })\) by crossplotting Eq. (20) against the variational profile \(\mu (\eta ;{\varPsi })\), Eq. (5). The plots in Fig. 6 display energy removal rates over the righthand half interface of GBG’s with various variational profiles.
Cooling distributions are uniquely determined by a GBG’s dihedral angle, with the highest rates of energy adsorption occurring well within their cusps either near, or at, their triple junction. Curiously, GBGs exhibit capillarymediated cooling without any heated regions. This behavior differs markedly from capillarymediated energy distributions examined previously for other interfacial profiles, where both cooling and heating were found in all cases [14]. Specifically, previous studies of capillary energy fields included the following interfacial forms: (1) closed conic sections, such as the circle and ellipses; (2) protuberant “open profiles,” such as the parabola and hyperbolas [5]; and (3) several transcendental shapes, such as the cellular forms of Saffman–Taylor “fingers”. You et al. recently examined the stability and patternformation dynamics on phasefield simulated periodic Saffman–Taylor cells and found quantitative agreement confirming their dynamic simulations of cellular growth with a biasfield analysis for tip stability of the same shape [61, 62].
Phasefield simulations
Diffuse and sharp interfaces
Simulations of capillarymediated energy fields were accomplished in the present study using a thermodynamically consistent multiphasefield model based on an entropy density functional [63]. In diffuseinterface simulations that employ the phasefield method, initial and boundary conditions are specified and the evolution of farfromequilibrium microstructures are usually computed. In this study, grain boundaries with prescribed orientation and energy densities intersected initially with planar solid–liquid interfaces. These simulations evolved constrained states over many time steps, as the simulated GBGs relaxed within their thermal gradient field to yield their steadystate equilibrated microstructures.
 (a)

Early in the equilibration process, the observed thermal field displays a grid of isotherms with orthogonal lines of heat flow that already suggest cooling of the melt within the groove cusp. Interface cooling causes upward deflection of the isotherms within the melt accompanied by bending, or “focusing,” of the lines of heat flow. The “jump,” or discontinuity, displayed by the thermal gradient is predicted by Eq. (4) and indicative of the presence of an active, persistent interface energy field.
 (b)

Several hundred thousand time steps later, isotherm deflection from interface cooling has spread further away from the triple junction, and inward bending of the heat flow lines is noticeable within the outermost five lines.
 (c)

Steadystate equilibration is eventually achieved after hundreds of thousands of numerical time steps, each one iteratively calculated by solving phasefield Eqs. (26) and (27).
 (d)

The phase image forms an equilibrated GBG consisting of a light gray region, the melt; a medium gray area, crystal 1, which forms a grain boundary with the dark gray area, crystal 2. All “interfaces” are, in fact, slightly diffuse: light gray/medium gray and light gray/dark gray borders represent smooth transition of the multiphasefield variables. Thermopotentials are measured postprocessing along the solidtoliquid isolines given by an average value of the phasefield order parameters corresponding to phases that are in contact.
In accepting this procedure to confirm the existence of interfacial bias fields resident on simulated GBGs, one notes that phasefield models numerically solve coupled partial differential equations that characterize the dynamic behavior of diffuse heterophase interfaces with timeevolved continuous phase indicators. Phasefield models, moreover—and this is important—are not coded with any explicit physics that admit autogenous capillary fields. In fact, quite to the contrary, bias fields were originally discovered by combining analytic constructs based on sharpinterface thermodynamics, omnimetric energy conservation, and classic field theory [5]. Biasfield theory (for sharp interfaces) and phasefield numerics (for diffuse interfaces) develop their respective thermodynamic descriptions of interfacial energetics using independent mathematical and physical approaches that avoid tautologies or hidden circularity between them; they are each consistent descriptions, applicable to different physical limits that describe the nature of equilibrated solid–liquid interfaces.
Notable consistency will be shown between independent approaches predicting interfacial behavior for stationary microstructures that captures the essentials of our findings: viz., phasefield equilibrated GBGs harbor active capillary energy fields equivalent to bias fields, which were predicted previously with sharpinterface thermodynamics and field theory.
Detecting interfacial energy fields
Potential change and energy rate
Variational GBGs, as discussed in “Variational grooves” section, lack interfacial energy fields. If the interface of an equilibrated groove did not support an active interface energy field, then its thermopotential distribution would remain linear with its depth coordinate, as already shown with Eq. (12) for variational GBGs. If, however, capillarymediated energy fields do persist along equilibrated GBGs, then their presence should modify the distribution of their thermopotentials. Detecting small nonlinear departures of nearly linear interface thermopotential distributions remains the central challenge in making phasefield measurements capable of uncovering the existence of capillary fields on GBGs, and determining distributions of their energy rates.
