35-EFT.WP.EDX.OrientedTension v1.0 | Chapter 8 Interfaces, Defects & Anisotropy

Chapter 8 Interfaces, Defects & Anisotropy

I. Abstract & Scope
This chapter formulates a minimal description of interfaces (phase/phase boundaries, interlayers), defects (orientational singularities/distortion cores), and spatial anisotropy in oriented systems. We define surface/interface energy W_surf, anchoring and jump conditions, defect core energy and topological conservation, and practical anisotropy metrics/diagnostics for engineering and physics. Procedural flows are provided for parameter inversion and map-based characterization. Symbols use English notation wrapped in backticks; SI units apply. No ToA terms appear here.

II. Dependencies & References

• Orientation geometry & distributions: Chapter 3 S80-1/2.
• Constitutive & dynamics: Chapter 4 S80-3/4, axioms P80-2/3/9/10.
• Metrology & inversion: Chapter 5 M80-1…4 (Q_ij, T_fil_ij, tau_relax, D_Q).
• Couplings & media: Chapter 6 S80-5/6 (W_cpl, anisotropic transport/waves).
• Energy accounting: Chapter 7 S80-7/8.
• Numerics & implementation: Chapter 10 (SimStack-OT), Chapter 12 (I80-*).

III. Normative Anchors (added in this chapter, P80-/S80-)

• P80-14 (Interface Anchoring Axiom): the surface/interface energy W_surf is built from objective invariants and respects material/geometric symmetries; interface conditions follow from energy minimization and flux continuity.
• P80-15 (Defect Topology Conservation Axiom): in a closed domain without interface creation/annihilation of defects, the topological charge is conserved; any numerical/experimental identification must report a bound on conservation error.
• S80-9 (Surface/Interface Energy & Anchoring):
  W_surf = κ_a (Q_ij, t_hat) + κ_b (Q_ij, n_b),
  where t_hat is the tangential basis and n_b the interface normal; κ_* are anchoring coefficients. Interface conditions from Euler–Lagrange variation and flux continuity:
  [[ Φ_E · n_b ]] = − ∂_t W_surf, [[ T_fil_ij n_{b,j} ]] = f^{(s)}_i (surface source/sink f^{(s)}_i; [[·]] denotes the jump between sides).
• S80-10 (Defect Core Energy & Gradient Penalty):
  W_defect = (1/2) K (∂_k Q_ij)(∂_k Q_ij) + U_core(Q),
  with K ≥ 0 and a positive core potential U_core. Defect indices derive from winding/half-winding invariants of Q_ij and serve as diagnostics.
• S80-11 (Anisotropy Metrics & Projections):
  S_dir(ê) = ê_i Q_ij ê_j, A_aniso = λ_max(Q) − λ_min(Q), R_aniso = ‖Q‖_F. Directional effective parameters are given by projections along ê.

IV. Body Structure

I. Background & Problem Statement

• Interfaces modify boundary conditions of orientation and tension, inducing jumps in birefringence/phase speed and diffusion tensors; defects alter energy budgets and observables through core energy and gradient penalties.
• A minimal, Chapter 4–7–consistent set of equations and metrology flows is required so that anchoring, defect identification, and anisotropy quantification are measurable and reproducible in simulation.

II. Key Equations & Derivations (S-series)

1. S80-9 (Anchoring & Jump Conditions)
   • Variational condition (schematic): n_b · ( K ∇Q_ij ) + ∂ W_surf / ∂ Q_ij = 0 (preserve symmetry and tracelessness).
   • Energy flux: [[ Φ_E · n_b ]] = − ∂_t W_surf.
   • Tension balance: [[ T_fil_ij n_{b,j} ]] = f^{(s)}_i (rhs =0 if no surface force).
2. S80-10 (Defect Core & Positivity)
   • Core energy: W_defect = (1/2) K ∂_k Q_ij ∂_k Q_ij + U_core(Q) ≥ 0.
   • Diagnostics: locate defects by eigen-direction fields of Q_ij and winding of S_dir(ê).
3. S80-11 (Anisotropy Indicators)
   • Directional projection: S_dir(ê) = ê_i Q_ij ê_j.
   • Scalars: A_aniso = λ_max − λ_min, R_aniso = ‖Q‖_F = (Q_ij Q_ij)^{1/2}.
   • Coupling consistency: under Chapter 6 weak coupling, Δn(ê) ∝ S_dir(ê), D_eff(ê) = D0 + D1 S_dir(ê).

