Optimization of a ternary geopolymer paste through orthogonal experimental design, ANOVA and entropy-based multi-objective decision analysis

1 Introduction

The swift expansion of the construction sector has sharply increased the demand for Portland cement-based products. The production of OPC involves significant energy consumption and considerable carbon dioxide (CO₂) emissions, exacerbating worldwide environmental issues. Cement manufacturing is expected to account for approximately 7%–8% of global CO₂ emissions, prompting researchers to investigate alternative sustainable binder systems for construction. In this context, geopolymer materials have emerged as viable, eco-friendly substitutes for traditional cementitious binders owing to their reduced carbon footprint and their ability to incorporate industrial by products such as FA and GGBS (McLellan et al., 2011; Cai et al., 2020a; Fu et al., 2020; Das et al., 2022; Ranjith and Sekar, 2025).

Geopolymers are inorganic aluminosilicate substances created by the interaction of aluminosilicate precursor materials with alkaline activators. When materials such as FA or GGBS interact with alkaline solutions, such as sodium hydroxide and sodium silicate, a three-dimensional polymeric network of aluminosilicate gel forms, imparting strength and durability to the resultant geopolymer matrix. Compared with traditional cement-based composites, geopolymer binders offer superior mechanical performance, greater chemical resistance, and greater durability (Bajpai et al., 2020; Wongsa et al., 2020; Akbar et al., 2021; Mayhoub et al., 2021). The integration of GGBS with FA has been shown to enhance geo polymerisation and promote early strength development because of the reactive calcium content (Aboulayt et al., 2018; Cai et al., 2020b; Pławecka et al., 2022). Fly Ash, a byproduct of coal combustion in thermal power plants, is extensively utilized as a principal aluminosilicate precursor in geopolymer synthesis. GGBS, a byproduct of the iron and steel industry, contains substantial amounts of calcium, silica, and alumina, thereby enhancing geopolymer gel formation and mechanical properties. Prior research has shown that the synergistic use of FA and GGBS can markedly improve the compressive strength, microstructural evolution, and durability of geopolymer materials (Jin et al., 2016; Jiang et al., 2020; Zhang et al., 2020; Marvila et al., 2021a; Mendes et al., 2021). Furthermore, the use of industrial byproducts in geopolymer technology enhances sustainable waste management and mitigates the environmental impacts associated with conventional construction materials (de Azevedo et al., 2020; 2021a; 2021b; Marvila et al., 2021b).

Despite the benefits of geopolymer materials, the efficacy of geopolymer paste is significantly affected by various factors, including the ratios of source materials, the concentration of the alkaline activator, the activator ratio, and the curing conditions. Enhancing these factors via traditional trial-and-error methods could be laborious and ineffective (Komnitsas and Zaharaki, 2007; Görhan et al., 2016; Fu et al., 2020; Liang et al., 2021; Zhang et al., 2021). Consequently, statistical Optimization methods, such as orthogonal experimental design, have been widely used to systematically assess the effects of various parameters while minimising the number of experimental trials. Orthogonal array methods, specifically L16 design, have been successfully employed in material Optimization research to determine the optimal parameter combinations that affect the properties of geopolymer systems.

Provis et al. (2010) highlights the critical influence of particle shape on the rheological behavior and water demand of geopolymer concretes: spherical FA particles tend to increase workability, whereas platy metakaolin particles have the opposite effect. These findings further emphasise that the current understanding of particle geometry effects in alkaline-activated systems remains limited, necessitating more detailed investigations into fresh geopolymer pastes. Additionally, the results indicate that the milling of FA can effectively reduce the particle size; however, it may simultaneously alter the particle shape, leading to varying effects on strength development depending on the milling technique employed. This study also revealed that the higher viscosity of alkaline-activated solutions than that of water complicates the rheological analysis of geopolymer concretes. Furthermore, the findings emphasise the importance of selecting appropriate aggregate gradations, as proper particle packing enhances density and improves resistance against external environmental agents, thereby contributing to the overall durability of the concrete.

