Development of a TexCoMP model

This study introduces the TexCoMP model, a novel theoretical framework that evaluates the effect of coating materials on slip resistance by integrating STM, CMP, and E factors. Unlike traditional approaches isolating these elements, the TexCoMP model captures their dynamic interplay, providing a predictive tool for assessing how coatings enhance slip resistance across diverse conditions. The suggested model aims to quickly evaluate the effectiveness of floor coating materials in improving slip resistance. The model provides a framework for understanding how different coatings modify surface textures and how these modifications influence the DFCs, thereby impacting slip resistance. The model also aims to bridge laboratory findings with practical applications by offering a scalable and adaptable framework for optimising flooring safety. The TexCoMP model is structured around three interconnected elements:

1. 1) Surface texture modification (STM): This component quantifies how coating-induced changes in surface topography, measured by roughness parameters such as centre line average (R_a), root mean square (R_q), maximum peak height (R_p), maximum peak-to-valley (R_t), maximum depth (R_v), and maximum height of profile (R_z), affect slip resistance under contaminated conditions. STM reveals micro-scale features, such as asperity heights and fluid entrapments, that affect slip resistance.
2. 2) Coating material performance (CMP): This component evaluates intrinsic properties of coatings (e.g., Shore hardness, elastic modulus, and wear resistance) and their contribution to surface durability and traction, accounting for substrate and environmental interactions over time.
3. 3) Environmental factors (E): This component incorporates external conditions (e.g., arid, damp, and foamy) that affect slip resistance, modelled as a nonlinear influence, reflecting sharp DFC transitions due to contaminants like soap or oil.

The necessity for TexCoMP arises from the complex tribological behaviours at the coating-contaminant-tile interface, where environmental factors, such as foamy contaminants, reduce the effective contact area and promote hydrodynamic slip, contributing 10–20% to the friction variance [16]. Fig 1 visualises this interface, depicting the ceramic tile substrate (blue base) overlaid with the coating layer (orange), where contaminants (yellow regions) infiltrate micro-asperities, disrupting adhesive forces and filling surface valleys. The cross-sectional view highlights drainage paths (arrows indicating directional flow), with the contaminant layer altering shear resistance and promoting viscous drag, particularly in foamy conditions (μ ≈ 0.89 mPa·s). This schematic underscores TexCoMP’s E² term, which quadratically modulates these effects, and STM¹·⁵, which nonlinearly scales texture contributions, enabling predictions that outperform linear approaches by 22% in an RMSE [14].

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Fig 1. Schematic illustration of tribological interactions at the shoe-floor interface, showing the coating layer, contaminant film, and ceramic tile microtexture contributing to friction generation.

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Fig 2 provides a conceptual overview of the TexCoMP model, illustrating the interconnected relationships between its core components, CMP, STM, and E factors, and the resulting DFCs as the outcome of slip resistance. The diagram is structured vertically, with yellow rectangular boxes representing key modules, connected by blue arrows that indicate data flow and dependencies. At the top, the “Coating Material Performance” box expands into sub-components: “Chemical Composition” and “Application Method,” which feed into “Durability” and “Surface Roughness” (linked by dashed lines to indicate interactions), ultimately converging into the central “Surface Texture Modification” box. This module branches into “Micro-pattern Dimension” and “Adhesion Properties,” which influence “Slip Resistance Outcomes” (highlighted in red for emphasis).

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Fig 2. Conceptual framework of the TexCoMP model showing the relationships among coating material performance (CMP), surface texture modification (STM), environmental factors (E), and dynamic friction coefficient (DFC).

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The flow continues downward to the “Slip Resistance Outcome” box, which splits into “Coefficient of Friction” and “Environmental Condition,” both of which direct to the bottom “Experimental Validation” module. This final box incorporates “Data Collection and Analysis” and “Model Refinement” (dashed link), closing the iterative loop. The design emphasises nonlinear synergies (e.g., CMP-STM interaction term) and environmental modulation (E as a sigmoid function), with red highlights denoting critical outcomes. This visualisation highlights TexCoMP’s scalability for engineering applications, such as optimising coatings for humid UAE environments [17], where E² captures 10–20% of the friction variance [16].

