Materials Grain Boundary Energy From Jcost

Grain boundary energy

A grain boundary is the interface where two crystals of the same material meet at different orientations. The energy of that interface, usually written γ_GB, is the extra free energy per unit area stored there, and it matters because it drives grain growth, recrystallization, and the mechanical properties of metals. Typical measured values for metals run from about 0.1 to 0.5 joules per square meter, while the free surface energy of a metal is roughly 1 J/m². The ratio between the two is often near 0.1, which is the empirical fact this research note aims at.

The classical picture comes from the 1950s and 1960s, when Read and Shockley derived a formula for low-angle grain boundaries based on the dislocation spacing, and later experiments mapped how γ_GB varies with the misorientation angle between the two crystals. A high-angle boundary can have nearly half the surface energy, while a low-angle boundary costs much less. The energy is not a single number but a function of the misorientation, the boundary plane, and the material's bonding. That is the standard surface a materials scientist expects.

In Recognition Science, the framework models this with its cost function J, which is forced by five plain conditions to equal J(x) = (x + 1/x)/2 - 1. The module defines domainCost as J(m/e), a ratio of two real numbers, and proves three general facts: the cost is zero when the two inputs are equal, it is nonnegative for positive inputs, and the constant phi - 3/2 is positive. The research note attached to the module proposes that γ_GB ≈ J(φ) × γ_surface ≈ 0.118 × surface energy, which for a metal with γ_surface ≈ 1 J/m² gives about 0.118 J/m², at the lower end of the measured range.

What the module itself establishes is narrower than that note. The formal code proves the three general properties of J(m/e) and nothing specific to grain boundaries, because the definition of domainCost never refers to a misorientation angle, a boundary plane, or a material. The module is a template shared verbatim with 2383 sibling modules; the universal content lives in one place, and what would make it a theorem about grain boundaries is a definition of m and e in the subject's own terms. The numerical prediction remains a research note, not a proved result.

The honest takeaway is that the framework supplies a candidate functional form for the ratio between grain boundary energy and surface energy, and the module certifies that the form has the basic properties a cost should have. Whether that form matches real boundaries is an empirical question the module does not answer, and the measured range is wide enough that the single number 0.118 sits comfortably inside it without being a sharp test.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module proves any specific numerical value for grain boundary energy in any real material. The framework derives the measured range 0.1 to 0.5 J/m² for metals. The prediction 0.118 J/m² is a tested empirical result rather than a research note.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/GrainBoundaryEnergyFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: