Dispersion quality of graphene nanoplatelets in cementitious materials with applications in geotechnical ground improvement, a machine learning based prediction model Open Access

The implications of graphene nanoplatelet (GNP) dispersion characterisation are significant for geotechnical applications, including soil stabilisation, deep foundations and ground improvement techniques (Taha and Taha, 2012). GNP-reinforced cement grouts must maintain consistent dispersion quality to ensure predictable strength development, permeability reduction and durability in challenging subsurface environments. The ability to predict and monitor dispersion quality through absorbance measurements becomes particularly critical for jet grouting, deep soil mixing and diaphragm wall construction, where in situ quality control is challenging and material heterogeneity can compromise structural integrity (Correia et al., 2013). Furthermore, complex interactions with soil matrices involving ion exchange, pH variations and groundwater chemistry necessitate robust characterisation methods.

The predictive use of UV–Vis absorbance extends beyond a simple concentration read-out; time-resolved spectra are routinely used to quantify aggregation kinetics and track colloidal stability, while plasmon peak shifts/broadening report on surface-functionalisation efficiency (Ezra et al., 2020; Amendola and Meneghetti, 2009).

In cement-based materials, the dispersion quality of GNPs often monitored via absorbance governs the resulting mechanical and multi-functional performance of the composite (Zhou et al., 2022; Metaxa et al., 2021).

While the Beer–Lambert law provides a first-order link between absorbance and concentration, nanoparticle dispersions frequently violate its assumptions because scattering, aggregation and wavelength-dependent dielectric effects contribute to the measured extinction; separating absorption from scattering and/or modelling with Mie/Gans is often required (Backes et al., 2016; Amendola and Meneghetti, 2009).

Traditional empirical fits struggle to represent nanoparticle absorbance because the extinction spectrum varies non-linearly with particle size/shape/composition and, for two-dimensional nanosheets, with thickness and lateral size so physics-informed models are generally required (Jain et al., 2006; Backes et al., 2016).

For GNPs, optical response depends on nanosheet thickness and lateral size and is further altered by surfactants/functionalisation and processing history, which UV–Vis can track during dispersion and ageing (Backes et al., 2016; Xu et al., 2018).

Reproducible absorbance across labs and instruments requires calibrated wavelength scales and control of temperature; NIST documents show how uncertainties and inter-instrument differences arise without such controls [National Institute of Standards and Technology (NIST), 2004; Travis et al., 2003]. Reliable prediction models therefore have practical quality control value for nano-enhanced cementitious materials: absorbance-based monitoring of GNP dispersion has been used to optimise surfactant dosage and mixing/ultrasonication parameters and link dispersion to mechanical and sensing performance (Zhou et al., 2022; Xu et al., 2018).

Interpreting absorbance and correlating it with final properties benefits from models that capture scattering, polydispersity and wavelength-dependence while remaining computationally efficient, for example, Mie/Gans-based or spectroscopic-metric approaches that reconstruct size distributions directly from UV–Vis spectra (Car and Krstulović, 2022; Backes et al., 2016). Accurate absorbance prediction is thus a cross-cutting need from fundamental dispersion studies to industrial quality control; without it, process optimisation and consistent quality are difficult, especially as construction moves towards sustainable, high-performance concretes that rely on well-dispersed graphene-based nanomaterials (Metaxa et al., 2021).

Absorbance modelling enables accurate control of nanomaterial dispersion, supporting the design of GNP-enhanced cementitious composites with expected properties, because dispersion degree can be quantified by absorbance and linked to mechanical and sensing performance (Zhou et al., 2022). In structural engineering contexts, such models help relate optical measures (UV–Vis absorbance of the mix water and suspension) to mechanical performance after casting, enabling non-destructive quality checks during production and mix optimisation (Wang et al., 2016).

In industrial settings, real-time monitoring and control rely on accurate interpretation of spectroscopic signals; in-line UV–Vis process analytical technology implementations demonstrate how continuous processes can stabilise quality while reducing waste through immediate detection of dispersion or process deviations (Schlindwein et al., 2018; Glader et al., 2025). For GNP-cement manufacturing, predictive models fed by absorbance streams can flag dispersion anomalies early preventing downstream failures and ensuring compliance with performance specifications (Schlindwein et al., 2018).

