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Crystallography on curved surfaces

Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved May 30, 2006 (received for review April 5, 2006)
Abstract
We study static and dynamical properties that distinguish 2D crystals constrained to lie on a curved substrate from their flatspace counterparts. A generic mechanism of dislocation unbinding in the presence of varying Gaussian curvature is presented in the context of a model surface amenable to full analytical treatment. We find that glide diffusion of isolated dislocations is suppressed by a binding potential of purely geometrical origin. Finally, the energetics and biased diffusion dynamics of point defects such as vacancies and interstitials are explained in terms of their geometric potential.
The physics of 2D crystals on curved substrates is emerging as an intriguing route to the engineering of selfassembled systems such as the “colloidosome,” a colloidal armor used for drug delivery (1), devices based on ordered arrays of block copolymers that are a promising tool for “soft lithography” (2, 3), and liquid–solid domains in vesicles (4, 5). Curved crystalline order also affects the mechanical properties of biological structures like clathrincoated pits (6, 7) or HIV viral capsids (8, 9) whose irregular shapes appear to induce a nonuniform distribution of disclinations in their shell (10).
In this article, we present a theoretical and numerical study of pointlike defects in a soft crystalline monolayer grown on a rigid substrate of varying Gaussian curvature. We suppose that the monolayer has a lattice constant of order, say, 10 nm or more. The substrate can then be assumed smooth, as would be the case for monolayers composed of diblock copolymers (2, 3). Disclinations and dislocations are important topological defects that induce longrange disruptions of orientational or translational order, respectively (ref. 11 and references therein and ref. 12). Disclinations are points of local 5 and 7fold symmetry in a triangular lattice (labeled by topological charges q = ±
These particlelike objects interact not only with each other, but also with the curvature of the substrate by a onebody geometric potential that depends on the particular type of defect (13, 14). These geometric potentials are in general nonlocal functions of the Gaussian curvature that we determine explicitly here for a model surface shaped as a “Gaussian bump.” An isolated bump of this kind models long wavelength undulations of a lithographic substrate, has regions of both positive and negative curvature, and yet is simple enough to allow straightforward analytic and numerical calculations. The presence of these geometric potentials triggers defectunbinding instabilities in the ground state of the curved space crystal, even if no topological constraints on the net number of defects exist. Geometric potentials also control the dynamics of isolated dislocations by suppressing motion in the glide direction. Similar mechanisms influence the equilibrium distribution and dynamics of vacancies, interstitials, and impurity atoms.
Basic Formalism
The inplane elastic free energy of a crystal embedded in a gently curved frozen substrate, given in the Monge form by r(x, y) = (x, y, h(x, y)), can be expressed in terms of the Lamé coefficients μ and λ (15):
where x = {x, y} represents a set of standard Cartesian coordinates in the plane and dA = dxdy
In Eq. 1 and the rest of this article, we adopt the ordinary flatspace metric (e.g., setting dA ≈ dxdy) and absorb all of the complications associated with the curved substrate into the tensor field A_{ij} (x) that resembles the more familiar vector potential of electromagnetism. Like its electromagnetic analog, the curl of the tensor field A_{ij} (x) has a clear physical meaning and is equal to the Gaussian curvature of the surface G(x) = −ε _{il} ε _{jk} ∂ _{l} ∂ _{k} A _{ij} (x), where ε _{ij} is the antisymmetric unit tensor (ε _{xy} = −ε _{yx} = 1) (14). The consistency of our perturbative formalism to leading order in α is demonstrated in Appendix 1, which is published as supporting information on the PNAS web site.
