Abstract
The Solid Surface Free Energy (SFE) measured by contact angle measurement instruments or water contact angle testers is a critical parameter for characterizing the wettability, adhesion, and interfacial interactions of materials. Its precise evaluation is essential for material science, industrial coating development, and biomedical applications. This article systematically reviews 14 SFE analysis models driven by contact angle measurement instruments, thoroughly explores the classification of testing methods, sources of error, optimization strategies, and industrial application cases, and proposes multidimensional accuracy enhancement solutions based on instrumental innovations.

1. Classification and Accuracy Comparison of Contact Angle Measurement Methods

The measurement method of a contact angle measurement instrument or water contact angle tester significantly impacts the accuracy of contact angle (θ) values. These methods are broadly categorized into two types:

1. Geometric Model Goniometry Methods

• Principle: Calculate θ directly from geometric features of the droplet contour (e.g., baseline tangent, curvature radius).

• Typical Methods:

  • Circle/Ellipse Fitting: Assumes an ideal circular or elliptical droplet contour; suitable for small contact angles (θ < 90°).

  • Polynomial Tangent Method: Fits droplet edges using quadratic or higher-order polynomials to adapt to asymmetric contours.

  • Spline Curve Method: Smoothes contours via spline interpolation to reduce noise.

  • TrueDrop® Asymmetric Dynamic Method: Analyzes dynamic droplet contours without axisymmetric assumptions, ideal for rough or heterogeneous surfaces.

• Advantages: Fast computation and low hardware costs.

• Disadvantages: Ignores droplet gravity and surface tension balance, leading to significant measurement errors, poor repeatability, and limited scientific validity. The data can only be used as estimates, with errors for low-surface-tension liquids (e.g., organic solvents) reaching up to ±5°.

2. Young-Laplace Equation Fitting Methods

• Principle: Solve θ numerically by iterating the Young-Laplace equation describing the mechanical equilibrium of droplet contours.

• Typical Methods:

  • Empirical Bond Number Method: Uses the dimensionless Bond number ($Bo=\frac{\mathrm{\Delta }\rho g{V}^{2\mathrm{/}3}}{{\gamma }_L}$) to simplify calculations but is limited to axisymmetric contours (e.g., German commercial instruments), narrow droplet volumes (1–5 μL), and exhibits >10% error for high-viscosity liquids.

  • ADSA®-RealDrop® Technology: Directly fits contours using dimensional Young-Laplace equations without axisymmetric constraints, supports asymmetric droplets (e.g., contact angle hysteresis analysis), extends droplet volume range to 0.1–10 μL, and achieves ±0.1° precision.

• Advantages: Rigorous physical principles and high accuracy.

• Disadvantages: High computational complexity requiring advanced hardware. The empirical Bond number method has low sensitivity and accuracy, with errors (Δθ) reaching up to ±3°.

2. SFE Analysis Models Driven by Contact Angle Measurement Instruments and Their Pros/Cons

1. Classical Single-Component Models

(1) Zisman Critical Surface Tension Method

• Principle: Extrapolate the $\cos \theta$ vs. ${\gamma }_L$ curve of multiple liquids to $\cos \theta =1$ to determine the critical surface tension ${\gamma }_C$.

• Pros: Simple operation, suitable for rapid screening of low-energy nonpolar materials (e.g., PTFE).

• Cons: Ignores polar interactions; errors >20% for high-energy surfaces; cannot differentiate SFE components [1].

(2) Berthelot Geometric Mean Method

• Formula:

  $${\gamma }_{SL}={\gamma }_S+{\gamma }_L-2\sqrt{{\gamma }_S{\gamma }_L}$$

• Pros: Simple formula for rough estimation of hydrocarbon materials.

• Cons: Assumes solid-liquid interactions are purely geometric mean; errors >30% for polar materials [2].

2. Two-Component Models

(3) Fowkes Dispersion Model

• Formula:

  $${\gamma }_{SL}={\gamma }_S+{\gamma }_L-2\sqrt{{\gamma }_S^d{\gamma }_L^d}$$

• Pros: Introduces dispersive component concept, enabling quantification for nonpolar materials (e.g., polyethylene).

• Cons: Completely ignores polar contributions; significant errors for polar surfaces (e.g., metal oxides) [3].

(4) OWRK Method (Owens-Wendt-Kaelble)

• Formula:

  $${\gamma }_{SL}={\gamma }_S+{\gamma }_L-2(\sqrt{{\gamma }_S^d{\gamma }_L^d}+\sqrt{{\gamma }_S^p{\gamma }_L^p})$$

• Pros: Separates dispersive (${\gamma }^d$) and polar (${\gamma }^p$) components; widely used for polymers and coatings.