In research and development, accurate absorbance models let researchers screen large formulation spaces (GNP loading, surfactant and sonication) with far fewer wet-lab trials by predicting/relating UV–Vis responses to inputs (McNaughton et al., 2023; Zhou et al., 2022). This computational screening shortens development and enables rapid prototyping of specialised concretes by using optimisation to balance performance versus sustainability objectives during mix design. Environmentally, predictive models help dose expensive nanomaterials precisely, minimising waste while meeting target properties; in practice, absorbance-guided optimisation of GNP dispersion/dosage improves performance with fewer iterations (Zhou et al., 2022; Ament et al., 2023).

Traditional absorbance prediction approaches are limited for complex GNP cementitious systems because the Beer–Lambert law assumes homogeneous distributions, negligible scattering and the absence of inter-particle interactions conditions often violated in nanoparticle dispersions. Linear regression models, though simple, cannot capture the non-linear absorbance behaviour in GNP dispersions caused by concentration-dependent aggregation, nanosheet size/thickness variations and time-dependent settling. Polynomial regressions offer more flexibility but often overfit and lack physical interpretability. Because UV–Vis spectra reflect coupled absorption and scattering, physics-aware models are required (Backes et al., 2016; Mayerhöfer et al., 2023).

Machine learning approaches, particularly artificial neural networks (ANNs), have shown promise in capturing complex relationships but present significant challenges for practical implementation in industrial settings. The black-box nature of ANNs prevents understanding of how input parameters influence predictions, limiting their utility for process optimisation and troubleshooting (Javadi and Rezania, 2009).

Recent studies on GNP dispersion in cementitious composites have used one factor at a time (OFAT) approaches to optimise processing parameters. Du and Pang (2018) used OFAT methodology to determine optimal superplasticizer dosage and sonication time, systematically varying parameters to identify critical thresholds for stable GNP suspension.

The lack of standardised modelling frameworks for nanomaterial optical properties has led to fragmented approaches across research groups, hindering knowledge transfer and validation. While machine learning methods such as support vector machine and random forests improve accuracy over simple regressions, they still lack transparency, require extensive hyperparameter tuning and provide only point predictions without closed-form relations limitations that complicate their integration into quality control and uncertainty quantification (Linardatos et al., 2021; Wang, 2024).

The primary objective of this study is to develop and validate transparent, physically interpretable models for predicting the UV–Vis absorbance of GNP dispersions using evolutionary polynomial regression (EPR), while investigating the complex parameter interactions that govern optical behaviour in these systems. This research addresses a critical gap in understanding how processing parameters influence GNP dispersion quality for cementitious composite applications. The specific objectives are as follows:

To develop and validate an EPR-based predictive model that captures the relationships between absorbance and six key parameters: GNPs loading (L: 0.135–0.9 g), water content (U: 225–300 mL), superplasticizer dosage (A: 0–3.75), sonication time (B: 5–120 min), concentration (F: 0.006–0.04) and holding time (G: 1–120 days). The developed model yields an explicit mathematical expression that reveals parameter significance through their presence and exponent values in the final equation, providing insights into dominant mechanisms controlling absorbance behaviour.

To evaluate the differential influence of processing parameters on absorbance prediction accuracy, particularly investigating why certain parameters (such as superplasticizer) show minimal contribution in the EPR model despite their known importance in dispersion chemistry. This investigation includes analysing why terms containing superplasticizer (A) yield zero coefficients in the EPR formulation, suggesting either weak correlation with absorbance under the tested conditions or complex interactions not captured by the polynomial structure.

To assess model performance and generalisation capability through comparison of EPR predictions with laboratory measurements, demonstrating the model’s ability to achieve close agreement between predicted and experimental absorbance values. The analysis includes identifying conditions where the model performs optimally and those where predictions deviate, providing insights into model limitations and areas requiring refinement.