Zhang et al. (2019) investigated the compressive strength behavior of low-calcium FA geopolymer concrete, and extensive research has been conducted, revealing the critical role of mix design parameters. It has been reported that the compressive strength reaches its maximum at a liquid-to-FA ratio of 0.44, indicating an optimum mixture proportion for strength development. Furthermore, among the influencing factors, the sodium hydroxide concentration was identified as the most significant parameter affecting the compressive strength, whereas the liquid-to-FA ratio exhibited only a marginal influence. In addition, the ratio of sodium silicate to sodium hydroxide solution (SS/SH) was observed to have a limited effect on early-age compressive strength but played a crucial role in governing long-term strength gain. This study also revealed a strong positive correlation between the compressive strength at 48 days and both the Na₂O/SiO₂ and Si/Al molar ratios, underscoring their importance in long-term strength development. Moreover, failure mode analysis indicated that the compressive strength of FA geopolymer concrete is closely associated with the strength of the mortar–aggregate interface zone, underscoring the importance of interfacial bonding in determining the overall mechanical performance of concrete.

Yuan et al. (2022) evaluated the compressive strength and microstructural characteristics of an MK-FA geopolymer using an orthogonal experimental design, quantifying the influence of the liquid-to-solid ratio, the sodium silicate-to-sodium hydroxide ratio, and FA substitution. Similarly, Ali et al. (2024) applied a Taguchi-Grey relational approach to multi-objective mix design of GGBS-based geopolymer concrete blended with natural mineral additions, while Driouich et al. (2023) used response surface methodology to optimize a metakaolin-slag geopolymer system. Abdulkareem et al. (2023) extended the optimization lens further, identifying optimal precursor combinations across different regional settings using environmental and economic criteria. Taken together, these studies confirm that orthogonal and statistical design methods are well suited to geopolymer mix optimization, but each is either confined to a single dominant response (typically compressive strength), restricted to a binary precursor system, or focused on precursor selection rather than the combined optimization of strength, flowability, and cost within a fixed ternary formulation.

A parallel and more recent strand of work has turned to machine learning and metaheuristic algorithms to handle this kind of multi-response optimization. Su et al. (2026) combined ensemble learning with metaheuristic search to optimize the mechanical, economic, and environmental performance of a silica-fume-activated industrial-waste geopolymer, and in related work developed a knowledge-guided, interpretable AI framework for designing geopolymer binders from dredged sludge (Su et al., 2026b). These approaches can capture complex, nonlinear relationships between mix variables and multiple objectives, but they generally depend on large training datasets and, even where interpretability tools such as SHAP are applied, remain comparatively opaque relative to a directly interpretable statistical design. Machine learning approaches have in fact become increasingly widespread across geopolymer research more broadly, extending beyond mix optimization to the dynamic prediction of in-service performance (Dai et al., 2026), for example, combined machine learning with multi objective optimization to jointly predict and optimize the mechanical and durability performance of geopolymer concrete under varying service conditions.

What remains largely unaddressed is a ternary FA-GGBS-MK geopolymer paste in which compressive strength, flowability, and material cost are treated as simultaneous, competing objectives within a compact, interpretable experimental design, with the resulting optimum then explained through direct microstructural evidence. The present study addresses this gap by developing a multi-objective-optimized ternary FA-GGBS-MK geopolymer paste and linking the selected composition to its underlying geopolymerization mechanism through XRD and SEM analysis.

Where Yuan et al. (2022) and comparable orthogonal-design studies optimize largely around a single mechanical response, the present work treats strength, flowability, and cost as jointly competing objectives from the outset. This is achieved by combining an L16 orthogonal experimental design with ANOVA and a synthetic weighted decision model that blends subjective and entropy-based objective weighting, allowing the individual and interactive contributions of each mix factor to be quantified without the scale of experimentation that a full factorial or machine-learning-based study would require. A central finding of this approach is that the mixture with the highest compressive strength is not automatically the best engineering choice once cost and workability are weighed alongside it.

The selected optimum is then examined through XRD and SEM analysis, which provide mechanistic support for the observed performance by linking the mixture’s gel formation, phase evolution, and matrix densification to its measured properties. Taken as a whole, this statistical-microstructural pairing offers a practical, data-efficient route to designing multi-objective-optimized alkali-activated binders, without requiring the large training datasets that computational approaches typically demand.

2 Materials and methods

2.1 Materials

In this study, low-calcium FA (class F) and GGBS were utilized as aluminosilicate precursor materials. The FA was obtained from the thermal power plant located in Ennoor, Chennai. The acquired FA was dry and gray and contained fine spherical particles characteristic of pulverised coal combustion byproducts. The GGBS was obtained from JSW Cement Private Limited in Chennai. The GGBS exhibited a light gray hue and a high calcium content and was designed to improve early-age reactivity under ambient curing conditions. The metakaolin (MK) was obtained from Astra Chemicals, located in Chennai. Throughout the manuscript, MK denotes metakaolin, while A/B ratio denotes the alkali-to-binder ratio, and these forms are used consistently for clarity and readability.

