## Abstract

Nano-faceted crystals answer the call for self-assembled, physico-chemically tailored materials, with those arising from a kinetically mediated response to free-energy disequilibria (*thermokinetics*) holding the greatest promise. The dynamics of slightly undercooled crystal–melt interfaces possessing strongly anisotropic and curvature-dependent surface energy and evolving under attachment–detachment limited kinetics offer a model system for the study of *thermokinetic* effects. The fundamental non-equilibrium feature of this dynamics is explicated through our discovery of one-dimensional convex and concave translating fronts (*solitons*) whose constant asymptotic angles provably deviate from the thermodynamically expected *Wulff* angles in direct proportion to the degree of undercooling. These *thermokinetic* solitons induce a novel emergent facet dynamics, which is exactly characterized via an original geometric matched-asymptotic analysis. We thereby discover an emergent parabolic symmetry of its coarsening facet ensembles, which naturally implies the universal scaling law *t* of the characteristic length

## 1. Introduction

Nano-faceting of material interfaces [1,2] is a paradigmatic, non-equilibrium self-assembly process which arises in a wide variety of physical settings; for example, high-efficiency photoelectrochemical cells yielding solar-energy storage through hydrogen production [3], and enantiomer-specific heterogeneous catalysts with application to biology [4]. Exotic nanocrystal motifs and unexpected orderings abound here, such as tetrahexahedra from electrochemical [5] synthesis, truncated *Wulff shapes* in solution-phase [6] synthesis, adorbate-induced nano-faceting of single-crystal surfaces [7], and the self-ordering of epitaxially grown *quantum dot* arrays [8], to name but a few. Since the pioneering low-energy electron diffraction work of MacRae [9], advances of *in situ* and *ex situ* instrumentation for nanoscale imaging [10–12] are permitting experimentalists to elucidate the essential physics and chemistry underlying this tantalizing range of nano-faceting phenomena, and the pace of discovery shows no signs of abating. However, theories which provide quantitative prediction of such shapes and shape distributions (*morphometrics*) are still in their scientific infancy. Since physico-chemical characteristics of nanocrystals are known to exhibit an exquisite sensitivity to their shape [13], morphometric theories will be crucially important for the scientific development of engineered materials with prescribed properties for applications ranging from bio-sensing [14] to *green* nanotechnologies [15].

Certain planar interfaces of macroscopic single crystals can be thermodynamically induced to spontaneously decompose into *fully faceted* morphologies by thermal [16,17], chemical [7,18–21] or electrochemical means [13,22]. Experimental observations of *spinodally decomposing* interfaces [23,24], such as those for thermally quenched Si(111) [17] or the oxygen-induced faceting of Cu(115) [19], reveal an ensuing coarsening of the resulting fully faceted interfaces, and the concomitant emergence of scaling regimes, wherein a *power law* governing the time *t* evolution of the characteristic length *n* that have been empirically discerned [25–27] suggest they depend on the dominant mass-transport mechanisms, and possibly even the symmetry group of the crystal's *Wulff facets*. Furthermore, normalized length distributions of such faceted morphologies within these scaling regimes (e.g. edge lengths) have also been empirically observed to be given by universal distributions [17].

Much of the non-equilibrium nano-faceting phenomena noted above lie definitively outside the predictive scope of purely energetic or purely kinetic considerations [28], but rather reflects the interface's kinetically mediated response to free-energy disequilibria [7,13]. The ensuing *thermokinetic* interfacial dynamics, which encodes a thermo-mechanical balance between the local rates of surface-free-energy/bulk-chemical-potential release and a kinetics-induced entropy production [29], finds expression in multiple length- and time-scale geometric partial differential equations (PDEs).

The isothermal growth of a nano-faceted crystal into its undercooled melt has been thermodynamically modelled at a continuum level by Herring, who adapted Wulff's assumption that the interfacial energy is (geometrically) non-convex with respect to the unit-normal **n** to the interface [30] by also including a dependence on the interfacial curvature [31]. Upon assuming *attachment–detachment* kinetics to be the dominant mass-transport mechanism, one concludes, upon suitable spatio-temporal scaling, that such a one-dimensional Herring interface **n** pointing into the melt (figure 1), evolves according to the hyperbolic–parabolic, geometric PDE [32]
*ϑ* and *κ*, respectively, denote the *normal velocity*, local-inclination angle and curvature of _{s} is the partial derivative with respect to the interfacial arc-length *s*, while the *surface stiffness* *ε* is a dimensionless constant that encodes the degree of undercooling. We note in passing that the Burton–Cabrera–Frank step-flow models for thin-film deposition on crystal substrates [33] may possess step-bunching instabilities that lead to *mound* formation, and the coarse-grained description thereof yields continuum models which qualitatively mimic equation (1.1): see Sec. 4.9.8 of Michely & Krug [33].