Specifically, the steady distribution of cooling found along a GBG’s solid–liquid interface, predicted by Eq. (20) and displayed in Fig. 6, should add a small nonlinear component to the GBG’s otherwise linear thermopotential distribution. Particularly useful is the fact that, at steadystate, the nonlinear additions to the thermopotential are locally proportional to the rate of thermal energy added to, or removed from, the interface. That is, if the interface were cooled, its thermopotential would be lowered in proportion to the cooling rate. If, instead, the interface were heated locally, then its potential at that location would rise by an amount proportionate to the heating rate. Lastly, if the interface were lacking a bias field, then its potential distribution would remain linear with depth.
This proportional linkage between the energy rate steadily added or removed on an interface, \(\delta \dot{Q}\), and the resulting steadystate change in local temperature, \(\delta {T}\), is based on a firstorder variation between conjugate state variables: viz., between entropy, S, and temperature T, the product of which is energy. The resulting “calorimetric” relationship at steady state provides the proportionality expected at constant pressure^{6} between small changes in temperature and the associated energy rate. Thus, in an open steadystate system at constant pressure, \(\delta \dot{Q}\propto \delta {T}\), indicating that capillarymediated interface energy rates are precisely proportional to the changes they induce in the local interface temperature, or thermopotential.
Finally, the steadystate relationship between interfacial energy rate and local interface thermopotential, to be simulated in “Phasefield model and results” section, also possesses a mathematical foundation derived from scalar potential theory. The strengths of point sinks, or sources, of energy (or any other extensive quantity) released along boundaries may be represented mathematically as line integrals of their Green’s function distributions [18]. Some specific examples are discussed in [16] of this proportionate response for both instantaneous and continuous diffusion or conduction sources, where Green’s functions are distributed along planar and circular boundaries in different spatial settings. Those cited examples also demonstrate that steadystate source rates of extensive quantities induce, respectively, proportionate changes on a boundary’s conjugate intensive parameters.
Exposure and measurement of interface fields
Phasefield simulations were carried out to probe the solid–liquid interfaces of GBGs, which upon full equilibration in a steady thermal gradient should spontaneously develop active interface energy fields. The opportunity to equilibrate a stationary microstructure presents a novel opportunity for using phase field to verify interface behavior consistent with predictions for sharp interfaces. Capillarymediated energy fields were exposed as thermopotential “residuals” and measured in situ along the phasefield isoline of a simulated GBG’s solid–liquid interface. The method revealing these interfacial perturbations is explained next.
Phasefield model and results
Multiphasefield model
The internal energy density, e, is related to the molar latent heat, \({\Delta {H}}_f\), and constant specific heat \(c_{v}\), by the relationship \(e = {\Delta {H}}_f\ h\left( \phi _{\alpha }\right) +c_{v}T\). The interpolation function \(h\left( \phi _{\alpha }\right) = \phi _{\alpha }^3\left( 6\phi _{\alpha }^2 15\phi _{\alpha } +10\right) \) satisfies the constraints \(h(1) = 1\) and \(h(0) = 0\).
The ratios of the grain boundary’s energy density to that of the crystal/melt boundary were selected to produce after steadystate equilibration a series of specific dihedral angles selected between \({\varPsi }=0\) and \(180^{\circ }\). Simulations started with either an initial configuration consisting of a flat crystal–melt interface intersected normally by a grain boundary, or, for greater computational efficiency, with a starting groove shape from a prior simulation run having a dihedral angle not too far from the new final value. Numerical iterations were extended well beyond the point where a desired equilibrium dihedral angle was initially achieved.
 1.
We employed an explicit finitedifference scheme for iteratively solving Eqs. (26) and (27) in a 2D domain of grid size \(2000{\Delta {X}}\) by \(400{\Delta {Y}}\).
 2.
We checked that the required steady 1D thermal gradient, G, a linear temperature distribution, develops fully along the Ygrid, as shown for the three thermopotential traces plotted in Fig. 8.
 3.
We measured the dihedral angle at every time step [66]. After this angle converged to the expected \({\varPsi }\)value, we continued equilibration for an additional \(10^4\) time steps, to rule out any chance of additional relaxation affecting the shape of the triplejunction region of the groove profile.
 4.
We initialized the numerical domain with the equilibrated phase and thermopotential fields corresponding to \({\varPsi } = 90^{\circ }\), \(120^{\circ }\), and \(180^{\circ }\), for subsequent simulations where the final angles were \({\varPsi } = 83.6^{\circ }\), \(150^{\circ }\), and \(165.5^{\circ }\), respectively. Adopting this strategy greatly reduced the time needed to attain full equilibration of the simulated GBG.
