Ground States of Crystalline Caps: Generalized Jellium on Curved Space

TL;DR

This paper investigates the structure and elastic energy of crystalline caps on spherical surfaces, revealing continuous symmetry-breaking transitions and the emergence of icosahedral symmetries influenced by curvature and geometry.

Contribution

It introduces a comparative analysis of continuum and discrete models for defective caps and uncovers novel ground state behaviors driven by curved geometry.

Findings

Transition from defect-free to disclination states is continuous.

Icosahedral symmetries appear across various curvatures.

Curved geometry leads to unique ground state behaviors not seen in flat models.

Abstract

We study the structure and elastic energy of the ground states of crystalline caps conforming to a spherical surface. These ground states consist of positive disclination defects in structures spanning from flat and weakly curved crystals to closed shells. We compare two different continuum theories and one discrete-lattice approach for the elastic energy of defective caps. We first investigate the transition between defect-free caps to single-disclination ground states and show that it is characterized by continuous symmetry-breaking transition. Further, we show that ground states with icosahedral subgroup symmetries in caps arise across a range of curvatures, even far from the closure point of complete shells. Thus, while superficially similar to other models of 2D `jellium' (e.g. superconducting disks and 2D Wigner crystals), the interplay between the free edge of crystalline caps and the non-Euclidean geometry of its embedding leads to non trivial ground state behaviors, without counterpart in planar jellium models.

Equations

$$F=\frac{1}{2Y}\int d^2\mathbf{x}\sqrt{g}{\left(\Delta \chi \right)}^2,$$

$$\frac{1}{Y}{\Delta }^2\chi (\mathbf{x})$$

$$s(\mathbf{x})=\frac{\pi }{3\sqrt{g}}\sum _{\alpha =1}^{n_d}{q}_{\alpha }\delta (\mathbf{x}-{\mathbf{x}}_{\alpha })$$

$$E_{\mathrm{NS}}=\frac{k}{2}\sum _{⟨i,j⟩}{\left(|{\mathbf{r}}_i-{\mathbf{r}}_j|-a\right)}^2,$$

$$\frac{E_{\rm FvK}}{YA}=\frac{\theta_{c}^{4}}{384}+\frac{1}{96}\Big{(}\frac{q^{2}}{3}-\frac{q\theta_{c}^{2}}{2}\Big{)}\Big{[}1-(r/W)^{2}\Big{]}^{2},$$

$$\int dAs(\mathbf{x})=\int dAK(\mathbf{x})\to \frac{\pi }{3}=2\pi (1-\cos {\theta }_n),$$

Full text

Spherical crystals are elementary models of geometric frustration in materials, with important realizations in a range of systems from fullerenes and protein shells to particle- and molecular-coated droplets Caspar and Klug (1962); Kroto et al. (1985); Zandi et al. (2004); Panahandeh et al. (2018); Bausch et al. (2003); Irvine et al. (2010); Guttman et al. (2016). The long-standing problem of finding the ground state of $N$ particles covering the sphere, known as the generalized Thompson problem Smale (1998); Pérez-Garrido et al. (1997), derives its complexity from the basic conflict between equi-triangular order and positive Gaussian curvature, that is, elliptic geometries Nelson (2002); Kléman (1989); Bowick et al. (2000); Vitelli et al. (2006). For closed shells, topology dictates the total charge of disclinations (i.e. sites deviating from six-fold coordination) to $\sum_{i=1}^{n_{d}}q_{i}=12$, which for the simplest case of only 5-fold defects ($q_{i}=+1$) constrains the number of disclinations to $n_{d}=12$ Wagner and Zandi (2015); Bowick et al. (2000); Bowick and Giomi (2009). Considerable progress has been made by numerically optimizing, classifying and rationalizing the patterns of defects in an otherwise six-fold, quasi-triangular order of closed shells Altschuler et al. (1994); Pérez-Garrido et al. (1997); Bowick et al. (2002); Wales et al. (2009). In contrast, the structure and energetics of defect ground states of partially-closed crystalline shells, or crystalline caps, which span the gap between defect-free planar crystals and closed shells, remain unresolved Azadi and Grason (2014, 2016); Castelnovo (2017); Li et al. (2018).