The alkaline activator solution was composed of sodium hydroxide (NaOH) and sodium silicate (Na₂SiO₃). Both substances were acquired from Astra Chemicals. Table 1 shows the procurement details and unit material cost of the aluminosilicate materials.

TABLE 1. Procurement details and unit material costs.

The sodium hydroxide flakes of laboratory-grade purity (≥98%) were utilized to formulate alkaline solutions of the specified molarity. A sodium silicate solution was used as one of the alkaline activators, with a solids content of 45%, a density of 1.52 g/cm³, and a silica modulus (SiO₂/Na₂O) of 2.0. NaOH solution of the desired molarity was prepared through dissolution of NaOH pellets in distilled water. This solution was permitted to cool to room temperature for 24 h before being used. Alkaline activator solution was prepared using sodium hydroxide solution and thereafter mixing with the sodium silicate solution in the specified ratio. The prepared alkaline activator solution was permitted to reach equilibrium for 24 h before mixing in order to stabilize the temperature and ensure dissolution of its constituents. The chemical compositions of FA, GGBS, and MK were determined by X-ray fluorescence (XRF) analysis, as shown in Table 2. These findings showed that the supply of silicon and aluminium atoms to the polymerisation system is more sufficient for the use of MK and FA. In this polymerisation system, the main source of calcium was GGBS (37.63 wt%), which is considered to positively influence the structural compactness and compressive strength of geopolymers (Zhao et al., 2019).

TABLE 2. Chemical composition of FA, GGBS, and metakaolin analysed by XRF.

2.2 Mix proportion and experimental design

Geopolymer paste formulations were prepared by combining FA, GGBS, and MK at specified ratios to achieve the total binder content. The alkaline activator solution, comprising sodium hydroxide and sodium silicate, was incorporated into the dry binder at the specified alkali-to-binder (A/B) ratio. Key influencing factors, including the FA–GGBS–MK ratio, alkaline activator (AA) ratio, and alkali-to-binder (A/B) ratio, were optimized using an orthogonal experimental design matrix. An orthogonal array was utilized to systematically assess the effects of various parameters while limiting the total number of experimental trials. The experimental method facilitated the evaluation of the importance of individual components and the influence of their interrelations. Statistical analysis utilized ANOVA to determine the contribution ratio and the significance level of each parameter for the performance indicators.

2.3 Single-factor experiments

The experimental program of this investigation was executed in two phases. In the first phase, oven temperature and curing duration were independently assessed to determine the optimal heat-curing conditions for geopolymerization, with sample performance evaluated on the basis of compressive strength; the resulting optimal curing conditions are presented in Table 3. In the second phase, preliminary single-factor experiments were conducted for the remaining five design factors. These trials established that compressive strength increased with GGBS content and with an alkaline activator ratio of 1:1, peaked at intermediate molarity (6M), and peaked at moderate MK content and alkali-to-binder ratio before declining at higher levels. On this basis, the final two-level ranges for the L16 multifactor design were selected as GGBS 50%/60%, MK 10%/15%, NaOH molarity 6M/8M, AA ratio 1:1/1:1.5, and A/B ratio 0.30/0.35. Detailed single-factor results are presented in Section 3.1.

TABLE 3. Levels of the single-factor variables considered in the two experimental series.

2.4 Multifactor experiment

To evaluate the combined influence of multiple parameters on the performance of geopolymer paste, a multifactor experimental approach was adopted. Unlike single-factor experiments, where one parameter is varied at a time, multifactor experiments consider the simultaneous variation of several influencing factors, enabling the study of both individual and interaction effects (Feng et al., 2021). In the present study, five key parameters were selected as influencing factors: GGBS content (A), MK content (B), molarity of the alkaline solution (C), alkaline activator ratio (D), and alkali-to-binder ratio (E). Each factor was considered at two levels on the basis of preliminary investigations. An orthogonal experimental design was employed to systematically reduce the number of experimental trials while maintaining analysis accuracy. A total of 16 experimental runs were conducted using an L16 orthogonal array, where each run represents a unique combination of factor levels listed in Table 4. Table 5 presents the mass-based mix proportions for all sixteen mixtures, derived from the binder percentages and A/B ratio specified in Table 4, using a fixed total binder mass of 600 g per batch.