Recursion
The recursive task of extracting precise isoline thermopotentials was achieved by scanning the groove’s computational domain and employing a numerical postprocessing algorithm. An additional difficulty encountered with equilibrated isolated GBGs was that the radii of curvature of their solid–liquid isolines gradually approach extremely large values in the flattest regions close to the domain’s lateral edges. We found that for interface locations in the immediate vicinity of a groove’s triple point, the postprocessing algorithm was capable of determining accurate isoline thermopotentials that correspond to \(\phi = 0.5\). However, achieving precise potential measurements (\(T_{\phi =0.5}\)) proved more challenging along the flatter portions of the solid–liquid boundary, remote from the groove’s triple junction. This difficulty arose due to the narrowing of the diffuse interface when almost flat, a condition requiring far fewer boundary points to span the diffuse interface between the bulk phases. We then chose to perform a linear interpolation between \(T_{\phi _1}\) and \(T_{\phi _2}\) with \(T_{\phi _3}\) to estimate the thermal residuals more precisely, by requiring that the isoline \(\phi _{i}\ (i = 1,2)< {1/2} < {\phi _3}\).
Thermopotentials on analytic profiles and phasefield isolines
Both the steadystate potential and curvature distributions along the analytic profile of a variational GBG, which has zero thickness, are perfectly linear in the variable \(\eta \). See again Eqs. (8) and (12). Therefore, if one were to subtract the value of the constraining (embedded) linear potential, \(4\times \eta \), from the interface thermopotential of a variational GBG, \(\vartheta ({\eta }(\mu );{\varPsi })\), one would obtain a trivial null residual: \(\vartheta ({\eta }(\mu );{\varPsi })4\eta =0\).
Now contrast what happens upon simulating a GBG subjected to phasefield equilibration, where superficial gradients of the Gibbs–Thomson interface potential appear and stimulate a divergent tangential flux. An interfacial energy sink, \({B(Y(X))}<0\), is created, by virtue of which the thermopotential distribution along the microstructure’s isoline, \(\phi =0.5\), is slightly, and nonlinearly, depressed from its linear “variational” form. Depression of the linear thermopotential is linked to the fact that, as explained in “Potential change and energy rate” section, thermodynamics and heat conduction theory show that the expected change in temperature, or thermopotential, caused by an interface sink of energy is proportional to its corresponding rate of energy withdrawal, B(Y(X)).
Consequently, the residuals, \({\mathfrak {R}}{(Y(X))} \equiv {T}_{\phi =0.5}G{\times {Y(X)}}\), calculated along the interfacial isoline, \(\phi =0.5\), equal the interfacial energy rate, B(Y(X)) times a ratio, \({\xi }\), that proportions the shifts in thermopotential (residuals) with their local capillarymediated energy rates along the isoline. Thus, one obtains the relationship \({\mathfrak {R}}{(Y(X))}=T_{\phi =0.5}G\times {{Y(X)}}= {B(Y(X))}\times \xi \).
Also important, the energy distributions measured for each specified dihedral angle, \({B(X(Y);{\varPsi })}\), that develop along simulated phasefield isolines are congruent with the theoretical biasfield distributions, \({\mathfrak {B}}(\mu (\eta );{\varPsi })\), calculated from Eq. (20) and plotted in Fig. 6.
In fact, the implied equality itself, viz. \({\mathfrak {R}}{(Y(X))}= \xi \times {\mathfrak {B}}(\eta (\mu );{\varPsi })\), supported by these measurements is slightly imperfect, because it ignores higherorder differences existing between the analytic shapes of variational profiles, viz. Eq. (5), and the simulated phasefield isolines. The shapes of equilibrated GBGs are modified slightly by selfinteraction with their capillary fields: (1) their local interface curvatures increase slightly from the additional interface cooling, and, consequently, (2) their cusp depths decrease slightly. The combination of increased curvature and decreased cusp depth allows the total rotation of a GBG’s normal angle between the triple junction and flat interfacial regions to remain constant at \({\varPsi /2}\), as required (in 2D) by topology.
Conclusions
 1.
Application of the Leibniz–Reynolds transport theorem shows that Stefan balances at solid–liquid interfaces—which exclude capillarity—do not satisfy omnimetric energy balance. Stefan balances lack higherorder capillarymediated terms needed to satisfy energy conservation on mesoscopic scales.
 2.
Previous studies showed that capillarymediated interface fields are capable of stimulating complex pattern formation on moving interfaces in lownoise environments [5]. The present study shows that stationary GBG microstructures support fields that cool their interfaces, allowing their measurement via multiphasefield simulations.
 3.