III. Methods & Flows (M-series)

1. M80-22 (Interface Segmentation & Parameter Inversion)
   • Segment interfaces Γ (image/field driven).
   • Estimate n_b, t_hat, and two-sided Q_ij, T_fil_ij.
   • Invert κ_a, κ_b and possible surface sources f^{(s)} via S80-9, with unit/dimension audits.
2. M80-23 (Defect Identification & Core-Parameter Estimation)
   • Localize singularities via Q_ij eigen-directions/S_dir(ê) maps.
   • Estimate {K, U_core} parameters or effective core radius using S80-10.
   • Produce a defect catalog and verify topological conservation (P80-15).
3. M80-24 (Anisotropy Mapping & Coupling Harmonization)
   • Build spatiotemporal maps of {S_dir(ê), A_aniso, R_aniso}.
   • Regress against Chapter 6 coupling parameters {D1, α_E, α_B, κ_s}.
   • Pass anisotropy masks to Chapter 7 for energy-ledger partitioning.

IV. Cross-References within/beyond this Volume

• Chapter 3: Q_ij and S^2 quadrature (M80-13/14) underpin interface/defect metrology.
• Chapter 4: interface/defect conditions feed back into constitutive/dynamics (S80-3/4).
• Chapter 5: Q_ij, T_fil_ij, and covariances required for inversion.
• Chapter 6: regression harmonization between anisotropy indicators and coupling parameters (S80-5/6).
• Chapter 7: consistency of W_surf, W_defect with energy flux/power terms (S80-7/8).
• Chapters 10/12: interface/defect kernels and benchmarks in SimStack-OT and I80-*.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • Adding W_surf/W_defect improves energy-closure residuals and evidence.
   • Interface alignment/phase direction correlates with S_dir(ê) as expected.
   • Jumps in D_eff, ε_ij/μ_ij, C_eff,ijkl match material symmetries; unit/dimension audits pass.
2. Negative criteria:
   • Removing interface/defect terms does not worsen fits.
   • [[ Φ_E · n_b ]] + ∂_t W_surf deviates significantly from zero.
   • The defect catalog violates P80-15 beyond stated confidence.
3. Contrasts:
   • Evidence/residuals for {with interface kernel, without}, {with defect kernel, without}.
   • Transport/propagation differences under {isotropic, anisotropic} models.
   • {weak anchoring, strong anchoring} effects on boundary-layer Q_ij.

VI. Deliverables & Figure List

1. Deliverables:
   • InterfaceCard.json (W_surf, κ_a/κ_b, boundary/jump fields, units/dimensions).
   • DefectCatalog.csv (locations, types, scales, CIs, and conservation checks).
   • AnisotropyMaps.npz (spatiotemporal maps & masks for S_dir(ê), A_aniso, R_aniso).
   • ConsistencyReport.md (energy closure, evidence, and contrast outcomes).
2. Figures/Tables (suggested):
   • Tab. 8-1 Interface/defect energy items & dimensional audits.
   • Fig. 8-1 Anchoring schematic and boundary-layer Q_ij distribution.
   • Tab. 8-2 Defect identification & parameter estimates (K, core radius, etc.).
   • Fig. 8-2 Anisotropy maps and regressions vs coupling parameters.
   • Tab. 8-3 Jump-condition and energy-flux consistency statistics.