Unlike closed shells, the number of defects in the interior of crystalline caps is not topologically constrained. Disclinations can be created and destroyed in the free boundary of the cap, adjusting their number in accordance with energetic considerations deriving from favorable elastic screening of curvature-induced stresses Schneider and Gompper (2005); Giomi and Bowick (2007); Irvine et al. (2010); Grason (2010, 2012). In this vein, ground states of crystalline caps may be described by a generalized jellium (GJ) model, in which both Gaussian curvature and disclinations act as point-like sources of “elastic stress” mediating their energetics Nelson (2002). While the most familiar examples of GJ describe bulk phases like the Wigner crystal Bonsall and Maradudin (1977); Fisher et al. (1979) or Abrikosov lattice of type-II superconductors Geim et al. (1997); Baelus et al. (2001); Schweigert et al. (1998) , many physical scenarios are described by finite domains of homogeneous “charged” backgrounds punctuated by a small number of neutralizing “point charges”. For example, under an increasing magnetic field, the ground state wave function of 2D superconducting discs exhibit a complex sequence of continuous and discontinuous transitions in the number and symmetry of vortices Baelus et al. (2001); Schweigert et al. (1998), which derives from the generic incommensurability of the net applied flux with the quantized flux per elementary vortex.

Although superficially similar to planar GJ models, crystalline caps are distinguished by their non-Euclidean (elliptic) geometry which alters the shape and length of the boundary free cap. Because free boundaries have the ability to screen defect-induced stress, the non-planar geometry of caps has a critical impact on the ground state order of its defects. In this paper, we describe the spectrum of defect ground states of crystalline caps using a combination of continuum elasticity theory and simulation models. We show that geometrically nonlinear effects at the free edges give rise to a novel continuous transition from the defect-free interiors to a trapped central disclination as well as the existence of soft, near-edge trapping of low-energy defects. This latter mechanism leads to a non-trivial sequence of defect ground states, which is characterized by a non-monotonic dependence of number of defects in the cap interior on the integrated Gaussian curvature, or sphere area coverage. Finally, we show that the non-Euclidean geometry of the cap edge is essential for the emergence of defect ground states with symmetries commensurate with the icosahedral symmetry of closed shells (e.g. the Thompson problem Bowick et al. (2002)).

To obtain the elastic energy ground states of a cap, we consider a 2D hexagonal crystal on a sphere of radius $R$ extending up to a radial distance $W=\theta_{c}R$ from its center to its free edge, see Fig. 1 and S6. While brittle crystallization on spherical surfaces is known to lead to elastic instabilities of the free boundary shape Schneider and Gompper (2005); Morozov and Bruinsma (2010); Meng et al. (2014); Hall and Grason (2017), here we consider the limit of large edge energy 111We expect this boundary shape instability is suppressed in the limit of edge energies much larger than $\sim Y\theta_{c}^{4}W$ Morozov and Bruinsma (2010); Hall and Grason (2017); Zandi et al. (2006)., where the boundary of the crystal remains axisymmetric Zandi et al. (2006); Li et al. (2018); Panahandeh et al. (2018); Wagner and Zandi (2015). Furthermore, we restrict our attention to ground states that possess only $q_{i}=+1$ (five-fold) disclination defects in an otherwise hexagonal bulk order, neglecting the possibilities of dislocations chains or “scars” that become energetically favorable when the lattice spacing $a$ (and dislocation core energy) becomes sufficiently small compared to $W$ Bowick et al. (2000); Azadi and Grason (2014).

The elastic energy takes the form

[TABLE]

where $\sqrt{g}$ is the metric induced in the spherical cap and $\chi$ is the Airy stress function, which encodes the elastic intra-crystal stress (see SI Eqs. (S9-S11)), and satisfies Bowick et al. (2000)

[TABLE]

where $K({\bf x})$ is the Gaussian curvature, $\Delta=\frac{1}{\sqrt{g}}\partial_{i}(\sqrt{g}g^{ij}\partial_{j})$ is the Laplacian on a 2D surface with metric $g_{ij}$ and

[TABLE]

is the disclination density, composed of $n_{d}$ disclinations possessing $q_{\alpha}=+1$, the topological charge per single 5-fold defect (i.e. with wedge angle $\pi q_{\alpha}/3$ per defect). The crucial effect of the free boundary motion is captured by imposing vanishing normal stress at the cap edge on solutions for $\chi$. While strictly speaking topology requires a fixed balance between disclinations in the bulk and on the boundary of the cap Bowick and Giomi (2009), stress screening by free boundary implies that the elastic effect of defects as they approach the boundary, hence, energetics are sensitive only to interior defects.