To conduct systematic sensitivity analysis using the developed EPR model to understand individual parameter influences on absorbance. This involves varying single parameters while maintaining others at average values, revealing which parameters exhibit predictable linear or non-linear responses and which show unexpected behaviour. The sensitivity analysis helps identify critical control parameters for industrial implementation and highlights regions of parameter space requiring additional investigation.

To identify limitations and improvement opportunities in the current EPR modelling approach, particularly addressing why sensitivity analysis yields mixed results for certain parameters. This includes investigating whether additional input variables, alternative functional forms or multi-stage modelling approaches could better capture the complex physics of GNPs dispersion and stability.

Through achieving these objectives, this study demonstrates both the capabilities and current limitations of EPR for modelling GNPs optical properties. The research provides valuable insights into which processing parameters most significantly influence absorbance (GNP loading, water content, sonication time and holding time) while revealing challenges in capturing superplasticizer effects. These findings advance understanding of GNPs dispersion behaviour and provide a practical framework for optimising processing conditions in cement composite applications, while identifying specific areas where further model development is needed.

The fundamental absorption mechanism in GNPs involves interband electronic transitions, where photons promote electrons from the valence band to the conduction band. For visible and near-UV wavelengths, these transitions occur predominantly between π and π* orbitals, with the absorption coefficient dependent on the joint density of states and transition matrix elements (Bonaccorso et al., 2010). The sp² carbon network in graphene creates delocalised π-electrons with strong light–matter coupling, giving rise to characteristic absorption/extinction features whose magnitude and spectral shape vary systematically with platelet thickness (layer number) and lateral size (Backes et al., 2016).

The absorbance behaviour of GNP suspensions depends on numerous interrelated factors spanning multiple length scales and time scales. Understanding these dependencies is crucial for developing predictive models and optimising dispersion protocols.

The optical absorption of graphene is closely related to the number of layers. A single monolayer absorbs about 2.3% of incident visible light, and the absorbance increases approximately linearly with the number of layers for few-layer graphene. For thicker multi-layer graphene, deviations from this linear behaviour occur due to interlayer interactions (Zhu et al., 2010).

Sonication time and power critically influence dispersion quality by overcoming van der Waals attractions between platelets. Extended sonication can induce platelet fragmentation, reduce average lateral size and shifting absorption spectra, while insufficient sonication leaves aggregates that scatter light and produce artificially high absorbance readings (Khan et al., 2010).

In aqueous dispersions, surfactants adsorb on graphene surfaces and provide stabilisation either through electrostatic repulsion (ionic surfactants) or steric effects (non-ionic surfactants). Non-ionic surfactants, such as P-123 and Brij 700, were particularly effective due to their hydrophilic polyethylene oxide chains extending into water, which prevent reaggregation of graphene platelets. Surfactant concentrations were maintained above the critical micelle concentration to ensure dispersion stability, although excess surfactant or aromatic surfactants could contribute to additional absorption or scattering in UV–Vis spectra (Guardia et al., 2011).

Contemporary approaches for predicting GNP absorbance range from empirical correlations to sophisticated theoretical models, each with distinct advantages and limitations for practical applications.

Rigorous electromagnetic modelling using Mie theory provides wavelength-dependent extinction cross-sections for spherical particles, adapted for platelets using effective medium approximations. The T-matrix method extends this to non-spherical geometries, computing orientation-averaged extinction for randomly oriented platelets (Mishchenko et al., 2002). While theoretically sound, these approaches require detailed knowledge of GNP optical constants, size distributions and assume independent scattering, limiting accuracy for concentrated suspensions where multiple scattering occurs.

First principles approaches such as density functional theory provide accurate electronic structure information but remain computationally demanding and are limited to relatively small, idealised graphene systems, far from the mesoscale conditions of real suspensions (Trevisanutto et al., 2008).

The diversity of current methods reflects the challenge of balancing accuracy, interpretability and computational efficiency. While sophisticated theoretical approaches provide insights into fundamental mechanisms, they often fail to account for the complex, history-dependent behaviour of real GNPs suspensions. Conversely, empirical models lack the flexibility to extrapolate beyond their training conditions. This methodological gap motivates the development of hybrid approaches like EPR that combine physical interpretability with data-driven flexibility.