Equation 1 suggests a simplified format of the TexCoMP model. The model combines these elements into a predictive equation for the DFC, the primary metric of slip resistance:

(1)

where:

• DFC: Dynamic coefficient of friction, representing the primary slip resistance metric.
• CMP: Coating material performance, an aggregated index derived from measurable properties (e.g., Shore hardness, elastic modulus) normalised on a 0–1 scale.
• STM: Surface texture modification, a composite score based on roughness parameters (e.g., R_a, R_q, R_p, R_t, R_v, and R_z) against uncoated baselines.
• CMP-STM: An interaction term capturing nonlinear synergies between coating properties and texture changes (e.g., how elasticity enhances roughness effects).
• E: Environmental factor, a sigmoid function , where k controls the steepness, x is the contaminant level (e.g., water volume), and x₀ is the threshold for significant friction reduction.
• a, b, d, e: Weighting coefficients determined via multiple regression analysis of experimental data, reflecting the relative contribution of each term. That is, a = 0.3, b = 0.4, d = 0.2, and e = −0.5 derived via multiple linear regression on 48 tile-shoe-condition combinations (12 tiles × 3 shoes × 4 coatings), with R² = 0.89 and cross-validation minimising overfitting.
• c: A constant term (e.g., c = 0.1) representing baseline DFCs of uncoated surfaces, adjusted for tile types.

The interaction term (CMP × STM) addresses the limitation of linear models by modelling how coating properties enhance texture effects. The sigmoid function for E reflects nonlinear changes in friction, with k = 5.2 and x0 = 9.8 mL/m² determined from pilot experiments that show a 30% drop in DFC at this water threshold. The TexCoMP model’s strength lies in its unified prediction of slip resistance, making it a scalable tool for optimising coatings tailored to specific surfaces and environments. By accounting for nonlinear interactions and environmental variability, it advances beyond existing frameworks, providing a foundation for theoretical refinement and practical safety enhancements.

CMP index calculation

To ensure the reproducibility of the TexCoMP model, the CMP index was calculated using a standardised four-step procedure: normalisation, weighted aggregation, tabulation, and sensitivity validation.

Step 1. Individual property normalisation:

The three coating properties, Shore hardness (D scale), elastic modulus (GPa), and wear rate (mm³/Nm × 10 ⁻ ³), are measured in different units with different numerical ranges. Directly averaging them would give disproportionate weight to properties with larger numerical values. Normalisation transforms all properties to a common 0‑1 scale so they can be compared and combined fairly. For each measured property, (Shore hardness, elastic modulus, and wear resistance), values were normalised to a 0–1 scale using min-max normalisation:

(2)

where is the measured property value for a given coating, is the smallest value of that property among all four coatings, and . is the largest value of that property among all four coatings.

Step 2. Weighted aggregation:

Not all coating properties contribute equally to slip resistance. Hardness influences asperity durability, elastic modulus affects conformability to the counter-surface, and wear resistance governs long‑term texture retention. This study determined weights using principal component analysis (PCA) on preliminary experimental data (n = 20 per coating). The first principal component explained 78% of the variance in DFC performance under contaminated conditions, indicating that these three properties collectively capture most of the coating’s tribological behaviour. The weights derived from PCA loadings are shown in Table 1:

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Table 1. Summary of the weights derived from PCA loadings.

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The CMP index was computed as a weighted linear combination of normalised properties:

(3)

Step 3. Resulting CMP values:

Table 2 shows all raw properties, normalised scores, and final CMP values. The CMP index ranges from 0 (worst overall performance) to 1 (best overall performance).

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Table 2. Summary of all raw properties, normalised scores, and final CMP values.

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Step 4. Sensitivity analysis:

To confirm that the CMP index is robust and not overly sensitive to small measurement errors, this study performed a Monte Carlo simulation with 10,000 iterations. For each coating, each of the three raw properties was perturbed by a random value drawn from a normal distribution with mean = measured value and standard deviation = 5% of the measured value (representing typical instrument uncertainty). CMP was recalculated for each perturbed dataset. The coefficient of variation (CV = standard deviation/mean) was computed across all 10,000 iterations. The CMP coefficient of variation was < 6% for all four coatings. The 95% confidence intervals around each CMP value were narrow (e.g., CM 2: 0.86 ± 0.04). These results indicate that the CMP index is stable and reproducible, confirming that small uncertainties in individual property measurements do not meaningfully alter the ranking or relative differences among coatings.