Minimization of the free energy in Eq. 1 with respect to the displacements u_{i} (x) naturally leads to the forcebalance equation, ∂ _{i} σ _{ij} (x) = 0. If we write σ _{ij} (x) in terms of the Airy stress function χ(x): then the force balance equation is automatically satisfied, because the commutator of partial derivatives is zero. However, we must be able to extract from χ(x) the correct u_{ij} (x) that incorporates the geometric constraint of Eq. 2 and accounts for the presence of any defects. These requirements are enforced by solving a biharmonic equation for χ(x) whose source is controlled by the varying Gaussian curvature of the surface G(x) and the defect density S(x) (11, 14):
In the special case S(x) = 0, we denote the solution of Eq. 5
as χ
^{G}
(x) where the superscript G indicates that the Airy function and the corresponding stress tensor σ _{ij}
^{G}(x) describe elastic deformations caused by the Gaussian curvature G(x) only, without contribution from the defects. The stress of geometric frustration, σ_{ij}
^{G}(x), is a nonlocal function of the curvature of the substrate and plays a central role in our treatment of curvedspace crystallography. The Young modulus Y =
The source S(x) for a distribution of N unbound disclinations with “topological charges” {q_{α}
= ±
To determine the geometric potential of a defect (on a deformed plane flat at infinity), we integrate by parts twice in Eq. 6 and use Eq. 5 to obtain Δ^{2}χ(x) and χ(x) in terms of the Green’s function of the biharmonic operator:
It is convenient to introduce an auxiliary function V(x) that satisfies the Poisson equation ΔV(x) = G(x) and vanishes at infinity where the surface flattens out. The geometric potential, ζ(x ^{α} ), of a defect at x ^{α} follows from integrating by parts the cross terms (involving the source and the Gaussian curvature) in Eq. 9 with the result:
This formula ignores defect selfenergies ^{§} and needs to be supplemented by boundary corrections, as discussed in Appendix 2, which is published as supporting information on the PNAS web site.
In the following sections, we explicitly show that our results derived from Eq. 10 can also be obtained by means of more heuristic arguments. According to this point of view, defects introduced in the curvedspace crystal are local probes of the preexisting stress of geometric frustration σ_{ij} ^{G}(x) to which they are coupled by intuitive physical mechanisms such as Peach–Koehler forces.
Geometric Frustration
We start by calculating the energy of a relaxed defectfree 2D crystal on a quenched topography. In analogy with the bending of thin plates we expect some stretching to arise as an unavoidable consequence of the geometric constraints associated with the Gaussian curvature (15). The resulting energy of geometric frustration, F _{0}, can be estimated with the aid of Eq. 5 , which, when S(x) = 0, reduces to a Poisson equation whose source is given by V(x): where H_{R} (x) is an harmonic function of x parameterized by the radius of the circular boundary R. H_{R} (x) vanishes in the limit R ≫ x _{0} if free boundary conditions are chosen. Using the general definition of the Airy function (Eq. 3 ), we obtain:
For a surface with azimuthal symmetry, like the bump, the only nonvanishing components of the stress tensor of geometric frustration σ_{ij} ^{G}(x) read: and Poisson’s equation can be readily solved upon applying Gauss’ theorem with the Gaussian curvature G(r) ≈ α^{2} e ^{−r} ^{2} (1 − r ^{2})/x_{0} ^{2} as a source (16):
The extra factors of x _{0} in the previous equations arise from expressing our results in terms of the dimensionless radial distance r.
Substituting Eq. 11 in Eq. 6 , we have:
The result in Eq. 16
is valid to leading order in α, consistent with the assumptions of our formalism after taking the limit R ≫ x
_{0}. For a harmonic lattice, Y = 2/
Geometric Potential for Dislocations
The energy of a 2D curved crystal with defects will include the frustration energy, the interdefect interactions (to leading order these are unchanged from their flatspace form), possible core energies, and a characteristic, onebody potential of purely geometrical origin that describes the coupling of the defects to the curvature given by Eq. 10 . The geometric potential of an isolated dislocation, ζ(x) ≡ D(x, θ), is a function of its position and of the angle θ that the Burger vector b forms with respect to the radial direction (in the tangent plane of the surface; see Fig. 1). Upon setting all q_{α} = 0 in Eq. 7 and substituting into Eq. 10 , we obtain, for an isolated dislocation, the resulting function D(r = x/x _{0}, θ):
In view of the azimuthal symmetry of the surface, Gauss’ theorem as expressed in Eq. 15 , was used in deriving the second equality in Eq. 17 , which is a function only of the dimensionless radial coordinate r. The first term in Eq. 17 corresponds to the R → ∞ geometric potential obtained from Eq. 10 , while the second term is a finite size correction arising from a circular boundary of radius R (see Appendix 2 for a detailed derivation). Eq. 17 is valid to leading order in perturbation theory, consistent with the small α approximation adopted in this work.
