<p>Surface roughness characterization lacks standardized protocols for representativity and uncertainty, limiting reproducibility and functional property prediction. We frame this as a convergence problem: determining the minimum measurement area, termed representative elementary volume (REV), such that descriptor estimates converge within tolerance and uncertainty falls below prescribed bounds. This study introduces a multi-metric REV framework with explicit convergence criteria and uncertainty thresholds for geometric (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">${S}_{a},{S}_{q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>a</mi> </msub> <mo>,</mo> <msub> <mi>S</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>), correlational (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">${L}_{c}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>), fractal (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$D$</EquationSource> <EquationSource Format="MATHML"><math> <mi>D</mi> </math></EquationSource> </InlineEquation>), Hurst scaling (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H$</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation>), and spectral (power spectral density, PSD) descriptors using corner-based two-dimensional expansion algorithms applied to high-resolution confocal microscopy data. All metrics exhibit pronounced scale dependence requiring distinct REVs, although not all provide equivalent diagnostic utility. The arithmetic roughness (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">${S}_{a}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation>) and Hurst exponent (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H$</EquationSource> <EquationSource Format="MATHML"><math> <mi>H</mi> </math></EquationSource> </InlineEquation>) achieve correlated convergence (<InlineEquation ID="IEq7"> <EquationSource Format="TEX">${REV}_{Sa}\approx {REV}_{H}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mi mathvariant="italic">REV</mi> </mrow> <mrow> <mi mathvariant="italic">Sa</mi> </mrow> </msub> <mo>≈</mo> <msub> <mrow> <mi mathvariant="italic">REV</mi> </mrow> <mi>H</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>), identifying transitions from multifractal to scale-invariant behavior. Correlation length (<InlineEquation ID="IEq8"> <EquationSource Format="TEX">${L}_{c}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>c</mi> </msub> </math></EquationSource> </InlineEquation>) displays strong scale dependence with material-specific convergence patterns—challenging its traditional treatment as intrinsic—while coarse surfaces often preclude reliable <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${REV}_{C}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mi mathvariant="italic">REV</mi> </mrow> <mi>C</mi> </msub> </math></EquationSource> </InlineEquation> determination. The global fractal dimension (<InlineEquation ID="IEq10"> <EquationSource Format="TEX">$D$</EquationSource> <EquationSource Format="MATHML"><math> <mi>D</mi> </math></EquationSource> </InlineEquation>) converges reliably but provides no material differentiation, demonstrating that statistical convergence does not guarantee interpretive value. Corner-based uncertainty metrics (<InlineEquation ID="IEq11"> <EquationSource Format="TEX">${\sigma}_{R},{\sigma}_{C},{\sigma}_{1/e}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>R</mi> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mi>C</mi> </msub> <mo>,</mo> <msub> <mi>σ</mi> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>e</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>) quantify spatial-method variability, enabling assessment of measurement reliability and undersampling costs at sub-REV scales. <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$PSD$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">PSD</mi> </mrow> </math></EquationSource> </InlineEquation> analysis shows spectral–spatial disconnect, where roll‑off wavelengths occur substantially below physical REV scales and exhibit monofractal behavior, contrasting with multifractal signatures detected through spatial methods. Collectively, these findings support a transition from fixed-value roughness characterization toward scale-aware, uncertainty-aware, and descriptor-specific frameworks in which representativity itself becomes a measurable surface property.</p>

From Scale Dependence to Scale Invariance: A Multi-metric REV Framework for Representative Surface Roughness Characterization

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2026年7月8日

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Kuldeep Singh,
Nitin Paliwal,
Gabrielle Gromofsky

摘要

Surface roughness characterization lacks standardized protocols for representativity and uncertainty, limiting reproducibility and functional property prediction. We frame this as a convergence problem: determining the minimum measurement area, termed representative elementary volume (REV), such that descriptor estimates converge within tolerance and uncertainty falls below prescribed bounds. This study introduces a multi-metric REV framework with explicit convergence criteria and uncertainty thresholds for geometric ( ${S}_{a},{S}_{q}$ $S_{a}, S_{q}$ ), correlational ( ${L}_{c}$ $L_{c}$ ), fractal ( $D$ $D$ ), Hurst scaling ( $H$ $H$ ), and spectral (power spectral density, PSD) descriptors using corner-based two-dimensional expansion algorithms applied to high-resolution confocal microscopy data. All metrics exhibit pronounced scale dependence requiring distinct REVs, although not all provide equivalent diagnostic utility. The arithmetic roughness ( ${S}_{a}$ $S_{a}$ ) and Hurst exponent ( $H$ $H$ ) achieve correlated convergence ( ${REV}_{Sa}\approx {REV}_{H}$ ${REV}_{Sa} \approx {REV}_{H}$ ), identifying transitions from multifractal to scale-invariant behavior. Correlation length ( ${L}_{c}$ $L_{c}$ ) displays strong scale dependence with material-specific convergence patterns—challenging its traditional treatment as intrinsic—while coarse surfaces often preclude reliable ${REV}_{C}$ ${REV}_{C}$ determination. The global fractal dimension ( $D$ $D$ ) converges reliably but provides no material differentiation, demonstrating that statistical convergence does not guarantee interpretive value. Corner-based uncertainty metrics ( ${\sigma}_{R},{\sigma}_{C},{\sigma}_{1/e}$ $σ_{R}, σ_{C}, σ_{1 / e}$ ) quantify spatial-method variability, enabling assessment of measurement reliability and undersampling costs at sub-REV scales. $PSD$ $PSD$ analysis shows spectral–spatial disconnect, where roll‑off wavelengths occur substantially below physical REV scales and exhibit monofractal behavior, contrasting with multifractal signatures detected through spatial methods. Collectively, these findings support a transition from fixed-value roughness characterization toward scale-aware, uncertainty-aware, and descriptor-specific frameworks in which representativity itself becomes a measurable surface property.