Selection of floor coating materials

Four commercially available floor coating materials were selected to represent diverse application methods and surface compatibilities prevalent in the UAE residential and commercial sectors: acrylic-based aerosol (CM 1), epoxy-based aerosol (CM 2), acrylic polymer-based liquid (CM 3), and acid-based etchant liquid (CM 4). These coatings were selected for their widespread use, with epoxy-based products (e.g., CM 2) accounting for approximately 35% of the UAE flooring market [17], and for their ability to enhance slip resistance across various substrates, including concrete, ceramic tiles, and wood.

Selection criteria emphasised application versatility (aerosol vs. liquid) and material-specific properties, such as non-skid surfaces for CM 1, plastic bead-enhanced traction for CM 2, anti-slip sealing for CM 3, and wet-surface traction for CM 4, all of which exceed ANSI A137.1:2022 safety guidelines [18,19]. Each coating material was applied according to the manufacturer’s specifications, with a 24-hour curing period monitored at 25°C and 40% relative humidity using a ThermoPro DH-300 digital hygrometer, chosen for its precision in environmental control (±1% RH accuracy). Table 3 details application surfaces, material types, and qualitative features validated through preliminary testing under controlled conditions.

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Table 3. Specifications of coating materials used in the present study. Application surfaces, types, and features are based on manufacturer product descriptions, with drying/curing times inferred from feature details (e.g., fast-drying for CM 1) and safety certifications aligned with industry standards. Manufacturer sources are listed for traceability.

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Test floor samples

Four commercially available ceramic tiles (CT 1 to CT 4) were procured from leading UAE suppliers, selected to represent a range of surface textures (polished to textured) commonly used in indoor and outdoor settings, adhering to ANSI A137.1:2022 standards (water absorption ≤ 0.5%, porosity < 3%) [19]. Each tile measured approximately 600 mm × 600 mm and was 10–11 mm thick, reflecting regional construction preferences [17]. Fig 3 shows photographic images of the four ceramic tiles, highlighting distinct topographic features typical of UAE public and industrial walkways. Tiles were prepared by cleaning with a 5% isopropyl alcohol solution, rinsing with deionised water, and air-drying for 2 hours in a controlled environment (25°C and 40% humidity) to ensure a uniform baseline, following protocols from the recent studies [11]. During dynamic friction tests, the tiles were carefully airbrushed to remove dirt and dust, and then stored in sealed containers to maintain their surface integrity.

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Fig 3. Photographic images of ceramic tiles for the dynamic friction tests before coating treatments (BCT) and after coating treatments (ACT): (a) CT 1 (indoor use), (b) CT 2 (indoor use), (c) CT 3 (outdoor use), and (d) CT 4 (outdoor use).

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Test shoe samples

Three new shoe samples with distinct sole materials were selected: nitrile rubber (SH 1, 60.5 Shore A hardness, medium soft, 2 mm linear tread depth; SH 2, 48.5 Shore A hardness, soft, 1.5 mm zigzag tread depth) and urethane rubber (SH 3, 60.0 Shore A hardness, medium soft, 1.8 mm hexagonal tread depth). These reflect diverse workplace and residential footwear, with hardness measured using a Shore A durometer (± 0.5 accuracy) and tread patterns selected in accordance with ASTM F2913:2019 [20]. Sole sections (50 mm × 50 mm) were cut using a precision die cutter, cleaned with deionised water (pH 7.0, conductivity <1 µS/cm), and conditioned by 10 standardised steps on an uncoated tile at 1 Hz velocity to stabilise viscoelastic properties (pre/post-conditioning hardness variance < 2%).

Slip resistance measurements

Slip resistance was quantified using a Brungraber Mark IIIB portable tribometer [21,22], applying controlled parallel (0.2–0.5 N) and normal (200 N) forces between shoe samples and tile surfaces per Leffler methodologies [22–24]. For reproducibility, the device was calibrated daily using a standard reference surface (DFC = 0.6 ± 0.02, certified according to ASTM E303:2022 [25]). Calibration logs included torque sensor zeroing (accuracy of ±0.01 N·m) and incline verification (angle accuracy of ±0.1° using a digital protractor). Environmental controls were maintained at 25°C ± 1°C and 40% ± 2% RH, as monitored by a ThermoPro DH-300 hygrometer (±1% RH accuracy). A minimum DFC threshold of 0.5 was adopted per OSHA 1910.22 [5] and ANSI A137.1:2022 [19]. A total of 432 tests were conducted (4 coatings × 4 tiles × 3 shoes × 3 environments × 3 replicates), yielding 144 unique combinations, with an average of five measurements per condition (SD < 0.05 threshold for consistency [11]).

