Nanochemistry — IUPAC-VII applied-chemistry at the 1–100 nm grain where quantum-size effects and high surface-to-volume ratios dominate over bulk-like behaviour. Foundations: the surface-to-volume ratio for a sphere is A/V = 3/R —…
nanochemistry
Sphere surface-to-volume ratio A/V = 3/R
Surface-to-volume ratio for a sphere is A/V = 3/R — inverse-linear in radius. At R = 1 nm, A/V = 3 nm⁻¹: ~30% of atoms in a 1-nm…
Quantum confinement (1D box): E_n = n²π²ℏ²/(2 m a²) ; E_n ∝ 1/a² (size scaling)
Quantum confinement: when the system dimension a approaches the de Broglie wavelength, electronic levels discretise. Infinite 1D square…
Scherrer equation: crystallite size t = Kλ/(β cos θ)
Scherrer equation (Scherrer 1918): for a crystallite of mean size t, an ideal narrow X-ray source at wavelength λ produces a Bragg peak of…
A/V at R=1 ⇒ 3; scale law A/V ∝ 1/R (SymPy-simplified sphere ratio)
Sympy-exact witness of the sphere surface-to-volume ratio. Canonical oracle pins: SA_V_form = '3/R' (after sp.simplify), SA_V_at_R1 = '3'…
E_1(a=1) = π²/2 ; E_1(a=2) = π²/8 ; halving ratio = 4 (1/a² scaling)
Sympy-exact witness of the 1/a² quantum-confinement scaling. Unit natural units (ℏ = m = 1, n = 1): E_1(a=1) = π²/2, E_1(a=2) = π²/8,…
Scherrer at (K, λ, β, θ) = (1, π, π, 0): t = 1 (unit crystallite-size canonical)
Sympy-exact witness of the Scherrer crystallite-size relation at a clean unit-parameter point. Setup: K=1 (dimensionless shape factor),…
Mie-Rayleigh framework: σ = (128π⁵R⁶/3λ⁴)·((n²-1)/(n²+2))² (R⁶/λ⁴ scaling)
The Rayleigh limit of Mie scattering (Rayleigh 1871; Mie 1908) — electromagnetic-scattering cross-section for a dielectric sphere of radius…
Ostwald-LSW framework: ⟨R⟩³(t) = R₀³ + K_LSW·t (t^{1/3} late-time coarsening)
The Lifshitz-Slyozov-Wagner (LSW) theory of Ostwald ripening (Lifshitz-Slyozov 1961; Wagner 1961) — the classical mean-field description of…
Kubo gap framework: δ = 4·E_F/(3·N) (nano-metal discrete-level energy spacing)
The Kubo formula for the mean level-spacing of a small metal cluster (Kubo 1962) — the nano-metal regime where the discrete-level…
Mie-Rayleigh: σ(2R)/σ(R) = 64 (R⁶); σ(λ)/σ(2λ) = 16 (λ⁻⁴)
Sympy-exact symbolic witness of the R⁶ and λ⁻⁴ scalings in the Rayleigh limit of Mie scattering. Setup: σ = (128π⁵R⁶/3λ⁴) ·…
Ostwald-LSW: ⟨R⟩³(t=0) = R₀³; late-time ⟨R⟩³(t₁)/⟨R⟩³(t₂) = t₁/t₂ (cubic growth)
Sympy-exact symbolic witness of the LSW cubic-growth law. Setup: ⟨R⟩³(t) = R₀³ + K_LSW·t as a sympy expression with R₀, K_LSW, t positive…
Kubo gap: δ(N=1) = 4E_F/3; δ(N=1)/δ(N=1000) = 1000 (inverse-N scaling)
Sympy-exact symbolic witness of the Kubo-gap inverse-N scaling. Setup: δ = 4·E_F/(3·N) as a sympy expression with E_F, N positive symbols.…
Quantum dot (Brus 1984)
L Brus 1984 size-dependent semiconductor bandgap E_g = E_bulk + h^2/(8 m R^2); CdSe Bawendi 1993 monodisperse synthesis; QD-LED commercial…
Graphene (Novoselov-Geim 2004)
Novoselov-Geim 2004 (Nobel 2010) scotch-tape exfoliation of single-layer carbon; massless Dirac electrons; record carrier-mobility 2x10^5…
Plasmonic nanoparticles (Mie 1908)
Mie 1908 LSPR theory for spherical metal nanoparticles; Au LSPR 520 nm + size-tunable; SERS 10^11 enhancement (Le Ru 2011);…
SAMs (Nuzzo-Allara 1983)
Nuzzo-Allara 1983 + Whitesides 1990s SAM: thiol-Au(111) headgroup chemisorption; hexagonal lattice 5 A; basis of surface-functionalization…
DNA origami (Rothemund 2006)
P Rothemund 2006 scaffold-staple DNA-origami: M13mp18 7249 nt + 200 short staples; arbitrary 2D shape at ~6 nm resolution; modern…
MOF (Yaghi 1995)
O Yaghi 1995 + 1999 MOF-5: Zn4O(BDC)3 secondary-building-unit + ditopic-linker; record SBET 7000 m^2/g (NU-1501-Al 2020); CO2 capture + H2…
QD (Brus 1984)
L Brus 1984 quantum-dot size-quantization E∝1/R²; modern Bawendi-Brus-Ekimov Nobel 2023 + commercial QLED + medical-imaging QD.
C60 (Kroto 1985)
H Kroto-R Smalley-R Curl 1985 (Nobel 1996) C60 buckminsterfullerene; modern modern carbon-nanotubes Iijima 1991 + graphene 2004.
CNT (Iijima 1991)
S Iijima 1991 multi-wall-CNT + 1993 single-wall; modern modern aligned-array + chirality-sorted + 2024 functional-CNT-FET semiconductor.
Nanocube (Seo 2009)
Y Seo 2009 + 2024 facet-controlled metal-nanoparticle-shape; modern modern catalytic-facet-effect + plasmonic-property-tuning.
UCNP (Haase 2011)
M Haase 2011 lanthanide-upconversion; modern modern theranostics + NIR-deep-tissue + 2024 Cre-encoded UCNP optogenetics.
EBL (Broers 1968)
A Broers 1968 electron-beam-lithography; modern modern EUV 13.5 nm + 2024 high-NA + nanoimprint-lithography mass-production.