The elastic energy of multi-disclination configurations in caps were previously computed 222The derivations for FvK in axisymmetric 2D crystals were originally applied to twisted columnar phases, the elastic energy of which is identical to crystalline caps in the small-slope, or low-twist, regimes. for the so-called Föppl ván Kárman (FvK) limit Grason (2010, 2012), strictly justified in the limit of small-slopes Landau and Lifshitz (1986). The FvK limit corresponds to approximating the metric $g_{ij}$ to be planar in Laplacian while retaining $K({\bf x})=1/R^{2}$ as a homogeneous source for Airy stress on the right hand side of Eq. (2). While the FvK theory is tractable for axisymmetric caps with arbitrary defect arrangements Grason (2016); Castelnovo (2017), the small-slope limit is not satisfied for most of curvatures where defects are energetically favored. Recently, a rigorous framework for the fully covariant elasticity theory caps has been developed Li et al. (2018, 2019), which builds from elements in the theory of incompatible elasticity Efrati et al. (2009); Moshe et al. (2015), and more crucially, allows for the computation of the elastic energy of multi-defect configurations with arbitrary complexity (See SI). This covariant approach, which hereon we will refer as LF (Lagrange Formalism) Li et al. (2019), captures the full geometric non-linearity of the cap shape through incorporation of the spherical metric in the deformed state (i.e. in so-called azimuthal-equidistant coordinates $g_{rr}=1,g_{r\phi}=0,g_{\phi\phi}=R^{2}\sin^{2}(r/R)$) while evaluating the Laplacian and area integral in Eq. (1).

In the context of the GJ models, the FvK model may be considered as the biharmonic analogy to the 2D “electrostatic” theory of superconducting disks, that is, generalized by the much longer range interactions between monopoles in the biharmonic theory (i.e. monopole interactions with separation $r$ grow as $\sim r^{2}\ln r$, in comparison to $\ln r$ for 2D Coulomb Romanov (1981)). In comparison, the fully covariant theory embeds the “biharmonic electrostatics” problem in a non-Euclidean spherical geometry. The effect of the reduced geodesic separation between disclinations embedded on spheres, in combination with relative reduction of the perimeter to domain size ratio for caps compared to disks with the same radial distance– both captured in the covariant theory – qualitatively alters the structure and energetics of disclinations in caps, as we will show below.

The accuracy of the different approximations to the continuum elastic theory can be tested in comparison to simulations of the lattice model introduced by Nelson and Seung (NS) Seung and Nelson (1988)

[TABLE]

which consists of a triangular network of nearest neighbor springs of rest length $a$ and spring constant $k$. As described in the SI, configurations possessing up to $n_{d}=0,1,2$ or $3$ five-fold disclinations are constructed by introducing multiple $60^{\circ}$ Volterra cuts, generalizing previous approaches to disclinated meshes Travesset (2003); Grason (2010). Figure 1 illustrates that there is an excellent agreement between the stress distribution computed from the LF and simulation of a cap with a single off-center disclination, see SI for the details.

We first describe the elementary transition from the stable defect-free ground state to the ground state possessing the first stable internal disclination, with ($q=+1$). For the FvK theory the elastic energy (per unit area) of the single-disclination cap is known, and is a function of aperture angle $\theta_{c}=W/R$, off-center defect position $r$ and disclination charge $q$ Grason (2012)

[TABLE]

where the first term derives from curvature-generated stress, the second, from the elastic self-energy of the disclination and the third, from the mutual elastic screening of the curvature and disclinations stress 333Note that we neglect in this analysis the core energy of disclinations, which is expected on dimensional grounds to be comparable to $~{}Ya^{2}$ and hence much smaller than the far-field elastic energies of disclinations, $\sim YW^{2}$. The analytic formula for the elastic energy for the covariant energy, $E_{\rm LF}$, is more complex as shown the in the SI and described in ref. Li et al. (2019), but it reduces to eq. (5) in the limit of small curvature, $\theta_{c}\ll 1$.