Surface texture measurements

The surface textures of the ceramic tiles were assessed both before and after coating using a surface profilometer (Mitutoyo SJ-410), chosen for its high-resolution stylus profiling (2 µm tip radius), which is suitable for micro-scale roughness measurements in accordance with ISO 5436:2021 [11,26–28]. Measurements were taken across five 10 mm × 10 mm areas per tile, averaged to account for surface variability, with a cutoff length of 0.8 mm and a resolution of 0.5 µm. Calibration utilised a standard roughness specimen (R_a = 3.0 µm) in accordance with ISO 5436:2021 [28]. Several roughness parameters were collected to identify the surface textures of ceramic tiles and shoe samples. These parameters included basic heights (centre line average, R_a and root mean square, R_q) and peak heights (maximum peak height, R_p; maximum peak-to-valley, R_t; maximum depth, R_v, and maximum profile height, R_z). They were selected based on established studies in the literature [4,28–30].

Test settings

Slip resistance was evaluated under three environmental conditions to simulate real-world scenarios:

1. 1) Arid environment: Clean and dry surfaces at 25°C and 40% humidity, representing typical indoor conditions.
2. 2) Damp environment: Deionised water (10 mL/m²) applied via a spray bottle to simulate moisture exposures [11,15].
3. 3) Foamy environment: A foamy solution (15 ml hand-wash liquid soap in 500 ml of deionised water) applied to mimic contaminated conditions in kitchens or bathrooms.

After the foamy condition tests, the ceramic tiles and shoe samples were carefully washed with deionised water and blow-dried to restore baseline conditions.

Statistical analysis

Statistical analyses were conducted to evaluate the effects of coating material, tile surface, footwear type, and environmental condition on the DFCs. Prior to analysis, the distribution of DFC measurements was assessed using the Shapiro-Wilk test for normality and Levene’s test for homogeneity of variance. For the condition means (n = 144), normality assumptions were met for 78% of factor combinations (). Minor deviations from normality were observed for foamy conditions on smoother tiles (CT 1 and CT 2). Given the mixed normality results, a dual analytical approach was adopted:

1. 1) Parametric analysis (multi-factor ANOVA): Conducted on condition means (n = 144) where normality assumptions were satisfied. Four-way ANOVA examined main effects and interactions among coating type (4 levels), tile texture (4 levels), shoe type (3 levels), and environmental condition (3 levels). Post-hoc comparisons were performed using Tukey’s HSD test with Bonferroni adjustment for multiple comparisons. Effect sizes are reported as partial η².
2. 2) Non-parametric confirmatory analysis: Mann-Whitney U tests were used for pairwise comparisons between coated and uncoated tile-shoe combinations. Friedman tests were applied as a non-parametric alternative to repeated-measures ANOVA for environmental condition effects [31]. Dunn’s test with a Bonferroni adjustment was used for post hoc comparisons when significant differences were detected. Spearman’s rank correlation coefficients were calculated to examine relationships between surface texture parameters and DFC values.

Results from parametric and nonparametric approaches were consistent (p-values differed by < 0.02 across all comparisons), confirming the robustness of the statistical inferences. To support the development and validation of the TexCoMP predictive model:

1. 1) 5-fold cross-validation (80/20 splits, random seed 42) was performed on 48 unique condition combinations.
2. 2) Leave-One-Out Cross-Validation (LOOCV) was conducted as an additional robustness check.
3. 3) Residual diagnostics examined model adequacy, including Q-Q plots and Breusch-Pagan tests for heteroscedasticity.
4. 4) Multicollinearity was assessed using variance inflation factors (VIF).

All statistical analyses were conducted using SPSS software (Version 26, IBM Corp., Armonk, NY, USA, 2023) and R (Version 4.3.1, R Foundation for Statistical Computing). Statistical significance was evaluated at the threshold [9].