In this article, we analytically characterize the *slow* facet dynamics that effectively govern the solutions to equation (1.1) in the slightly undercooled regime 0<*ε*≪1. To determine this effective dynamics, which mathematically emerges from equation (1.1) by letting *ϵ*↘0, we develop novel geometric matched-asymptotic methods; these significantly extend the analysis [34] previously applied to the *convective Cahn–Hilliard equation* [35,36]. The emergent facet dynamics we uncover is highly non-trivial since it does not coincide with simply setting *ε*=0 in equation (1.1), which only yields a mathematically ill-posed (*backward parabolic*) equation when

After deriving equation (1.1) in §2, we proceed in §3 to establish the existence of convex (+) and concave (−) translating fronts that asymptote to angles *thermokinetic angles*), which both depend on the degree of undercooling *ε* and also deviate from the thermodynamically expected Wulff angles ±*ω*; a *thermokinetic* effect. In §4, we develop a matched-asymptotic analysis of equation (1.1) which analytically demonstrates that this thermokinetic effect, *Peclet length*, below which a parabolic-scaling symmetry operates; this is the geometric extension of the emergent Peclet length for the convective Cahn–Hilliard equation [34]. In §5, we turn to an analysis of the coarsening dynamics of solutions to equation (1.1) that originate from a small random perturbation of a thermodynamically unstable, macroscopic facet orientation *ϑ*=0. We theoretically predict and numerically validate that the time *t* evolution of the characteristic (effective) *facet* length

## 2. Herring's oriented interface dynamics

The surface free energy of a crystal is classically presumed to depend solely on its local crystallographic orientations [30], and thereby permits edges to arise between minimum-energy facets of equilibrium crystals. Reflecting the configurational abhorrence of an atom to sitting at such an edge, *Herring* augmented *the Wulff energy* with a curvature-dependent term [31] to enforce, over a prescribed nanoscopic length scale, *α*, a smooth transition between the surface normals of adjacent equilibrium facets; we henceforth refer to *α* as the *bending length*. For a one-dimensional interface *γ*_{0} sets the interfacial energy scale, the dimensionless function *θ*, relative to a prescribed crystallographic axis, and *s*, *ϑ* and *κ*:=∂_{s}*ϑ* denote the arc-length, local-inclination angle relative to a prescribed polar axis, and signed curvature of the interface *surface stiffness*, ^{′} denotes the derivative), determines the stability of a facet orientation. We assume the orientation *θ*=0 both is thermodynamically unstable, *ω*; i.e.

Now consider such a one-dimensional single-crystal interface *μ*]] that is induced by the melt's temperature *T* being below the melting temperature of the crystal *T*_{m}, namely *T*<*T*_{m}, is given by
*L* is the specific latent heat of fusion. We presume that attachment–detachment kinetics is the sole mass-transport mechanism, and let *ρ*_{m} and *ρ*_{c}, coincide; namely, *ρ*_{m}=*ρ*_{c}=*ρ*. In this case, no convective flow field is induced during solidification/melting of the crystal. Choosing the orientation of the interface *normal interfacial velocity*, *the surface stiffness* ^{′} denotes the derivative), and ∂_{s} denotes the partial derivative with respect to the interfacial arc-length *s*.

The ratio of the surface energy scale *γ*_{0} to the jump in chemical potential per unit volume, *ρ*[[*μ*]], sets the characteristic length scale *kinetic* velocity scale *ε*≠0 measures the ratio of the bending length *α* to the length scale *L*=2×10^{5} J kg^{−1}, *ρ*=8.5×10^{3} kg m^{−3}, *γ*_{0}=1.75 J m^{−2} and estimating *α*≈1 nm, one obtains, for a dimensionless undercooling of 1%, the value *ε*≈0.01.