In Fig. 1 we present a detailed comparison between the theoretical predictions for the geometric potential D(r, θ)/Ybx
_{0} plotted versus r = x/x
_{0} as continuous lines and numerical data from constrained minimization of an harmonic solid on a bump with α = 0.5, under conditions such that R ≫ x
_{0} ≫ a. (See Appendix 3 and Figs. 7 and 8, which are published as supporting information on the PNAS web site, for a discussion of our numerical approach.) The lower and upper branches of the graph are obtained from Eq. 17
by setting θ = ±
The analogy between the geometrical potential of the dislocation and the more familiar interaction of an electric dipole in an external field can be elucidated by regarding the dislocation as a charge neutral pair of disclinations whose dipole moment qd_{i}
= ε
_{ij}b_{j}
is a lattice vector perpendicular to b that connects the two points of 5 and 7fold symmetry. The geometric potential, U(r), of a disclination of topological charge q interacting with the Gaussian curvature satisfies the Poisson equation ΔU(r) = −qV(r) as can be seen by substituting the source in Eq. 7
with all b
^{β} = 0 into Eq. 10
. For small r, positive (negative) disclinations are attracted (repelled) from the center of the bump by the integrated background source V(r), which increases for r ≤ 1 like α^{2}
r
^{2} and is multiplied by the 2D electric field
The physical origin of the dislocation potential can be understood heuristically by considering dislocation climb, or the motion of the dislocation in the radial direction (Fig. 2). According to standard elasticity theory, a dislocation in an external stress field σ _{ij} (x) experiences a Peach–Koehler force, f(x), given by f_{k} (x) = ε _{kj}b_{i} σ _{ij} (x) (19, 20). Similarly, a dislocation introduced into the curved 2D crystal will experience a Peach–Koehler force as a result of the preexisting stress field of geometric frustration σ_{ij} ^{G}(x ^{α}) whose nondiagonal components vanish. This interpretation is consistent with the geometric potential derived in Eq. 17 , provided we use Eq. 11 to write D(x) = b_{i} ε _{ij} ∂ _{j} χ ^{G} . With b along its minimum orientation (azimuthal counterclockwise), we obtain a radial Peach–Koehler force of magnitude f(r) = bσ _{φφ} ^{G}(r) that matches: (see Eq. 13 ).
Dislocation Unbinding
If the 2D crystal is grown on a substrate that is sufficiently deformed, the resulting elastic strain can be partially relaxed by introducing unbound dislocations into the ground state (14, 16). Here, we present a simple estimate of the threshold aspect ratio, α _{c} , necessary to trigger this instability. Boundary effects will be ignored in what follows by letting R → ∞ in Eq. 17 .
Consider two dislocations located at x _{1} and x _{2} a distance of ≈x _{0} from the center of the bump on opposite sides (see Fig. 3 a Inset). Their disclination–dipole moments are opposite to each other and aligned in the radial direction so that the two (antiparallel) Burger’s vectors are perpendicular to the separation vector in the plane x _{12} ≡ x _{1} − x _{2} . In this case, the interaction between the dislocations reduces to V_{12} ≈ (Y/4π)b ^{2} ln(x _{12} /a) (11, 12). The instability occurs when the energy gain from placing each dislocation in the minima of the potential D(x) given by Eq. 17 outweighs the sum of the work needed to tear them apart plus the core energies 2 E_{c} . The critical aspect ratio at the threshold that results is given by: where b′ = (b/2) e ^{−8πEc/Yb2} and c ≈ 1/2. Note that the core energy E_{c} is determined by the microscopic physics of the particular system under study.