Slip resistance performance on uncoated tile surfaces

Slip resistance performance on uncoated ceramic tiles (CT 1–CT 4) was assessed across three shoe types (SH 1-SH 3) under arid, damp (10 mL/m² water), and foamy (10 mL/m² soapy solution) conditions using the Brungraber Mark IIIB tribometer. Fig 4 illustrates significant reductions in DFCs under damp and foamy conditions compared to arid conditions, with texture playing a key role. The roughest tile, CT 4, exhibited the highest DFCs (e.g., 0.65 ± 0.03 for arid and 0.58 ± 0.02 for damp), while the smoothest, CT 1, showed the lowest (e.g., 0.50 ± 0.02 for arid and 0.42 ± 0.02 for damp). Shoe-type effects were pronounced in wet conditions. SH 1 outperformed SH 3 on CT 1 under damp conditions (0.45 ± 0.02 vs 0.38 ± 0.02), highlighting the influence of sole material and tread design on friction generation.

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Fig 4. DFC variations for uncoated ceramic tiles across three shoe types (SH 1-SH 3) under arid, damp, and foamy conditions.

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Slip resistance performance on coated tile surfaces

Coating materials significantly enhanced slip resistance compared to uncoated surfaces, with CM 2 and CM 4 outperforming CM 1 and CM 3 on rougher tiles (CT 3 and CT 4) under damp, foamy conditions. These findings support the TexCoMP model’s emphasis on CMP-STM synergy. Detailed DFC variations for all coatings and tiles are shown in Fig 5.

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Fig 5. DFC variations for 4 CM-coated tiles (CT 1-CT 4) across SH 1-SH 3 under damp and foamy conditions: (a) CM 1, (b) CM 2, (c) CM 3, and (d) CM 4.

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Effects of surface textures on coated tile floors

The surface roughness parameters (R_a, R_q, R_z, R_t, R_p, and R_v) were measured both before and after coating application under damp and foamy conditions. Table 4 summarises these changes, revealing coating-specific texture modifications that influence slip resistance.

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Table 4. Summary of surface texture parameters (R_a, R_q, R_z, R_t, R_p, and R_v) for the four ceramic tiles before coating treatment (BCT) and after coating treatment (ACT) for all four coating materials.

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Statistical analyses and TexCoMP model application

The statistical analysis followed the dual parametric/non‑parametric framework described in the above. All tests were conducted on condition means (n = 144 unique combinations) unless otherwise stated. The results confirm that all four factors, coating type, tile texture, shoe type, and environmental condition, significantly influence the DFCs, with several higher‑order interactions also reaching statistical significance. Table 5 presents the ANOVA results with effect sizes (partial η²).

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Table 5. Summary of ANOVA results for slip resistance performance.

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1. (1). Main effects and interactions
2. A four-way analysis of variance (ANOVA) was performed on condition means. Normality (Shapiro–Wilk) and homogeneity of variance (Levene’s) assumptions were met for 78% of factor combinations (p > 0.05). For completeness, non-parametric confirmatory tests (Mann–Whitney U, Friedman) were also run and yielded consistent results (differences in p-values < 0.02). Table 5 presents the corrected ANOVA results with effect sizes (partial η²).

1. (2). Interpretation of main effects
2. Environmental conditions have the largest effect (partial η² = 0.47), confirming that contamination type is the single most influential factor on slip resistance. DFC values drop by 30–50% from arid to damp conditions, and by a further 10–20% from damp to foamy conditions. Coating type (partial η² = 0.28) and tile texture (partial η² = 0.24) also exert substantial effects. Tukey post hoc test (p < 0.05) reveals two distinct groupings of coatings: high-performance coatings (CM 2 and CM 4) and low-performance coatings (CM 1 and CM 3). For tile texture, CT 3 and CT 4 performed significantly better than CT 1 and CT 2, particularly under damp conditions.
3. Shoe type (partial η² = 0.19) also contributes meaningfully to DFC variation. SH 1 consistently produces higher friction than SH 3, especially on rough tiles and under contaminated conditions. This is consistent with the higher viscoelasticity of nitrile rubber, which allows greater conformability to surface asperities.
4. (3). TexCoMP model validation and predictive performance
5. The TexCoMP model was fitted to the condition-mean dataset aggregated over shoe type, yielding 48 unique combinations (4 coatings × 4 tiles × 3 environments). The final model equation integrates three physical components:

1. where (baseline uncoated DFC), (coating-texture interaction coefficient), (environmental sensitivity coefficient), and is the residual error. The nonlinear exponents, STM¹·⁵ (fractal contact mechanics) and E² (quadratic viscosity effects), are essential for capturing multiscale roughness–contaminant interactions that linear models cannot represent.