• Cons: Underestimates polar components for high-energy surfaces (e.g., metals); requires strict liquid selection [4].

(5) Wu Harmonic Mean Method

• Formula:

  $${\gamma }_{SL}={\gamma }_S+{\gamma }_L-\frac{4{\gamma }_S^d{\gamma }_L^d}{{\gamma }_S^d+{\gamma }_L^d}-\frac{4{\gamma }_S^p{\gamma }_L^p}{{\gamma }_S^p+{\gamma }_L^p}$$

• Pros: Enhances polar interaction calculations; suitable for high-polarity materials (e.g., organic pigments).

• Cons: Sensitive to θ measurement errors; 1° deviation in θ leads to >5% SFE error [5].

(6) Girifalco-Good Equation

• Formula:

  $${\gamma }_{SL}={\gamma }_S+{\gamma }_L-2\mathrm{\Phi }\sqrt{{\gamma }_S{\gamma }_L}$$

• Parameter: $\mathrm{\Phi }$ (polarity matching factor, 0.5 < $\mathrm{\Phi }$ < 1.2).

• Pros: Corrects geometric mean via $\mathrm{\Phi }$, partially explaining polarity differences.

• Cons: $\mathrm{\Phi }$ requires experimental calibration; poor universality [6].

3. Three-Component and Multi-Component Models

(7) Acid-Base Model (van Oss Method)

• Formula:

  $${\gamma }_{SL}={\gamma }_S+{\gamma }_L-2(\sqrt{{\gamma }_S^{LW}{\gamma }_L^{LW}}+\sqrt{{\gamma }_S^{+}{\gamma }_L^{-}}+\sqrt{{\gamma }_S^{-}{\gamma }_L^{+}})$$

• Pros: Distinguishes Lewis acid (${\gamma }^{+}$) and base (${\gamma }^{-}$) interactions; ideal for biomaterials and drug carriers.

• Cons: Requires three liquids for simultaneous equations; may yield negative parameters with ambiguous physical interpretation [7].

(8) Schultz Dual-Liquid Method

• Principle: Solves polar/dispersive components using two liquids with distinct polarities (e.g., water, diiodomethane).

• Pros: Reduces liquid quantity (only two liquids); suitable for rapid testing.

• Cons: Relies on high-precision surface tension data; limited practical applicability [8].

(9) Chibowski Thermodynamic Method

• Formula:

  $${\gamma }_S=\frac{{\gamma }_L(1+\cos \theta )}{2}+\sqrt{{\gamma }_L^d{\gamma }_S^d}$$

• Pros: Single-liquid estimation for mixed-polarity surfaces; simplifies experiments.

• Cons: Requires preset dispersive component ratio; errors >15% for strongly polar materials [9].

4. Equation of State and Dynamic Wetting Models

(10) Neumann Equation of State (EQS)

• Formula:

  $$\cos \theta =-1+2\sqrt{\frac{{\gamma }_S}{{\gamma }_L}}\cdot e^{-\beta ({\gamma }_L-{\gamma }_S{)}^2}$$

• Parameter: $\beta \approx 0.0001247{\text{m}}^2\mathrm{/}{\text{mJ}}^2$.

• Pros: Single-liquid rapid estimation; avoids multi-liquid selection.

• Cons: Empirical parameter $\beta$ lacks universality; errors >20% for high-energy surfaces (${\gamma }_S>50{\text{mJ/m}}^2$) [10].

(11) Kwok-Neumann Modified Equation

• Improvement: Introduces temperature-dependent $\beta$ optimization.

• Pros: Reduces errors for metal/ceramic surfaces to ±10%.

• Cons: Requires multi-liquid calibration; increased operational complexity [11].

(12) Dynamic Wetting Method (Wenzel Equation Correction)

• Formula:

  $$\cos {\theta }_{\text{apparent}}=r\cos {\theta }_{\text{intrinsic}}$$

• Parameter: $r$ (roughness factor).

• Pros: Incorporates roughness factor to reflect real-world wetting behavior.

• Cons: Requires simultaneous surface morphology characterization (e.g., AFM); high cost and time consumption [12].

5. Emerging Multi-Scale Models

(13) Molecular Dynamics (MD) Simulation Combined with Contact Angle Method

• Procedure: Simulates droplet spreading to derive ${\gamma }_S$.

• Pros: Avoids experimental interference; <5% error for nanoscale superhydrophobic materials.

• Cons: High computational resource demands; sensitive to force field parameters [13].

(14) Electrochemical Correction Model (CQC Model)

• Principle: Introduces surface charge to correct SFE calculations.

• Pros: Integrates surface charge effects; suitable for ionic liquid-modified materials.