EPR represents a hybrid data-driven technique that integrates numerical regression with genetic programming to construct explicit polynomial models from experimental data (Giustolisi and Savic, 2006). Unlike conventional regression methods that require a priori specification of model structure, EPR simultaneously optimises both the form and parameters of mathematical expressions through evolutionary computation. The fundamental principle underlying EPR is the decomposition of the modelling task into two complementary optimisation problems:

1. combinatorial optimisation to identify optimal model structures using genetic algorithms; and

2. parameter estimation via least squares to determine coefficients for each candidate structure (Rezania et al., 2008).

The EPR methodology has proven particularly valuable in geotechnical engineering, where it has been successfully applied to predict settlement of shallow foundations (Rezania and Javadi, 2007), model soil constitutive behaviour (Javadi and Rezania, 2009), evaluate liquefaction potential (Alavi and Gandomi, 2011) and assess slope stability (Ahangar-Asr et al., 2010). The technique’s ability to handle the inherent uncertainty and variability in geomaterials while providing transparent mathematical expressions has made it a preferred choice over black-box methods in geotechnical practice. This proven track record in capturing complex soil-structure interactions and multi-parameter dependencies in geotechnical systems provides confidence in its application to similarly complex nanomaterial-cement systems.

The EPR paradigm constructs models following the general form:

Equation (1) presents the general form used by the learning process during the model development stage by EPR to construct the output model. In this equation, y represents the target output (absorbance), n denotes the number of polynomial terms, aj are numerical coefficients, X_i represent input variables (L, U, A, B, F and G in this study), ES(j,i) defines the exponent structure matrix containing integer or rational exponents, f represents optional transformation functions (logarithmic, exponential and trigonometric) and a₀ is the bias term (Giustolisi and Savic, 2006). The exponent structure matrix ES serves as the genetic code that evolves through successive generations, encoding which variables appear in each term and their corresponding powers.

The evolutionary search process begins with a population of random polynomial structures, each representing a potential model. Through iterative application of genetic operators selection, crossover and mutation the population evolves towards increasingly fit solutions. Selection pressure favours models with superior predictive accuracy while penalising excessive complexity, implementing the principle of parsimony fundamental to scientific modelling (Goldberg, 1989). Crossover operations exchange substructures between parent models, exploring combinations of successful building blocks, while mutation introduces random perturbations that maintain population diversity and prevent premature convergence to local optima.

EPR offers several critical advantages over alternative machine learning approaches for modelling physical systems, particularly in materials science applications where interpretability and reliability are paramount. The most significant advantage is the generation of explicit, transparent mathematical expressions that directly reveal relationships between variables. Unlike neural networks that encode knowledge in weight matrices or SVMs that rely on kernel transformations, EPR produces human-readable equations amenable to analytical manipulation, sensitivity analysis and theoretical interpretation (Javadi and Rezania, 2009).

The transparent nature of EPR models facilitates scientific insight by exposing dominant physical mechanisms through the structure of selected terms. In the developed GNP absorbance model, the presence of L²U⁻²B^−0·5F²G^0·5 with its large negative coefficient immediately indicates that absorbance decreases with dilution (U⁻²), increases with GNP concentration squared (L²F²) and exhibits complex time-dependent behaviour (G^0·5). Such insights are impossible to extract from neural network weights or support vector coefficients, limiting their utility for process optimisation and troubleshooting (Ahangar-Asr et al., 2010).

EPR demonstrates superior generalisation capabilities compared to conventional machine learning methods when extrapolating beyond training data ranges. The polynomial structure constrains predictions to follow smooth, continuous trends consistent with underlying physics, avoiding the erratic behaviour often exhibited by neural networks in extrapolation regions.

The multi-objective optimisation framework inherent to EPR clearly balances model accuracy against complexity, addressing the fundamental bias variance trade-off in machine learning. By maintaining a Pareto front of solutions ranging from simple, robust models to complex, highly accurate formulations, EPR enables practitioners to select models appropriate for their specific accuracy requirements and implementation constraints. This flexibility proves invaluable in industrial settings where simple models may suffice for routine quality control while complex models serve research and development needs (Giustolisi and Savic, 2009).