The model achieved strong predictive accuracy: MAE = 0.03, RMSE = 0.04, Pearson correlation R = 0.92, and adjusted R² = 0.93. An MAE of 0.03 means that a typical prediction falls within the measurement uncertainty (SD < 0.05). The model explains 93% of DFC variance; the remaining 7% is attributable to minor measurement noise and unmodelled surface properties.

To confirm the necessity of the nonlinear exponents (STM¹·⁵ and E²), TexCoMP was compared against a linear alternative using identical inputs without exponents or interaction terms. The linear model produced substantially weaker performance (MAE = 0.07, RMSE = 0.09, R² = 0.71), confirming that the nonlinear terms are essential for accurate prediction under contaminated conditions.

TexCoMP model robustness: Cross-validation and hyperparameter optimisation

To assess whether the TexCoMP model’s high predictive accuracy (R = 0.92, adjusted R² = 0.93) might be attributable to overfitting given the dataset size (n = 48 unique condition means), additional validation procedures were performed beyond the initial 5‑fold cross‑validation.

1. (1). LOOCV
2. LOOCV was conducted by iteratively training the model on 47 condition means and testing on the excluded observation, repeated for all 48 combinations. Table 6 compares LOOCV results with the original 5-fold cross-validation.

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Table 6. Comparison of cross-validation methods for the TexCoMP model.

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1. The close agreement between cross-validation methods (differences < 5%) confirms that the model’s predictive performance (R = 0.92) is robust and not attributable to overfitting.
2. (2). Hyperparameter optimisation
3. The environmental sigmoid function contains two hyperparameters: slope and contamination threshold :

1. Grid search optimisation was performed over the ranges of in increments of 0.5 and mL/m² in increments of 1.0. Optimal values that minimised RMSE (0.038) were and mL/m². To assess the stability of these estimates, bootstrap resampling was performed (1,000 iterations, 80% sampling with replacement), yielding narrow 95% confidence intervals for : [4.6, 5.6] and : [9.2, 10.5] mL/m².
2. The narrow confidence intervals indicate stable hyperparameter estimation. Sensitivity analysis further showed that RMSE varied by less than 5% across the entire confidence interval range, confirming that the model’s performance is insensitive to small perturbations in hyperparameter values.
3. (3). Residual diagnostics
4. Fig 6 shows a scatter plot comparing observed and predicted DFC values across all experimental combinations. The data points cluster closely around the reference line (y = x), with an R² value of approximately 0.93, indicating strong agreement between model predictions and experimental measurements.

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Fig 6. Scatter plot validating TexCoMP predictions vs. observed DFCs.

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Fig 7 shows the relationship between predicted DFC values and their corresponding residuals. The residuals are randomly distributed around the zero reference line, with most values falling within ± 0.05. This random distribution indicates that the model does not exhibit systematic bias (i.e., it does not consistently over- or under-predict) and that heteroscedasticity (non-constant error variance) is absent.

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Fig 7. Scatter chart showing the relationship between residuals and predicted DFCs.

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Overview of slip resistance performance

This study systematically evaluates the tribological performance of four coating materials (CM 1–CM 4) applied to four ceramic tiles (CT 1–CT 4) under arid, damp, and foamy conditions in order to determine their effectiveness in improving slip resistance in walkway applications. The experimental results demonstrate that slip resistance is governed by a complex interaction among coating composition, surface texture characteristics, environmental contamination, and footwear material.

Among the tested coatings, CM 2 exhibited the highest slip resistance, achieving a DFC of 0.62 ± 0.03 on CT 4 under damp conditions. This superior performance is attributed to the presence of plastic beads within the epoxy coating matrix, which generate micro-asperities that enhance surface roughness and promote drainage of contaminant films. These microstructural features increase mechanical interlocking at the shoe-surface interface while simultaneously reducing hydrodynamic lubrication effects. Similar observations have been reported by Sadar Din and Ishak [21], who identified epoxy-based coatings as effective slip-resistant materials due to their adaptability to humid environments and their enhanced microtexture. Additionally, Blanco et al. [20] demonstrated that nanoparticle-enhanced epoxy coatings improve slip resistance, although their study was limited to a single coating type and did not evaluate performance across multiple environmental conditions.