## 3. *Thermokinetic angles* of translating fronts

We recall that a *translating front* is a time-evolving curve **V** is a constant vector and *t* obtained by translating the fixed curve *t***V**; we naturally refer to **i** and **j** denote the standard unit basis vectors of a prescribed stationary two-dimensional Cartesian reference frame, we note for future reference that *s* parametrization of

We now turn to the analysis of translating fronts which solve Herring's oriented interface dynamics (2.5) when 0<*ε*≪1: what we henceforth call the *slightly driven* regime. We focus in particular on identifying those with symmetric unbounded profiles *Ω* will shortly be shown to depend on *ϵ* and the sign of their curvature (+ or −). When we need to, we will later use the notation

Note that reflection in the Cartesian *y*-axis, (*x*,*y*)↦(−*x*,*y*), is a symmetry of equation (2.5) since *θ*↦−*θ*. It is therefore natural to seek travelling-front solutions to equation (2.5) which also respect this symmetry, and so we take *y*-axis, and set the velocity **V** to be parallel to the *y*-axis, *s* of the oriented curve *s*=0 the corresponding local angle of *polar axis* *θ*=0. Denoting the corresponding arc-length parametrizations of the local angle *ϑ* of

Anticipating the appearance of the inner-length scale *S*:=*s*/*ε*, over which a sharp transition in the interfacial angle of *S*, one concludes *S* serves as an arc-length coordinate on *Θ* of *s* we find
*ϑ* of *inner-scale* *S*:=*s*/*ε*:

To determine whether the translating front (3.3) with *κ*, normal velocity *Θ* of its scaled profile

By incorporating the asymptotic flatness

In the case where _{s}*Θ*>0 (∂_{S}*Θ*<0), then its local angle *Θ* is monotone increasing (decreasing) with respect to its arc-length *S*. Therefore, the local angle of *Θ*, may serve as natural parametrization of *Θ* at each _{S} is related to *the angular derivative* ∂_{Θ} through the chain rule by

In summary, given the profile *Ω*. Last, we may naturally view the solutions to equation (3.20) with 0<*ε*≪1 as regular perturbations in *ε* of the symmetric curvature profiles

There is a natural converse to the above whereby, given a function *Ω* which solve equation (3.20), one may readily construct associated travelling front solutions to equation (2.5). First, let *Θ*→*Ω*. It then follows that the travelling front

### (a) Exact translating fronts and thermokinetic angles Ω ε + and Ω ε −

Here we present exact convex and concave travelling front solutions to equation (2.5) that arise for the particular surface energy
*a*>0 and *ω*∈(−*π*/2,*π*/2) given. One readily verifies that this choice of surface energy meets the conditions expressed in equation (2.2) with Wulff angles ±*ω*. Our approach to finding these exact solutions is to use the converse construction noted around equation (3.22). To that end, we seek a positive (negative) solution to equation (3.20), *Ω*^{+}>0 (*Ω*^{−}>0), within the ansatz
*Ω*^{+}>0 and *Ω*^{−}>0 denote generic positive and negative angles.

Rather remarkably, one finds that both *Ω*=*Ω*^{+} and *Ω*=*Ω*^{−} each solve the second-order PDE appearing in equation (3.20) precisely when the corresponding angles *Ω*^{±} are solutions of the trigonometric-polynomial equation

The range of *ε* for which there exist solutions to equation (3.26) is readily determined by considering the graph *Ω*^{+}>0 (*Ω*^{−}<0) of equation (3.26). Furthermore, it is only the least positive (negative) solution *Ω*^{+}(*Ω*^{+}) of equation (3.26) for which *Ω*=*Ω*^{−}) meet the non-zero condition equation (3.20); namely *Θ*∈(−*Ω*^{±},*Ω*^{±}). For each
*Ω*^{+}_{ε}(*Ω*^{+}_{ε}) denote the least positive (negative) solution of equation (3.26) and introduce the corresponding curvature functions

Now recall the travelling front solutions to equation (2.5) constructed from solutions to equations (3.20) via equations (3.22) and (3.24). In the case of equations (3.27) with *ε* chosen in the appropriate range, we are able to solve equations (3.22) exactly, and thereby explicitly characterize the corresponding convex and concave profiles, here denoted by *Θ*, respectively. Now choosing the arc-length *S* for which *Θ*=0 at *S*=0, we may identify the arc-length parametrization of the local angle of

In conclusion, the scaled oriented profiles *ω* are determined through a free-energy minimization, the deviation from ±*ω* noted in (3.33) marks the profiles *thermokinetically* determined. Furthermore, an examination of the *ε*-expansion of these *thermokinetic* angles *thermokinetic* offset from Wulff angles, as exhibited in the *ε*-expansion (3.34), will now be shown to be universally true for the convex and concave travelling front solutions of the Herring equation (2.5) at small driving 0<*ε*≪1, provided only that the surface energy meets the natural faceting conditions expressed in (2.2).