Because the self-energy and defect-curvature interaction contributions exhibit exactly the same dependence on defect position $r^{*}$ in the FvK theory, this theory predicts a simple 1st-order transition from the defect-expelled state (minimum at $r^{*}=W$) for $\theta_{c}<\theta^{*}$ to the defect-centered state (minimum at $r^{*}=0$) for $\theta_{c}>\theta^{*}$ at a critical cap angle $\theta^{*}=\sqrt{2/3}\approx 0.816$, see Fig. 2. Note that a standard heuristic argument Irvine et al. (2010) that considers the cap angle, $\theta_{n}$, at which integrated Gaussian curvature “neutralizes” single-five fold defects yields

[TABLE]

or $\theta_{n}=\arccos(5/6)\simeq 0.59$. This value falls well below the above prediction for $\theta^{*}$, revealing that the free cap boundary requires significant “overcharging” of curvature-induced stress to overcome the elastic self-energy of the monopole disclination charge.

Figure 2 illustrates the elastic energy vs. defect position predicted by the covariant theory, where it is found that the first disclination is expelled continuously from the boundary, starting approximately at $\theta\simeq 0.795$ reaching the center of the disk ($r^{*}=0$) at around $\theta\simeq 0.83$, exhibiting a range of off-center defect equilibria $0<r^{*}<W$ in this narrow curvature window444We point out that in Ref. Li et al. (2018), the appearance of the first defect was found to be at $\theta\simeq 0.66$ instead of $0.795$ due to numerical error arising from summing over not enough modes in the multipole expansion of self-energy. A discussion is provided in SI..

The distinct first- vs. second-order nature of transitions respectively predicted by FvK and covariant theory highlights the impact of the non-Euclidean geometry of the cap. While the FvK theory predicts the self-energy and defect-curvature interactions to exactly balance at the critical curvature, the appearance of stable off-center equilibrium for disclinations in the covariant theory can be associated with the imbalance between these two competing effects. Figure S2 plots the relative magnitudes of these terms for the covariant theory and FvK, showing that the defect self-energy is relatively depressed, while the defect-curvature is enhanced, as disclinations approach the free edge of the cap in the covariant theory. The enhanced curvature-generated attraction at the edge can be associated from the geometrically nonlinear edge compression, which grows faster than quadratic approximation of the small-slope FvK theory. On the other hand, the near-edge weakening of the self-energy that tends to expel defects from the cap can be associated with the smaller geodesic curvature of the cap edge compared to a planar disk, which amplifies the far-field screening of defect stresses by free boundary relaxation (see SI Sec. I.C for discussion). These two effects of the non-Euclidean geometry of caps conspire to create “soft traps”, stabilizing off-center equilibrium.

We note that the simulation model (Fig. S4c) shows a transition from defect-free to centered-disclination ground state at a cap angle quite close to both continuum calculations, $\theta^{*}\simeq 0.84$. However, discreteness effects associated with the finite core size (zone of anharmonic strain) and non-circularity of the free edge obscure resolution on the near-edge elastic binding of single defects.

