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 capillary-mediated 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.
Thus, the Leibniz–Reynolds transport theorem can be reduced to four independent rates that enter omnimetric energy conservation for isotropic interfaces [
5]:
$$\begin{aligned} k_c{\varvec{\nabla}}[T_c({\mathbf {r}})]\cdot {{{\mathbf {n}}}} +k_{\ell }{\varvec{\nabla}}[{T_{\ell }}({\mathbf {r}})] \cdot {{{\mathbf {n}}}}+\frac{{{\Delta {H}}}_f}{\varOmega } {{\mathbf {v}}_{LR}({\mathbf {r}})}\cdot {{\mathbf {n}}} -{\varvec{\nabla} }_{{\tau }}[{{\phi }}_{\tau }({\mathbf {r}}) \cdot {{\varvec{\tau }}}]=0. \end{aligned}$$
(1)
The first two terms in Eq. (
1) are local rates of thermal conduction driven by macro-gradients within the bulk crystalline (
c) and liquid (
\(\ell \)) phases surrounding a moving interface,
\({\mathbf {r}}(x,y,z,t)\). The third term represents the rate of latent heat produced (or absorbed) by normal motion of the interface during phase transformation, including—by adjustment of the volumetric enthalpy change—energy stored (or released) from interface “stretching.” All the energy rates described by the first three terms in Eq. (
1) are associated with gradients or velocities directed along the unit normal,
\({\mathbf {n}}\), to the interface. The subscript “
LR” appended to the latent heat rate designates the local interface velocity consistent with the presence of the fourth energy rate term included in Eq. (
1) that varies, point to point, over the interface. The average rate of latent heat released, when integrated over the solid–liquid interface, agrees with Stefan’s balance, as capillary modulations, which arise from a conservative vector field, tend to cancel over large scales on moving interfaces. (For proof, see Ref. [
5], Section 5.2.2.)
The fourth energy rate, appearing in Eq. (
1), is responsible for local capillary-based energetic addition and subtraction. It represents the interfacial energy rate associated with the surface divergence of the capillary-mediated 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 thermo-potential. 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 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.
In general, we argue, capillary-mediated 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 multi-scale 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 phase-field theories and their numerical models.
Now, applying similar notation to that used in Eq. (
1), we write the Stefan condition as used in conventional interface balances describing solidification and crystal growth as [
24,
25],
$$\begin{aligned} k_c{\varvec{\nabla} }[T_c({\mathbf {r}})]\cdot {{\mathbf {n}}} +k_{\ell }{\varvec{\nabla} }[{T_{\ell }}({\mathbf {r}})] \cdot {{\mathbf {n}}}+\frac{{{\Delta {H}}}_f}{\varOmega } {{\mathbf {v}}_{St}}({\mathbf {r}})\cdot {{\mathbf {n}}}=0. \end{aligned}$$
(2)
Note that the interface velocity appearing in Eq. (
2), by contrast with
\({\mathbf {v}}_{LR}({\mathbf {r}})\) in Eq. (
1), now denotes the Stefan interface velocity,
\({\mathbf {v}}_{\mathrm{St}}({\mathbf {r}})\), as the normal speed of the solid–liquid interface, without consideration of capillarity.
If one subtracts Stefan’s condition, Eq. (
2), from the Leibniz–Reynolds energy balance, Eq. (
1), and rearranges terms, an interesting expression appears that estimates the normal speed difference expected between moving interfaces with and without capillarity:
$$\begin{aligned} \frac{-{\varvec{\nabla} }_{\tau }[{{\phi }}_{\tau }({\mathbf {r}}) \cdot {\varvec{\tau }}]}{{\Delta {H}}_f/\varOmega } =\left( {\mathbf {v}}_{St}({\mathbf {r}})-{\mathbf {v}}_{LR} ({\mathbf {r}})\right) \cdot {{\mathbf {n}}}. \end{aligned}$$
(3)
The right-hand side of Eq. (
3) equals the interfacial speed difference, or “modulation” (
\({\hbox {m}}\,{\hbox {s}}^{-1}\)), caused by capillary energy fields. Its left-hand side is the ratio of the scalar bias field
\({\mathrm {(W\,{m}^{-2})}}\) to the system’s volumetric enthalpy density
\({\mathrm {(J\,{m}^{-3})}}\). Also of interest in the case of moving interfaces are any discrete locations where this field reverses sign. Sign reversals occur at the roots (zeros) of the capillary bias field. Roots cause speed modulations that simultaneously increase and decrease over a small region surrounding the roots. The juxtaposition of opposing speed modulations causes inflections to form. Inflections, amplified by the transport fields surrounding the moving interface, can develop into bumps, branches, or “fingers” that protrude from the interface and enhance pattern complexity [
5].
Connections between pattern formation, interfacial speed modulations, and the presence of fourth-order capillary fields were overlooked in early mathematical formulations of dendritic growth [
26], in subsequent pattern-formation models for diffusion-limited 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 multi-scale energy balance, rather than the Stefan balance, might avoid mathematical singularities introduced by capillarity in sharp-interface models of pattern-forming 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 “
Phase-field 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, non-trivial setting on which to measure capillary-mediated 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.
The Leibniz–Reynolds energy balance, Eq. (
1), for a stationary interface in the form of a grain boundary groove constrained by an applied thermal gradient,
\({\mathbf{G}}\), simplifies still further as,
$$\begin{aligned} -\nabla _{\tau }[{{\phi }}_{\tau }({\mathbf {r}})\cdot {\varvec{\tau }}] =k_{\mathrm{th}}{\mathcal {J}}({\mathbf{G}}), \end{aligned}$$
(4)
where
\({\mathcal {J}}({\mathbf{G}})\) is the jump that develops in the norm of the thermal gradient across the interface from the divergence of the tangential flux, namely
\(|{\mathbf{G}}|_{\ell }-|{\mathbf{G}}|_{c}\). Equation (
4) describes the effect of the interfacial bias field on a
stationary solid–liquid interface. The interface’s shape, separating phases with identical thermal conductivities,
\(k_{\mathrm{th}}\), is described by the position vector
\({\mathbf {r}}(x,y,z)\). The presence of an interfacial energy field causes a “jump,”
\({\mathcal {J}}({\mathbf{G}})\), to appear in the normal components of the thermal gradient across that interface. Note, that absent the presence of a bias field, for equal phase thermal conductivities surrounding a stationary solid–liquid interface, the thermal gradient would remain perfectly uniform and continuous, and would not change magnitude when crossing the interface.
Equation (
4), moreover, also suggests a basis for detecting bias fields that manifest their presence as small nonlinear shifts in the interface’s thermo-potential. The change in thermo-potential along static solid–liquid interfaces is also proportional to the local energy rate of the bias field [the left-hand side of Eq. (
4)], and proportional to the jump developed in the gradient across the interface [the right-hand side of Eq. (
4)]. The basis for proportionality between shifts in interface thermo-potential and field energy rate is discussed in detail in “
Detecting interfacial energy fields” section.