### (b) General asymptotic analysis of Ω ε + and Ω ε −

Here we present an analysis of convex and concave travelling front solutions to equation (2.5) for a generic non-convex surface energy, subject only to the natural thermodynamic conditions appearing in equation (2.2). Given the *ε*-family of exact solutions of this type which we have explicitly constructed for the particular surface energy equation (3.25), it is natural to assume the existence of such an *ε*-parametrized family of solutions for the general case equation (2.2), at least for 0<*ε*≪1.

We assume there exists for each 0<*ε*≪1 a corresponding unbounded, symmetric, convex (concave) profile, *ε*-family of solutions to equation (3.37), namely *ε*≪1, as regular asymptotic expansion in *ε* of the convex and concave curvature profiles *ε*=0 in equation (3.37), namely

Viewing equation (3.37) as being of the form *w* and *f* are symmetric in *Θ*, and also noting that the only symmetric solutions of the associated homogeneous equation ( *f*=0) are scalar multiples of *variation of constants*.

Recalling the asymptotic flatness of the curvature expressed in equation (3.21), namely *Θ* to yield

Setting *ε*=0 first in (3.46), we find
*ε*=0 in equation (3.42), we immediately conclude *equilibrium profiles* previously identified in [37]. Now recalling equation (3.48), we conclude that the *ε*-expansion of *ε*=0 assumes the form
*η*^{±}=*η*>0, where the universal *Herring offset* *η* is given by

## 4. Emergent dynamics

Figure 4 presents a representative numerical solution of equation (2.5) starting from a thermodynamically unstable facet *ϑ*=0 subject to a small initial random perturbation, which models the *spinodal* decomposition of a crystal facet [1,23]. After an initial transient instability, which is naturally characterized through a linear instability analysis of equation (2.5) around the unstable interface *ϑ*=0, one observes the emergence of a hill valley-type profile. It comprises O(1)-length segments with an orientation approximated by the Wulff angles ±*ω* to leading order. These essentially linear O(1)-length segments (*psuedo-facets*) are seen to meet and merge through O(*ε*)-length rounded profiles which smoothly interpolate between the disparate angles ±*ω* of the joining segments: this is consistent with the behaviour of solutions to approximate theories that have been derived from a small-angle assumption [34–36]. We also note that the characteristic length of the interface grows (*coarsens*) in time as psuedo-facets are extinguished. Furthermore, our numerical data suggest that there is a unique coarsening mechanism, namely a concave neighbouring pair of psuedo-facets are extinguished as their lengths simultaneously shrink to zero. We figuratively describe this unique coarsening motif as a vanishing *hillock*. Last, we see that the leading-order kinematics of our computed solution *slowly evolving* Wulff-faceted interface

To comprehend the slow dynamics of *ε*-families of profiles *i*=0,1,…, that are *framed* on *ε*^{i+1}); *i*+1 being the *residual order* of the family *ε*-family of approximate solutions *ε*^{2}) necessarily imposes an intrinsic dynamics on

Let *slow time* *τ*=*εt*, and let *s* parametrization. We sequentially order its facets, between critical events, in the direction of increasing arc-length, adopting the convention that the *i*th facet will have angle (−1)^{i}*ω*, and denoting the arc-length of the *i*th facet's *left* and *right* edges at time *τ* by *s*_{i−1}(*τ*) and *s*_{i}(*τ*), respectively. It is notationally convenient to introduce the arc-length *s* parametrization of the indicator function for the *i*th facet, namely *θ*_{0} of *υ*_{i} denotes the normal velocity of its *i*th segment (facet) (figure 5).

Consider the *ε*-families of profiles *normally* deforming *ψ*(*ξ*,*τ*) is a given *normal deformation*. We look to choose *ψ*(*ξ*,*τ*) so that the advected front, *ε*^{2}) residual.