In Fig. 3 a we present a comparison between Eq. 18 and numerical results. For each value of x _{0}/b, the corresponding α _{c} is obtained numerically by comparing the energy of a lattice without defects to the configuration with the two dislocations in their equilibrium positions. This interpretation for the origin of the instability is corroborated by the (numerical) strain energy density plots of Fig. 3 b and c, where it is shown that introducing the pair of dislocations reduces the strain energy density on the top of the bump at the price of creating some large, but localized, strains around the dislocation cores where u_{ij} diverges. In the continuum limit b ≪ x _{0}, very small deformations are enough to trigger the instability. This is the regime in which our perturbation treatment applies. As α is increased even further, a cascade of dislocationunbinding transitions occurs involving larger numbers of dislocations and more complicated equilibrium arrangements of zero net Burger vector. For sufficiently large aspect ratios, we expect that the dislocations tend to line up in grain boundary scars similar to the ones observed in spherical crystals (1, 21). This scenario is consistent with preliminary results from Monte Carlo simulations in which the fixedconnectivity constraint is lifted and more complicated surface morphologies are considered (A. Hexamer, personal communication).
Glide Suppression
The dynamics of dislocations proceeds by means of two distinct processes: glide and climb. Glide describes motion along the direction defined by the Burger’s vector; in flat space glide requires a very low activation energy and is the dominant form of motion at low temperature (see Fig. 4 a Inset). Climb, or motion perpendicular to the Burger vector, requires diffusion of vacancies and interstitials (Fig. 2) and is usually frozen out relative to glide that involves only local rearrangements of atoms. On a curved surface, the geometric potential D(r,θ) imposes constraints on the glide dynamics of isolated dislocations, in sharp contrast to flat space where only small energy barriers are present because of the periodic Peierls potential (20).
As the dislocation represented in Fig. 4
a Inset moves in the glide direction, it experiences a restoring potential generated by the variation of the (scaled) radial distance and the deviation from the radial alignment of the dislocation dipole. For a small transverse displacement, y, the harmonic potential U(y) =
The harmonic potential
The resulting thermal motion in the glide direction of dislocations in this binding harmonic potential can be modeled by an overdamped Langevin equation for the glide coordinate y, leading to 〈Δy ^{2}〉 = (1 − e^{−2μKdt} )/βk_{d} , where μ is the dislocation mobility (18, 22). In the case of a bump geometry, the effective spring constant k_{d} can be evaluated with Eq. 19 . We emphasize, however, that the glide suppression mechanism considered here is not caused by the interaction of the dislocation with other defects but purely by the geometric interaction with the curvature of the substrate.
Vacancies, Interstitials, and Impurities
We now turn to a derivation of the geometric potential, I(x), for interstitials, isotropic vacancies, and impurities. Inspection of Fig. 5 reveals that an interstitial (vacancy) can be viewed either as the product of locally adding (removing) an atom to the lattice or as a composite object made up of three disclination dipoles. To derive I(x), we substitute the source term of Eq. 8 into Eq. 10 and integrate by parts twice. The result reads: where boundary terms and a positionindependent nucleation energy have been dropped. The constant Ω represents the area excess or deficit associated with the defect. In Fig. 5 a comparison of Eq. 20 with the results obtained from mapping out I(r) numerically is presented. The area changes Ω _{i} and Ω _{v} for interstitials and vacancies, respectively, were fit to the numerical data. We find that Ω _{v} is negative and greater in magnitude than Ω _{i} . The large r behavior of I(r) indicates that the core energy of a vacancy in flat space is greater than the one of an interstitial for the harmonic lattice. Interstitials tend to seek the top of the bump, whereas vacancies are pushed into the flat space regions. From the definition of V(r) in Eq. 15 , we deduce that an interstitial (vacancy) is attracted (repelled) by regions of positive (negative) Gaussian curvature similar to the behavior of an electrostatic charge interacting with a background charge distribution given by −G(r). The function V(r) controls the curvaturedefect interaction for other types of defects such as disclinations in liquid crystals and vortices in ^{4}He films (16, 23). The expression for V(r) in Eq. 15 reveals that I(r) is indeed a nonlocal function of the Gaussian curvature determined as in electrostatics by the application of Gauss’ law. Thus, the vacancy potential on a bump does not reach a minimum at the point where the Gaussian curvature is maximally negative but rather at infinity where the integrated Gaussian curvature vanishes.