Isotropic GBGs provide wellstudied examples of stable microstructures that remain in thermodynamic equilibrium in the presence of steady thermal gradients. GBG profiles predicted from variational calculus are linear minimizers of the groove microstructure’s free energy and differ only slightly from simulated equilibrated GBGs that are nonlinear minimizers, because of selfinteraction with their persistent interfacial energy fields.
 4.
The distribution of interfacial energy rates on GBGs may be calculated from their variational profiles as surface Laplacians of their Gibbs–Thomson thermopotential, or, equivalently, from (minus) the divergence of the tangential flux. Variational GBGs with isotropic solid–liquid energy density yield simple cubic expressions for their capillary energy fields.
 5.
Multiphasefield simulations were performed on simulated GBGs with different dihedral angles to verify the existence of active capillary fields. Measurements of isoline thermopotentials permit calculation of residuals by subtracting from the potentials the linear distribution imposed by the applied thermal gradient. Residuals are shown to be proportional to the time rate of the capillary energy field acting along the interface isoline.
 6.
Phasefield residuals confirm, in every case tested, quantitative agreement with interfacial bias fields predicted from sharpinterface thermodynamics. Selfconsistency and independence of simulated phasefield measurements and sharpinterface theory support the existence of bias fields on stationary crystal–melt interfaces in agreement with results found earlier for moving solid–liquid interfaces [5].
 7.
The physical interface mechanism explored in this study shows that capillarymediated fields provide perturbations capable of initiating diffusionlimited patterns. These include patterns in nature exhibited by snowflakes and crystallized mineral forms, as well as microstructures of cast alloys. Capillarymediated interface fields might provide new approaches toward achieving improvements in solidification processing, welding, and crystal growth by control of microstructure at mesoscopic scales.
Footnotes
 1.
The term “bias field” is chosen for these thermodynamic fields, as they modulate, or bias, the local velocity on a moving interface to be slightly above or below the mean speed predicted by the Stefan condition.
 2.
Details of GBG functionals and the mathematics of “extremizing” their profiles via the Euler–Lagrange equation are available in Ref. [43].
 3.
Isotropy is approximated by some crystalline materials that exhibit weak variations (\(<1\%\)) of their crystal–melt interface energy, \(\gamma _{s\ell }\), around some particular crystallographic zone axis. For GBGs in BCC succinonitrile, an example of which is provided in Fig. 3, the fourfold anisotropic variation of the solid–liquid interfacial energy density is only \(\pm\, 0.5\)% around its three cubic [100] zone axes, with its mean value \({\gamma }_{s\ell }=(9.0\pm 1)\times 10^{3} ({\hbox {J}}\,{\hbox {m}}^{2}\)).
 4.
Arc length, \(\hat{s}\), increases as the unit tangent vector, \({\varvec{\tau }}\), advances anticlockwise along a GBG’s solid–liquid interface. The thermopotential falls with increasing arc length along a GBG’s right profile, whereas this potential increases with increasing arc length along the left profile. Thus, the tangential gradient of a GBG’s thermopotential switches from antiparallel to parallel with the local unit tangent vector and reverses sign when crossing its triple junction.
 5.
The derivatives \({\mathrm{d}}y/{\mathrm{d}}s={\mathrm{d}}\eta /{\mathrm{d}}\hat{s}\) are positive along the groove’s left profile and negative along its right profile. The depth coordinates, y or \(\eta \), become increasingly negative into the groove cusp. See Fig. 1.
 6.
See also Ref. [64], equation (4.42), and its discussion on pp. 52–55 that provide formal proportional relationships among heat content, enthalpy, and temperature changes in pure substances at constant pressure.
Notes
Acknowledgements
Authors acknowledge their preliminary discussions with Dr. Britta Nestler, Karlsruhe Institute of Technology, Institute of Applied Materials–Computational Materials Science, Germany. Author KA acknowledges support from the German Research Foundation (DFG) under Grant Number AN 1245/11.s and from the College of Engineering, Arizona State University. Author MEG gives special thanks for the Allen S. Henry endowed Chair of Engineering that provided financial support through the Florida Institute of Technology, Melbourne, Florida. Both authors thank the reviewer for several useful suggestions improving our manuscript and wish to express our heartfelt gratitude for the brave crew aboard shuttle Columbia, on flight STS87, flown in November, 1997, one of whom, our IDGE Mission Specialist, Dr. Kalpana Chawla, perished when Columbia, upon reentering Earth’s atmosphere on February 1, 2003, was destroyed with the tragic loss of all its seven crew members.
Compliance with ethical standards
Conflict of interest
The authors declare no conflicts of interest. Sponsors had no role in the design of the study, or in the collection, analyses, or interpretation of data, or in the writing of the manuscript, and in the decision to publish the results.
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