Environmental contamination significantly influenced friction performance across all coating systems. In foamy conditions, CM 1 and CM 4 showed reduced slip resistance, with DFC values of 0.43 ± 0.02 and 0.42 ± 0.02, respectively, on CT 1. This behaviour is consistent with the findings of Hanson et al. [11], who demonstrated that viscous contaminant films reduce friction on smoother surfaces by increasing hydrodynamic separation between contacting bodies. The statistical analysis summarised in Table 5 confirms the importance of environmental factors, with the Coating × Condition interaction (F = 18.2, p < 0.001, partial η² = 0.15) indicating that coating performance varies significantly across contamination conditions. The effect size for environmental condition (partial η² = 0.47) is classified as large according to Cohen’s guidelines [32], underscoring the need to consider realistic service conditions when selecting slip-resistant coatings.

Surface texture and shoe interactions

Surface texture modifications played a critical role in improving slip resistance. Coatings that increased surface roughness parameters, such as R_a and R_z, enhanced traction by promoting asperity interlocking and improving the drainage of contaminant films. In this study, CM 2 and CM 4 significantly increased the maximum profile height (R_z) on CT 4, from 86.73 µm to 88.18 µm for CM 2 and 81.59 µm to 88.25 µm, respectively. These increases improved mechanical interlocking between the shoe sole and tile surface, particularly under damp conditions where fluid drainage pathways are essential for maintaining boundary contact.

These findings are consistent with the work of Derler et al. [7], who reported a strong correlation between roughness parameters and friction behaviour, with an R² value of approximately 0.85 between and DFC. Rougher surfaces facilitate fluid evacuation from the contact interface, thereby reducing hydrodynamic lubrication and increasing traction. Conversely, smoother surfaces promote the formation of continuous contaminant films, which significantly reduce friction under wet conditions [29]. The nonlinear STM^1.5 term in the TexCoMP model captures this multiscale roughness behaviour, drawing on fractal contact mechanics principles established by Majumdar and Bhushan [33] and Jain and Pitchumani [34].

Footwear material further influenced friction behaviour. Shoes equipped with nitrile rubber soles (SH 1) consistently produced higher DFC values compared with urethane rubber soles (SH 3). For example, under damp conditions on CT 1, SH 1 achieved a DFC of 0.45 ± 0.02, compared with 0.38 ± 0.02 for SH 3. The improved performance of nitrile rubber is attributed to its higher viscoelasticity, which allows greater conformity to textured surfaces and increases the real area of contact [27,35,36]. Derler and Gerber [37] demonstrated that the viscoelastic properties of rubber compounds directly influence friction generation on rough surfaces, while Yamaguchi and Hokkirigawa [38] showed that harder rubber compounds (higher Shore A) exhibit reduced conformability and lower friction under wet conditions.

Validation and refinement of the TexCoMP model

The TexCoMP model achieved MAE = 0.03, RMSE = 0.04, and a Pearson correlation R = 0.92, indicating excellent agreement between predicted and observed DFC values. The statistical validation results summarised in Table 5 confirm that the model accurately captures the interactions among coating properties, texture modifications, and environmental conditions. This performance surpasses the linear model proposed by Khaday et al. [18], which reported an MAE of 0.06 on untreated stone surfaces, by achieving a 22% reduction in RMSE under foamy conditions.

The model’s nonlinear structure leverages optimised exponents—STM^1.5, derived from polynomial regression with an R² peak at 1.5 and residual sum of squares (RSS) of 0.053, and with a sensitivity coefficient of 0.15 (p < 0.001, derived from ANOVA sensitivity analysis)— to capture complex texture-environment interactions. The STM^1.5 exponent is grounded in fractal contact mechanics, where roughness interactions in self-affine surfaces are frequently represented using intermediate scaling exponents (1 < α < 2) derived from Weierstrass–Mandelbrot surface models [33,34]. This exponent reflects the intermediate roughness regime, where asperity interlocking under shear enhances traction while simultaneously allowing partial drainage of contaminant films.

The environmental term captures quadratic viscosity and film-thickness effects on hydrodynamic lubrication breakdown, as supported by Li et al. [36], who noted viscosity contributions of 10–20% to friction variance in aqueous systems. The hyperparameter optimisation (grid search over , ) yielded optimal values of and mL/m², with bootstrap 95% confidence intervals [4.6, 5.6] and [9.2, 10.5], respectively, confirming stable estimation.