TABLE 4. Orthogonal array for (L16) with factor obligation for the experiments.

TABLE 5. Mass-based mix proportions of the ternary geopolymer paste mixtures.

The L16 orthogonal array employed in this study is a fractional factorial design in which main effects A (GGBS content), D (alkaline activator ratio), and E (alkali-to-binder ratio) are aliased with each other’s two-factor interactions (A≡D × E, D≡A × E, E≡A × D) and cannot be statistically separated using the present dataset; main effects B (MK content) and C (molarity) remain independently interpretable.

2.5 Preparation of geopolymer paste

The dry precursor materials, FA, GGBS, and MK, were initially homogenized to ensure uniform dispersion. The preprepared alkaline activator solution was thereafter incorporated into the dry binder and mixed using an AIMIL digital mortar mixer at the high-speed setting for 120 s, producing a uniform paste free from discernible agglomerations, as shown in Figure 1. Following mixing, the paste was transferred into cube moulds (50 × 50 × 50 mm) and allowed to set sufficiently for demoulding. The specimens were then demoulded and cured in an oven at 60 °C for 24 h.

2.6 Testing methods

A flow table test in compliance with ASTM C1437 is shown in Figure 2. The test involved placing fresh paste in two layers on a 25 cm diameter table, compacting it appropriately, lifting the mould to allow free flow, and then dropping the flow table 25 times in 15 s, as required by the standard. The spread diameter was measured along two perpendicular directions in centimetres, and the average value was reported as the flow spread.

The compressive strength of the geopolymer paste samples was assessed in accordance with ASTM C109/C109M. For each mix proportion, three 50 × 50 × 50 mm cubes were cast as subsamples from a single mixed batch, as shown in Figure 3. Following casting, the samples were retained in moulds for 24 h and demoulded and then heat-cured at 60 °C to facilitate geopolymerization, as shown in Figure 4a. After the curing period, as shown in Figure 4b, the samples were allowed to cool to ambient temperature before testing. The compressive strength test was conducted using a compression testing machine, with a continuous loading rate of 1400 N/s until failure. The maximum failure load was recorded, and the compressive strength was subsequently computed. The mean value derived from three samples was recorded as the compressive strength of each geopolymer paste mixture, with standard deviation and coefficient of variation reported in Table 6 to indicate reproducibility.

TABLE 6. Results of the geopolymer paste property indices (PIs).

“#” represents the optimal value of each PI; “*” represents the maximum/minimum value of each PI., The cost was calculated as ₹/kg of binder, accounting for the degree of NaOH, solution preparation.

2.7 X-ray diffraction (XRD) analysis

The phase composition and mineralogical characteristics of the geopolymer paste mixtures were investigated using X-ray diffraction (XRD). Prior to analysis, the hardened geopolymer specimens were crushed and ground into fine powder (<75 μm) using an agate mortar to ensure homogeneous particle size and minimize preferred orientation effects. The powdered samples were analysed using a PANalytical Empyrean diffractometer equipped with a 2.2 kW Cu anode ceramic X-ray tube, a PIXcel detector, and a Ni-β filter. Diffraction patterns were recorded over a 2θ range of 10°–90° at a step size of 0.02° and a scan rate of approximately 2°/min to identify both crystalline and amorphous phases. Diffractograms were background-subtracted and intensity-scaled for consistent visual presentation using High Score software, with final pattern plotting and comparison performed in Origin Pro. As precursor proportions, and consequently unreacted-quartz content, differ between R2 and R15 by mix design, amorphous-hump intensity comparisons between the two mixtures are interpreted qualitatively rather than as a rigorously quantified measure of amorphous content. The obtained diffraction peaks were indexed using the International Centre for Diffraction Data (ICDD) Powder Diffraction File (PDF) database. The XRD analysis was performed to identify the crystalline phases inherited from the precursor materials and to evaluate the formation of geopolymer reaction products and the degree of geo-polymerisation.