A direct calculation shows that the arc-length and parametric derivatives of *θ*_{1}:=∂_{ξ}*ψ*(*ξ*,*τ*). We thereby deduce the following *ε*-expansion of the curvature of *ε* provided on each interval

To construct from *outer scale*) pseudo-facets of the profile *ε*)-length-scale (*inner scale*) translating fronts (3.32). Note that the slowly evolving part of *s*−*s*_{i}|≫*ε*, *s*∼*s*_{i}, where the outer structures of *i* denotes − or + depending on whether *i* is even or odd. Comparing the inner and outer expansions, namely equations (4.10) and (4.9), on an intermediate scale, *ζ*=*ε*^{1/2}(*s*−*s*_{i}), while recalling that equation (3.53) provides the exact O(*ε*^{2})-expansion of *ε*) terms here yields *θ*_{1} only if the following compatibility condition for the facet velocities of *l*_{i}:=*s*_{i}−*s*_{i−1} (the length of the *i*th facet of *Peclet length*

In figure 7, we compare numerically computed periodic solutions of equation (2.5) with matched asymptotic profiles, *ε*) and O(*ε*^{2}), respectively. We note that *ground state* orientations *θ*=±*ω*.

## 5. Scaling laws

Numerical simulation of the facet dynamics prescribed by equation (4.12) suggests that the generic singularity (*critical event*) of any solution *critical* time *t*^{⋆}. That this behaviour mimics the vanishing-*hillock* coarsening motif observed in the numerical simulations of the original dynamics given in equation (2.5), as exhibited in figure 4, not only explains this motif, but provides us with the means to infer how to resolve the facet dynamics at such a critical time *t*^{⋆}. Namely, we generate the *daughter* faceted interface *parent* interface disappears, thereby joining the two adjacent, and now neighbouring, facets. Possessing two fewer facets than its parent, this daughter interface is thereby morphologically coarser. By continuing to evolve this daughter interface *coarsening events* as they arise, the average facet length 〈*l*(*t*)〉 of

We now turn to a study of the length statistics of solutions to our derived coarsening facet dynamics. Starting from Wulff-faceted interfaces *sub-Peclet* initial lengths, i.e. 0<*l*_{i}(0)≪*L*_{p}, our numerical studies show that, after a short transient, the average facet length 〈*l*(*t*)〉 displays the *universal* scaling law 〈*l*(*t*)〉∼*t*^{1/2}. This scaling persists until 〈*l*(*t*)〉 enters the *Peclet* regime, 〈*l*(*t*)〉∼*L*_{p}, wherein a crossover to logarithmically slow coarsening ensues thereafter. Furthermore, within this sub-Peclet scaling regime, the distribution of the lengths of the solution

Given our indexing convention is such that the *i*th facet of ^{i}*ω*, it follows that the purely kinematic relationship between the facet length *l*_{i} of *sub-Peclet regime**emergent symmetry* since it is not present in the original dynamics equation (2.5) with 0<*ε*≪1, but rather emerges as *ε*↘0 within its effective dynamics as expressed by equation (5.2).

Following an argument appearing in [34], we may conclude from equation (5.2) that the essentially sole critical event of solutions to equation (5.2) will involve concave pairs of lengths simultaneously shrinking to zero at a corresponding critical time *in situ* experimental test of theory.

Given the *universal statistical behaviour* of the distribution of facet lengths that we have already empirically observed, it is natural to presume universal behaviour on the expected (average) length *τ*=0 are uniformly small, *τ*=0, we have adopted a distinguished, albeit arbitrary, choice of *initial time* from which to start to observe the evolution of the distribution of lengths.

The parabolic scaling *parabolic-equivariance* relation that is equation (5.5), one thereby deduces the power law

Note that to develop a mean-field theory to predict the structure of the empirically determined scaling function exhibited in figure 8 would require an appropriate interpolation between the famous Lifshitz–Slyozov–Wagner (LSW) theory [38] for the small-phase fraction limit of Ostwald ripening, and the theory of von Smoluchowski [39] for the time evolution of the number density of aggregating particles: mean-field theories incorporating such interpolated transport and coagulation mechanisms for faceted-interface dynamics have already begun to be explored [40], and we hope to develop in future a comprehensive theory to predict the length distribution appearing in figure 8.

## 6. Conclusion

In conclusion, we have analytically uncovered a parabolic scaling symmetry that emerges from driven faceting crystal–melt interfaces governed by equation (2.5) through a novel matched-asymptotics analysis. Key to our analysis is the discovery of two non-equilibrium defects—convex and concave translating fronts, which display asymptotic orientations *ω*. We thereby predict that while the characteristic facet length *L*_{p} then the temporal power law *driven* phase-ordering systems with generic ground state manifolds [41–43].

- Received July 22, 2014.
- Accepted November 14, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.