• Cons: Requires additional electrochemical testing equipment; high industrial application barriers [14].

3. Optimization of Contact Angle Measurement Accuracy

1. Probe Liquid Purity Verification

• Water Purity Standard: Surface tension ${\gamma }_L\ge 72.8\text{mN/m}$ (25°C); repurify if measured value <70 mN/m.

• Organic Solvent Screening: Diiodomethane, formamide, etc., must be verified via gas chromatography (purity ≥99%).

2. Solid Surface Contamination Detection

• Criterion: Theoretical surface tension of pure water is 72 mN/m. If the measured value on a solid surface is ≤69 mN/m (error >3 mN/m), organic contamination is confirmed.

• Detection Method:

  1. Measure surface tension ${\gamma }_L^{\mathrm{\prime }}$ of droplets on the solid using a contact angle instrument.

  2. Compare ${\gamma }_L^{\mathrm{\prime }}$ with theoretical ${\gamma }_L$; contamination is confirmed if $\mathrm{\mid }{\gamma }_L-{\gamma }_L^{\mathrm{\prime }}\mathrm{\mid }>2\text{mN/m}$.

3. Droplet Contour Fitting Technology Breakthrough

• ShangHai Solon RealDrop®/TrueDrop® Contact Angle Instruments:

  • ADSA®-RealDrop® Technology: Directly fits contours using Young-Laplace equations without Bond number assumptions, extending droplet volume range to 0.1–10 μL with ±0.1° resolution.

  • Integrated Surface Tension Monitoring: Equipped with high-precision pressure-sensitive sensors and the Wilhelmy plate method based on the ADSA® algorithm, enabling real-time detection of surface tension changes and direct identification of contamination effects [15].

4. Industrial Applications and Case Studies

1. Coating Industry

• Scenario: Optimizing hydrophobicity of automotive coatings via SFE analysis.

• Solution: OWRK method combined with ADSA®-RealDrop® technology achieves ±3% error control.

2. Biomedical Field

• Scenario: Detecting the impact of surface contamination on cell adhesion for implant materials.

• Solution: Wilhelmy plate method monitors surface tension changes with contamination sensitivity of 1 ng/cm² [16].

5. Technical Challenges and Future Directions

1. Current Challenges

• Model Parameter Universality: Parameters like $\beta$ (Neumann EQS) and $\mathrm{\Phi }$ (Girifalco-Good) require material-specific calibration.

• Dynamic Interface Characterization: Theoretical links between dynamic wetting and SFE remain incomplete.

2. Future Directions

• Multi-Technique Integration: Combine contact angle instruments with AFM/XPS for synchronized surface chemistry-wettability analysis.

• AI-Driven Parameter Optimization: Train models on historical data to predict $\beta$, $\mathrm{\Phi }$, etc.

• Standardized Validation Protocols: Establish probe liquid databases and contamination thresholds (referencing ISO 19403-7).

Keywords: Contact angle measurement instrument, solid surface free energy, surface tension analysis, Young-Laplace equation, wettability testing, surface contamination detection, ADSA technology, TrueDrop dynamic method, SFE calculation models, industrial quality control

References

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2. Berthelot, D. Comptes Rendus 126, 954 (1898).

3. Fowkes, F. M. J. Phys. Chem. 67, 2538 (1963).

4. Owens, D. K., & Wendt, R. C. J. Appl. Polym. Sci. 13, 1741 (1969).

5. Wu, S. J. Polym. Sci. C 34, 19 (1971).

6. Girifalco, L. A., & Good, R. J. J. Phys. Chem. 64, 561 (1960).

7. van Oss, C. J. Interfacial Forces in Aqueous Media, CRC Press (2006).

8. Schultz, J., et al. J. Colloid Interf. Sci. 59, 277 (1977).

9. Chibowski, E. Adv. Colloid Interf. Sci. 103, 227 (2003).

10. Neumann, A. W. Adv. Colloid Interf. Sci. 4, 105 (1974).

11. Kwok, D. Y., & Neumann, A. W. Langmuir 12, 5551 (1996).

12. Wenzel, R. N. Ind. Eng. Chem. 28, 988 (1936).

13. Wang, J., et al. J. Phys. Chem. C 122, 25521 (2018).

14. Chang, Q., et al. Langmuir 31, 10751 (2015).

15. ShangHai Solon Scientific Instrument Inc. RealDrop® Technical Manual (2023).

16. Liu, G., et al. ACS Appl. Mater. Interfaces 12, 5678 (2020).

Declaration: RealDrop®, TrueDrop®, and ADSA® are registered trademarks of ShangHai Solon Information Technology Co., Ltd.