Materials Grain Boundary Energy Rs

Grain boundary energy in the framework

Grain boundary energy is the thermodynamic cost, per unit area, of the interface where two crystals of the same phase meet at different orientations. Typical measured values for metals fall between about 0.5 and 1.0 J/m². The boundary is a defect: atoms there are not in either grain's perfect lattice, so the system pays an energy penalty that drives grain growth and recrystallization.

In Recognition Science, the module models this energy as a fraction of the surface energy. The defining choice is cost, the price reality pays to register a recognition event; the framework proves a unique dimensionless cost function J(x) = (x + 1/x)/2 - 1. The module sets the grain boundary energy to J(phi) times the surface energy gamma_s, which evaluates to 0.118 * gamma_s. For steel with gamma_s about 2 J/m², that gives roughly 0.236 J/m², a lower bound consistent with the measured range.

The machine-checked library of formal theorems proves three general facts about this cost function, not about grain boundaries specifically. First, the cost vanishes when the two inputs are equal: domainCost r r = 0 for any nonzero r. Second, the cost is nonnegative for positive inputs. Third, the threshold phi - 3/2 is positive. These are properties of the universal cost function, instantiated here with the ratio m/e.

What the module does not do is derive the grain boundary energy from first principles. The docstring is explicit: the numerical value 0.118 is a research note, not a theorem. The formal content is a template shared verbatim with 2383 sibling modules, each applying the same cost function to a different domain. To make this a theorem about grain boundaries, one would need a definition of m and e in terms of the physical system, which the module does not provide.

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

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]

What this page does not claim

The numerical value 0.236 J/m² for steel is not a derived result. The module does not prove that grain boundary energy equals J(phi) times surface energy. The formal theorems hold for the cost function, not for any specific material.

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

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