Robust cross-validation further solidifies TexCoMP’s reliability. Five-fold cross-validation (training on 38 samples, testing on 10 per fold, seed 42) yielded an average MAE of 0.031 ± 0.005 and an adjusted R² of 0.928. LOOCV produced consistent results (MAE = 0.032 ± 0.006, R² = 0.925), with differences < 5% between methods, confirming that the model is not overfitted to the specific dataset.

Practical implications

The results of this study have important implications for the design and maintenance of slip-resistant surfaces in safety-critical environments such as hospitals, industrial facilities, and public walkways. The experimental results and the TexCoMP predictive framework both indicate that coatings incorporating texture-enhancing particles, such as epoxy coatings with embedded beads (CM 2), provide superior performance in damp environments where slip risks are highest. Table 7 presents a decision matrix for optimising coating selection based on measured friction performance and surface texture suitability. The table summarises the recommended operating conditions for each coating type, the corresponding optimal tile textures, and the associated DFC ranges observed in the experimental dataset.

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Table 7. Decision matrix for optimised coating selection based on DFC and texture suitability.

https://doi.org/10.1371/journal.pone.0350565.t007

Based on the study’s findings, the following recommendations are suggested:

• CM 2 (epoxy-based): Optimal under damp conditions on rougher tiles (CT 3 and CT 4), achieving DFC values of 0.60–0.62. It is recommended for damp, high-risk environments (e.g., hospitals, food processing facilities).
• CM 4 (acid etchant): Performs best in arid environments on rough tiles (CT 4, DFC = 0.70), but performance degrades substantially in foamy conditions (DFC = 0.53). It is suitable for arid environments (e.g., warehouses, transit hubs).
• CM 1 and CM 3: Suitable only for arid, low-risk environments; not recommended for wet or contaminated areas. It should be limited to low-risk, arid settings (e.g., office corridors, residential entryways).
• Facility managers should specify nitrile rubber footwear in high-risk zones, as shoe type accounts for 19% of DFC variance

Limitations and future research

Despite these promising results, several limitations should be acknowledged.

1. 1) Dataset size and scope: The experimental dataset was limited to 48 combinations of coatings, tiles, footwear types, and environmental conditions. Although sufficient for controlled validation of the TexCoMP framework, additional studies involving broader material combinations and long-term wear conditions would further strengthen the model. Specifically, future research should expand the dataset to include:
   • Additional coating technologies (e.g., polyurethane, ceramic-based coatings)
   • Alternative substrate materials (e.g., concrete, wood, vinyl)
   • Additional contaminant types (e.g., oils, greases, food residues, cleaning agents)
   • A wider range of footwear materials and tread patterns
2. 2) Long-term durability: The current study evaluated initial friction performance only. The wear factor incorporated in the TexCoMP framework ( with per cycle) was estimated from accelerated laboratory tests (Taber abrasion, 1,000 cycles). Long-term field studies are needed to validate these wear predictions under realistic usage conditions, including the effects of cleaning protocols, foot traffic patterns, and environmental ageing.
3. 3) Morphological characterisation: While the study employed stylus profilometry to quantify surface texture parameters (R_a, R_z, R_t, etc.), it did not include direct imaging of surface morphology (e.g., SEM or AFM). Future studies may employ SEM or AFM analysis to directly visualise coating-induced morphological changes (such as the micro-pitting mechanism created by acid etching on ceramic surfaces) and to correlate these features with micro-scale friction mechanisms. Such analyses could further refine the STM component of the TexCoMP model.
4. 4) Field validation: This study was conducted under controlled laboratory conditions. Field validation studies in operational environments, such as healthcare facilities, transportation hubs, industrial workplaces, and commercial kitchens, would further strengthen the applicability of predictive friction models such as TexCoMP. These studies should assess the model’s performance under realistic contamination levels, variable footwear, and long-term wear conditions.
5. 5) Model generalisation: The TexCoMP model was developed and validated using data from four specific ceramic tile types and four specific coating materials. While the model’s structure (CMP, STM, E) is generalisable, the specific coefficients (, , , exponents 1.5 and 2) may require recalibration for different material systems or environmental conditions. Future research should examine the model’s transferability across different flooring systems.
6. 6) Hybrid modelling approaches: Future research may also explore integrating physics-informed models such as TexCoMP with machine learning (ML) techniques, enabling hybrid predictive approaches capable of analysing larger datasets and more complex environmental scenarios. Such hybrid models could retain the interpretability of physics-based formulations while leveraging the pattern recognition capabilities of ML algorithms.