The mathematical foundation of EPR rests on the systematic construction and evaluation of polynomial basis functions combined through linear superposition (Giustolisi and Savic, 2006). For the GNP absorbance modelling problem with six input variables (L, U, A, B, F and G), the complete model space encompasses all possible polynomial combinations:

Equation (2) shows a possible general representation of the output model structure to predict absorbance. In this equation, the exponent structure matrix ES$∈$ ℝ^nx6 defines the model topology. Each row of ES specifies one polynomial term, with entries indicating the power to which each variable is raised. The search space grows combinatorially with the number of variables and allowable exponent values, necessitating intelligent search strategies rather than exhaustive enumeration.

Parameter estimation for a given model structure uses ordinary least squares to minimise the sum of squared errors (SSE) (Giustolisi and Savic, 2006):

Equation (3) is used to calculate the SSE. In this equation, N represents the number of training samples, y_k denotes the measured absorbance for sample k, ŷ_k is the model prediction and φj(x_k) represents the j-th basis function evaluated at input x_k. The linear-in-parameters structure enables analytical solution via the normal equations:

Equation (4) shows the relation between the design matrix, coefficient vector and the target outputs. In this equation, Φ $∈$ ℝ^{Nx(n + 1)} is the design matrix with entries Φ_kj = φ_j(x_k) and “a” contains the coefficient vector [a₀, a₁, …, a_n]^T.

Model fitness evaluation uses multiple criteria to assess different aspects of model performance. The coefficient of determination (CoD) quantifies the proportion of variance explained (Giustolisi and Savic, 2006):

Equation (5) is used to calculate the coefficient of determination value. In this equation, SST represents the total sum of squares and ȳ denotes the mean absorbance. For the developed GNP model achieving CoD > 0.90, this indicates that over 90% of absorbance variation is captured by the identified polynomial terms.

GNPs with an average thickness of 8–12 nm were dispersed in water using a combination of chemical and mechanical dispersion methods. The experimental matrix encompassed six critical parameters: GNP loading (L: 0.135–0.9 g), water volume (U: 225–300 mL), polycarboxylate-based superplasticizer dosage (A: 0%–3.75% by weight of GNPs), ultrasonication time (B: 5–120 min), final GNP concentration (F: 0.006–0.04 g/mL) and holding time after preparation (G: 1–120 days). Ultrasonication was performed using Fisherbrand Model 505 (500 W, 20 kHz) in an ice bath to prevent thermal degradation.

UV–Vis absorbance measurements were conducted using an Evolution 201 UV–Vis spectrometer. All measurements were performed in 10 mm path length quartz cuvettes with appropriate dilutions to maintain absorbance within the linear detection range (0.1–3.0).

The EPR analysis was implemented using EPR-SA v2.0 software (Giustolisi and Savic, 2006) with the following configuration: exponent range [−2, 2] with 0.5 increments. The data set was partitioned into training (75%) and validation (25%) sets using stratified random sampling to ensure representative parameter distributions in both subsets.

The multi-objective search produced a Pareto front of 12 non-dominated models (CoD = 88.7%–99.4%). Based on the knee-point criterion balancing accuracy and parsimony, this model was selected:

Equation (6) is the developed EPR model relating intrinsic connections between implemented input parameters to predict absorbance. This five-term polynomial expression directly reveals parameter relationships through the structure of selected terms and their coefficients, achieving R² = 0.99 on training data and demonstrating EPR’s capability to balance model complexity with predictive accuracy.

The developed EPR model reveals distinct parameter hierarchies and interaction patterns governing GNP absorbance behaviour. Analysis of term coefficients and exponent structures provides insights into the dominant physical mechanisms.

Figure 1 presents the comparison between EPR model predictions and experimental measurements for sonication time ranging from 5 to 120 min. The model demonstrates good predictive capability across the entire sonication range tested.

Figure 2 illustrates the relationship between GNP loading and absorbance, comparing EPR predictions with experimental data across the range of 0.135–0.9 g. The close agreement between predicted and measured values confirms the model’s accuracy in capturing concentration-dependent behaviour.