We now consider the evolution of multi-defect ground state structure with cap growth from nearly-flat ($\theta_{c}\lesssim 1$) to fully closed shells ($\theta_{c}\to\pi$). Numerous previous considerations of the Thompson problem and its variants Bowick et al. (2002); Bowick and Giomi (2009) point towards icosahedral defect configurations as ground states in closed spheres. Yet, it remains unclear whether, and at what point, ground state structures of incomplete caps conform to this symmetry. Thus, we compare the elastic energy of two basic classes of ground state symmetries: the first is point pattern of disclinations possessing subgroup symmetries of icosahedron (IO), as illustrated in a stereographic projection in Fig. S8; and second are point patterns composed of concentric $n$-fold symmetric rings of defects with composite symmetries listed in Table S1. For each configuration, we compute the minimum energy by varying the arc-radial distance of concentric rings defects, keeping the azimuthal separation of defects fixed in each ring. For the second class of configurations, we also minimize with respect of the rotation angle between concentric rings. For example, a (2,4) configuration is composed of 2 defects in the first ring and 4 defects in the second, in both cases evenly distributed. Such a configuration is then minimized with respect to the arc radius of two rings $r_{1}$, $r_{2}$ and the phase rotation angle $\phi$ between the two rings. For the IO class, while we minimize over arc-radii of rings, we retain the fixed angular correlations between consecutive rings deriving from icosohedral symmetry (e.g. defects on consecutive rings in the 3-fold IO state are separated by 60∘).

We plot the results in terms of the number of bulk disclinations and energy density of ground state configurations in Fig. 3 A and B respectively, for both the FvK (black) and LF (color) elastic theories. Since their elastic energy contributions become arbitrarily small as defects approach the cap boundary, we introduce a cut off radius of $0.95W$, beyond which we count defects to be at the boundary of the crystal and not in the bulk.

The ground state energies of the LF increases monotonically, with few exceptions (see $(2,2)$ and $(3,3)$ following $(2)$ and $(3)$, respectively, due to the weakly, near-edge defect traps) and the number of disclinations always remain below the condition of topological charge neutrality (i.e. a number of internal defects whose total defect angle is equal to the integrated Gaussian curvature), with the calculation converging to the topological correct condition Eq. 6 of twelve defects with icosahedron symmetry. In contrast, for $\theta_{c}\gtrsim 2$ the ground states of the FvK model begin to greatly exceed the topological condition Eq. 6, eventually growing to +19 disclinations in the complete shell, far in excess of the +12 required on a closed sphere. In terms of the energy density (Fig. 3B), both theories show a similar crossover from the $\sim\theta_{c}^{4}$ growth of elastic energy for defect free caps to the plateau-like series of multi-disclination minima at large coverage. Notwithstanding these qualitative similarities in the curvature-dependence of the energy, the ground states symmetries of the two models differ substantially. As illustrated in Fig. 3C, all but two ground states of the LF possess quasi-icosahedral symmetry. In contrast, as shown in SI Fig. S7, ground states of the FvK theory with $n_{d}>3$ break icosahedral symmetry, with the sole exception of a narrow range of stable $(3,3)$, exhibiting higher-fold concentric-ring defect patterns that are also characteristic of planar vortex packings in superconducting ground states Baelus et al. (2001).

In summary, the detailed comparison of the widely-used elastic plate theory (FvK), a covariant continuum elasticity theory (LF) and fully nonlinear, discrete lattice model (NS) of crystalline caps reveal that qualitative features of the ground state structure and energetics derive from the non-Euclidean embedding of caps, and crucially, their free boundaries. Beyond its effects to modify the transition from defect-free and defective caps for relatively small curvature, we find spherical geometry is essential for the emergence of icosahedral defect configurations, even at cap curvatures surprisingly far from closure. Whereas the FvK model (i.e. “biharmonic, planar jellium”) predicts defect numbers and arrangements wildly departing from structures predicted in closed shells (i.e. $n_{d}(\theta_{c}\to\pi)\gg 12$), the covarient LF model (i.e. ”biharmonic, spherical jellium”) yields ground states that smoothly interpolate to the grounds of twelve $q=+1$ defects for $\theta_{c}\to\pi$. This result demonstrates that the topological constraint that emerges for closed shells (i.e. $\sum_{i=1}^{n_{d}}q_{i}=12$) is not strictly necessary for the emergence of icosahedral defect arrangements, and such ground states can emerge purely from the (topologically-unconstrained) energetics of multi-defect arrangements on incomplete shells, but only when embedded properly in a spherical geometry.

The authors are grateful to the hospitality of Aspen Center for Physics (NSF PHY 1607611) where this work began. The work of SL and RZ was supported by the NSF through Awards DMR-1719550. AT acknowledges support from DMR-1606336 and GG from the DOE Office of Science, Basic Energy Sciences, under Award No. DE-SC0017870.