Characterization of self-affinity in the global regime

Methods for characterization of self-affine surfaces and measurements of their roughness exponents H are developed. It is shown that for smoothed surfaces, which underwent particular coarse graining or averaging of the small-scale fluctuations, the excess surface area Sₑₓ and the mean square root radius of curvature ac are related by two distinct asymptotic power laws if ac is well below or well above a certain crossover scale acr. In the local regime of self-affinity, when ac{}acr, Sₑₓ{∝}(ac/acr{)}^{{{-}}(1{{-}}H)}$. In the global regime of self-affinity, when ${a}c{a}cr$, ${S}ₑₓ∝({a}c$/${a}cr)^-2(1-H)/(2-H). The former scaling relationship is consistent with the well known definition of local fractal dimensions dloc=dₜₒₚ+1-H. The latter scaling relationship offers alternatives for characterization of self-affinity over large scales by means of excess dimensions defined as dₑₓ=dₜₒₚ+2(1-H)/(2-H) and can be used for determination of roughness exponents from the measurements provided in the global regime. The thermodynamic method of fractal analysis, proposed earlier for self-similar surfaces (A.V. Neimark, Pis'ma Zh. Eksp. Teor. Fiz. 51, 535 (1990) [JETP Lett. 51, 607 (1990)]; Physica A 191, 258 (1992)), is extended for self-affine surfaces for determination of fractal dimensions and roughness exponents from adsorption and capillary experimental data.