Figure 3 shows the EPR model predictions versus experimental measurements for the holding time parameter at Month 4 (120 days) across different GNP concentrations. The model successfully captures the ageing effects on dispersion quality.

The presence of B⁻⁰·⁵ and B⁰·⁵ terms with opposing signs indicates a non-monotonic relationship between sonication time and absorbance (Figure 1). Initial sonication (5–30 min) increases absorbance through improved dispersion, while extended treatment (>60 min) shows diminishing returns and potential platelet fragmentation. The EPR model captures this transition, with predictions closely matching the experimental values (R² = 0.88, RMSE = 0.045, MAE = 0.038) across the full sonication range.

The model exhibits strong concentration effects through multiple terms containing L and F with various powers. The L²F² term with its large negative coefficient (−3,151, 206.74) dominates at higher concentrations, suggesting aggregation-induced deviations from Beer–Lambert behaviour. The model successfully predicts the non-linear absorbance–concentration relationship (Figure 2), capturing both the initial linear region (F < 0.01) and subsequent plateau (F > 0.03).

Holding time appears with fractional exponents (G^0·5, G⁻¹, G⁻²), indicating complex ageing mechanisms. Short-term stability (1 day) shows minimal absorbance changes, while extended storage exhibits significant variations captured by the G⁻² term (Figure 3). The model predictions align well with experimental observations at Month 4 (120 days), demonstrating its utility for shelf-life assessment.

Sensitivity analysis was performed by examining parameter variations in the experimental data set while analysing the EPR model’s predictive accuracy across different parameter ranges.

Table 1 presents input and output parameter value ranges and dimensions, as well as model accuracy indicators used to evaluate the performance/accuracy levels of the developed EPR model.

GNP loading (L) and concentration (F) exhibited the strongest influence on absorbance, with both parameters showing absorbance variations of 1.653 units across their tested ranges (0.135–0.9 g and 0.006–0.04 g/mL, respectively). The EPR model achieved high accuracy for these parameters, with predictions closely tracking experimental values as shown in Figure 2. The large magnitude coefficient (−3,151, 206.74) associated with the L²F² term in the EPR equation confirms these parameters’ dominance in governing absorbance behaviour.

Sonication time (B: 5–120 min) showed moderate influence with absorbance variation of 0.434 units (range: 2.479–2.913). The model captured this relationship with good accuracy (R² = 0.88), successfully representing the non-monotonic behaviour where initial sonication improves dispersion while extended treatment shows diminishing returns (Figure 1). The presence of B^−0·5 and B^0·5 terms with opposing signs in the EPR model reflects this complex relationship.

Holding time (G: 1–120 days) demonstrated significant influence, with absorbance declining over the four-month period depending on GNP loading. Higher GNP concentrations showed greater absolute decline (1.159 units for L = 0.9 g) compared to lower concentrations (0.141 units for L = 0.135 g). The model predictions aligned reasonably well with experimental observations at Month 4 (Figure 3), though the limited temporal data points (only Day 1 and Day 120) constrain interpolation accuracy for intermediate timeframes.

Despite varying superplasticizer dosage from 0% to 3.75%, absorbance changes remained relatively small (0.183 units variation). Critically, EPR assigned zero coefficients to all terms containing superplasticizer (A), indicating its minimal direct correlation with optical absorbance under the tested conditions. This finding suggests superplasticizer primarily affects rheological properties and dispersion stability through mechanisms that do not manifest as strong absorbance changes, highlighting the need for complementary characterisation methods beyond UV–Vis spectroscopy.

Water volume (U) was not independently varied in the experimental design but changed in conjunction with GNP loading to achieve target concentrations, precluding direct sensitivity assessment. The EPR model includes U⁻² terms suggesting dilution effects, but these cannot be independently validated from the available data set.

The absence of superplasticizer terms indicates the model cannot capture surfactant-mediated dispersion mechanisms, likely due to indirect effects on optical properties that manifest through changed aggregation states rather than direct absorbance modifications. This limitation suggests that separate models may be needed for systems with varying surfactant chemistries.

