Read--Shockley formula for a general Bravais lattice in two dimensions

1. Problem Statement

The paper addresses the mathematical modeling of small-angle grain boundaries in two-dimensional crystals. Specifically, it seeks to derive the energy scaling of a boundary separating two crystal grains with a small misorientation angle $\theta$.

• Context: The authors utilize a nonlinear semi-discrete energy functional introduced by Lauteri and Luckhaus. This model treats the crystal lattice as a continuum except for the immediate vicinity of dislocations (defects), where the energy is concentrated.
• Goal: To construct an explicit, physically motivated deformation field for a general Bravais lattice (not just square lattices) that satisfies specific boundary conditions and proves that the total energy scales according to the Read–Shockley formula:
  $$E \sim \varepsilon \theta (A - \log \theta)$$
  where $\varepsilon$ is the microscopic scale (lattice spacing), $\theta$ is the misorientation angle, and $A$ is a constant depending on the lattice geometry and core radius.
• Challenge: Previous works (e.g., [9], [4]) established this scaling for square lattices or required complex constructions. The authors aim to simplify the construction and generalize it to any Bravais lattice while satisfying a stricter quantization condition on the strain circulation.

2. Methodology and Construction

The authors propose a novel, piecewise affine construction for the strain field $\beta$ and the set of dislocations $S$.

A. Domain Splitting

The domain $\Omega$ is divided into four vertical strips:

1. Outer Strips ($\Sigma_{-\theta}, \Sigma_{\theta}$): These regions match the boundary conditions, where the strain is a rigid rotation $R_{-\theta}$ and $R_{\theta}$ respectively.
2. Inner Strips ($\Sigma_1, \Sigma_2$): These strips contain the grain boundary. They are further subdivided into a grid of squares ($Q_{i,k}$) of side length $2\ell_i \approx \varepsilon/\theta$.

B. Micro-Structure of the Strips

Inside each square $Q_{i,k}$, the domain is partitioned into:

• Central Core ($D_0$): A small square of size proportional to the dislocation core radius $\lambda \varepsilon$.
• Zig-zag Regions ($T^a, T^b$): Trapezoidal regions used to transition between different Burgers vectors and accumulate the displacement.
• Annular Regions ($\Delta^{a,n}, \Delta^{b,n}$): Dyadic square annuli surrounding the core.

C. Strain Definition

The strain field $\beta$ is defined as piecewise constant (the gradient of a piecewise affine deformation):

• Rotations: In the outer strips and specific sub-regions, $\beta$ equals the rotation matrices $R_{\pm \theta}$.
• Dislocation Accommodation: In the annular regions, the strain is an interpolation between the rotation and a translation. The translation accumulates the Burgers vector ($b_1$ or $b_2$) as one moves down the strip.
  • Specifically, the construction ensures that the circulation of the strain around a dislocation core equals exactly one lattice vector ($\tau \varepsilon b_i$).
• Rank-One Connections: The construction is designed so that adjacent regions are rank-one connected (compatible) along most interfaces, minimizing the energy penalty associated with discontinuities. Discontinuities (dislocations) are concentrated only in the small core regions $D_0$.

D. Parameter Determination

The width of the inner strips ($\ell_i$) and the number of dyadic layers ($\bar{n}_i$) are not arbitrary. They are determined by enforcing rank-one connectivity across the interface between the two inner strips ($\Sigma_1 \cap \Sigma_2$). This yields a system of equations linking the lattice vectors $b_1, b_2$ and the misorientation angle $\theta$, allowing the authors to solve for the optimal strip widths explicitly.

3. Key Contributions

1. Generalization to Arbitrary Bravais Lattices: Unlike previous works restricted to square lattices, this construction works for any 2D Bravais lattice generated by vectors $b_1, b_2$.
2. Simplified Construction: The authors replace the three-strip construction used in prior literature with a two-strip construction.
3. Piecewise Constant Strain: The proposed strain field is piecewise constant, forming a polygonal partition (Caccioppoli partition). This is a stronger condition than previous models and makes the construction amenable to purely matrix-based analysis.
4. Strict Quantization: The construction satisfies a stronger version of the circulation condition (H2), ensuring that the Burgers vector of each dislocation is exactly a generator of the lattice, rather than just a linear combination.

4. Results and Energy Estimates

The authors compute the upper bound for the total energy $F_\varepsilon(\beta, S)$, which consists of elastic energy and core energy.

• Elastic Energy: The energy density in the transition regions scales as $1/r_n^2$. Summing over all annuli and strips yields a term proportional to $\varepsilon L \theta \log(1/\theta)$.

• Core Energy: The energy associated with the dislocation cores scales with the number of dislocations ($\sim L/\theta$) and the core area. This yields a term proportional to $\varepsilon L \theta$.

• Total Energy Bound:
  $$F_\varepsilon(\beta, S) \leq C \varepsilon L E_0 \sin \theta (A - \log_2(\sin \theta))$$
  For small $\theta$, $\sin \theta \approx \theta$, recovering the Read–Shockley scaling:
  $$E \sim \varepsilon \theta (A - \log \theta)$$

• Explicit Constants:

  • $E_0$: The leading-order coefficient, derived explicitly in terms of the lattice vectors and misorientation angle. For a square lattice, this matches the classical result by Read and Shockley exactly.
  • $A$: The lower-order constant. The authors note that while the leading term is optimal, the constant $A$ differs slightly from the classical result due to the specific geometric modeling of the dislocation core (square vs. circular) and the simplification of the core energy term.

5. Significance

• Mathematical Rigor: The paper provides a rigorous proof of the upper bound for the Read–Shockley formula for general lattices using a semi-discrete energy model, bridging the gap between atomistic and continuum descriptions.
• Physical Insight: The construction offers a clear physical picture of how dislocations arrange themselves to accommodate misorientation in arbitrary lattices, specifically highlighting the role of rank-one connections in minimizing energy.
• Foundation for Future Work: The authors suggest that this simpler, piecewise constant construction is a robust framework that can be extended to more complex scenarios, such as multi-well energies (modeling twins) or laminates, which were beyond the scope of this specific note.

In summary, Scardia and Tolotti successfully generalize the Read–Shockley energy scaling to arbitrary 2D Bravais lattices by introducing a streamlined, piecewise constant strain construction that explicitly accounts for lattice geometry and dislocation core effects.