We now argue heuristically how a localized point defect can couple nonlocally to the curvature and in the process we provide an alternative justification of Eq. 20 analogous to the informal derivation of the dislocation potential. The energy cost of a defect at x ^{α} , I(x ^{α} ), caused by local compression or stretching in the presence of an arbitrary elastic stress tensor σ _{ij} (x) is given by I(x ^{α} ) = p(x ^{α} )δV, where δV is the local volume change and p(x ^{α} ) is the local pressure related to the stress tensor by σ _{ij} (x ^{α} ) = −p(x ^{α} )δ _{ij} for the case of an isotropic stress (15). In two dimensions, we have I(x ^{α} ) = −σ _{kk} (x ^{α} )Ω ^{α} /2. We recover the result in Eq. 20 , by assuming that the local deformation Ω^{α} (induced by the nucleation of a point defect) couples to the preexisting stress of geometric frustration σ_{kk} ^{G} = −YV(x) (see Eq. 12 ), which is a nonlocal function of the Gaussian curvature. Elastic deformations created by the geometric constraint throughout the curved 2D solid are propagated to the position of the point defect by force chains spanning the entire system. The point defect can then be viewed as a local probe of the stress field that does not measure the additional stresses induced by its own presence.
Note that the geometrical potential of an isotropic point defect is unchanged if we swap the 5 and 7fold disclinations comprising it (corresponding to a rotation of the point defect by
An arbitrary configuration of weakly interacting point defects will relax to its equilibrium distribution by diffusive motion in a force field f(r) = −∇I(r). In overdamped situations, this geometric force leads to a biased diffusion dynamics with drift velocity where β = 1/k_{B}T and D is the defect diffusivity (20). Eventually, a dilute gas of point defects equilibrates to a nonunform spatial density proportional to e ^{−βI} ^{(r} ^{)}.
Acknowledgments
We thank G. Chan, B. I. Halperin, A. Hexemer, Y. Kafri, R. D. Kamien, E. Kramer, A. Travesset, and A. M. Turner for stimulating conversations and the CrimsonGrid Initiative (Harvard University) for access to its computational resources. This work was supported by National Science Foundation Grant DMR0231631 and Harvard Materials Research Laboratory Grant DMR0213805. J.B.L. was supported by the Hertz Foundation. V.V. was supported by National Science Foundation Grants DMR0129804 and DMR0547230.
Footnotes
 ^{‡}To whom correspondence should be addressed. Email: vitelli{at}sas.upenn.edu

Author contributions: V.V., J.B.L., and D.R.N. designed research, performed research, analyzed data, and wrote the paper.

Conflict of interest statement: No conflicts declared.

This paper was submitted directly (Track II) to the PNAS office.

↵ ^{§}For dislocations, vacancies, and interstitials, the positiondependent selfenergies can be ignored compared with ζ(x ^{α}), because they are proportional to higher powers of the lattice spacing a.

Freely available online through the PNAS open access option.
 © 2006 by The National Academy of Sciences of the USA
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