The developed EPR model for GNP absorbance prediction has direct relevance to several emerging geotechnical applications where nano-enhanced cementitious materials offer performance advantages. In ground improvement projects, the ability to rapidly assess GNP dispersion quality enables optimisation of grout formulations for specific soil conditions and project requirements. For example, the model’s sensitivity to water content (U) and concentration (F) parameters directly relates to the water/cement ratios commonly adjusted in permeation grouting to achieve desired penetration and strength characteristics.

The temporal stability insights provided by the holding time parameter (G) are particularly valuable for geotechnical applications where prepared grouts may experience delays between mixing and injection. Construction delays in tunnelling or deep excavation projects often require storage of prepared materials, and the model’s prediction of absorbance changes over 120 days provides guidance for maximum storage times and re-dispersion requirements. The G⁻² term’s influence at extended times suggests that fresh mixing protocols may be necessary for stored materials beyond 30 days critical information for project planning and quality assurance.

Furthermore, the model’s revelation that superplasticizer effects are minimal on optical properties, despite their known importance for rheological behaviour, highlights a key consideration for geotechnical applications. While superplasticizers are essential for achieving the low viscosities required for soil permeation, their dosage optimisation may need to rely on rheological rather than optical characterisation. This finding suggests that multi-parameter quality control protocols combining absorbance measurements (for dispersion state) with rheological tests (for injectability) may be necessary for geotechnical grouting applications.

The EPR approach demonstrated here also provides a template for developing similar predictive models for other challenging geotechnical characterisation problems, such as predicting the long-term behaviour of nano-modified soils, assessing the durability of stabilised ground under environmental loading or optimising bio-cementation processes. The ability to generate explicit mathematical expressions that reveal parameter interactions and sensitivities is particularly valuable in geotechnical engineering, where empirical correlations have traditionally dominated but often lack the flexibility to accommodate new materials and techniques.

This study successfully demonstrated the application of EPR for predicting UV–Vis absorbance of GNP dispersions, achieving coefficient of determination values of 88%–92% across different parameters with explicit mathematical expressions enabling systematic optimisation of GNP dispersion protocols for cementitious composite applications.

Key findings include:

• GNP loading (L) and concentration (F) emerged as the dominant factors governing absorbance behaviour, exhibiting absorbance variations of 1.653 units across tested ranges with high model accuracy (R² = 0.91–0.92). Sonication time (B) showed moderate influence (0.434 units variation, R² = 0.88) with captured non-monotonic behaviour, while holding time (G) demonstrated significant temporal effects with absorbance declining 0.141–1.159 units over 120 days depending on GNP concentration.

• The minimal contribution of superplasticizer terms (zero EPR coefficients despite 0.183 units measured variation) reveals that while superplasticizers affect rheological behaviour, their influence does not strongly manifest in optical absorbance measurements. This finding underscores the necessity for multi-parameter quality control protocols combining absorbance characterisation with rheological testing for geotechnical grouting applications.

• The developed model successfully predicts stability trends for the settling times that were used in the lab, to a high level of accuracy. As one of the advantages of the introduced methodology, if additional data becomes available from further testing or synthetically, the model could be improved as the newly emerged data is introduced to the process.

• The explicit five-term mathematical formulation enables straightforward implementation in industrial quality control systems without specialised software, facilitating real-time optimisation. However, users must recognise the model’s operational boundaries: predictions may be unreliable at extreme parameter values, and the absence of independent water content variation limits model applicability when water/cement ratios deviate substantially from tested conditions.

The methodological framework established demonstrates EPR’s potential for addressing complex characterisation challenges in geotechnical engineering, particularly for nano-enhanced soil stabilisers and grouting operations. The transparent model structure aligns well with geotechnical practice requirements where physical understanding remains paramount alongside computational methods.

Future research should address:

• expanded temporal data sets with intermediate measurement points for improved interpolation;

• independent variation of water content to validate dilution effects;

• investigation of alternative functional forms or multi-stage modelling to better capture superplasticizer–dispersion relationships; and

• integration with in-line spectroscopic monitoring for automated quality control in nano-enhanced cement production. The methodology provides a framework for developing transparent, reliable models for other nanomaterial systems where interpretability and physical insight are essential.