2.8 Scanning electron microscopy (SEM) analysis

The microstructural morphology of the geopolymer paste mixtures was examined using scanning electron microscopy (SEM). Hardened geopolymer specimens were prepared and carefully sectioned to expose representative internal surfaces. The specimens were oven-dried to eliminate residual moisture and mounted on aluminum stubs using conductive carbon adhesive; oven-drying prior to imaging may introduce shrinkage-related microcracking, which is considered when interpreting the microcrack features reported in Section 4. To prevent surface charging during electron beam exposure, the samples were sputter-coated with a thin layer of gold prior to examination. Microstructural observations were carried out using an EVO 18 Research Scanning Electron Microscope (CARL ZEISS, Germany) operating at an accelerating voltage of 10.00 kV under high-vacuum conditions, using a secondary electron (SE1) detector. Images were captured at magnifications of ×500, 2,000×, and 5,000× to examine features ranging from overall matrix morphology to fine gel-scale detail. SEM analysis was employed to evaluate the morphology of the geopolymer gel, matrix densification, pore distribution, unreacted precursor particles, microcrack formation, and the interfacial bonding characteristics within the geopolymer matrix.

3 Results and discussion

3.1 Single-factor experiments

Single-factor experiments were conducted before the L16 (2¹⁵) orthogonal array to determine appropriate two-level ranges for each of the five independent factors governing the fresh and hardened properties of ternary FA–GGBS–MK geopolymer paste. Each parameter was varied independently, while all the others were held at a fixed baseline, and the compressive strength (CS) measured after hot-air-oven curing at 60 °C for 24 h was used as the primary response variable. The parameters investigated included curing temperature, curing duration, GGBS content, MK content, NaOH molarity, alkaline activator ratio, and alkali-to-binder ratio. The two-factor levels subsequently adopted in the orthogonal experiment for each parameter were selected directly from the outcomes of the corresponding single-factor study, as detailed in Sections 3.1.1–3.1.3.

3.1.1 Effect of curing temperature on compressive strength

The curing-temperature and curing-duration trials were conducted on a binary FA-GGBS system, to establish the optimal curing regime prior to introducing the ternary FA-GGBS-MK mix design. The effects of curing temperature (40, 60, and 80 °C) on the compressive strength of the paste at three FA:GGBS ratios (40:60, 50:50, and 60:40) with a fixed curing duration of 24 h are shown in Figure 5. MK was subsequently incorporated at the ternary mix-design stage (Section 2.4), using the curing conditions established here. All three mix proportions exhibit a consistent inverted-U relationship: the compressive strength increases from 40 °C to 60 °C but then decreases toward 80 °C. The 40:60 mix attained the highest peak strength of approximately 61 MPa at 60 °C, followed by the 50:50 mix (57 MPa) and the 60:40 mix (47 MPa). At 40 °C, the geopolymerization reaction rate is insufficient to fully dissolve aluminosilicate precursors within the curing period, resulting in an incompletely cross-linked gel network and reduced strength. At 60 °C, the thermal energy accelerates both the dissolution of FA, GGBS particles by the alkaline activator and the subsequent polycondensations to form dense N-A-S-H and C-A-S-H gels, resulting in the maximum achievable strength within 24 h. At 80 °C, however, rapid evaporation of pore water occurs before the gel network has fully formed, inducing shrinkage of microcracks and a more porous microstructure that reduces strength. Furthermore, the consistent ranking of the three mixed lines at all temperatures confirms that the GGBS content independently governs strength: higher GGBS provides more CaO, which accelerates reaction kinetics and contributes to the formation of additional C-A-S-H gels. On the basis of these results, a curing temperature of 60 °C was selected as the fixed curing condition for all L16 orthogonal array mixes.

3.1.2 Effect of curing duration on compressive strength

The compressive strength of the FA–GGBS–MK ternary geopolymer paste clearly shows a consistent inverted-V trend with increasing curing duration: the strength increases from 8 h to a universal maximum at 24 h and then decreases at 48 h for all binder compositions (Figure 6). From the results, it can be inferred that the optimum curing temperature for curing is 60 °C, which corresponds to 24 h since the strength plateaus after which any further increase in curing time would have no significant effect on strength. However, this inference is based only on compressive strength without independent confirmation of the degree of geopolymerization at different curing temperatures. Beyond this point, prolonged curing leads to dehydration of the gel structure, resulting in shrinkage and microcracking that reduce strength. Additionally, the 40:60 (GGBS-rich) mix consistently has the highest strength, followed by 50:50 and 60:40, indicating that the GGBS content governs the overall strength, whereas the curing duration controls the strength evolution pattern. On this basis, 24 h of curing was adopted as the fixed condition for all mixtures in the subsequent multifactor investigation.

3.1.3 Effect of GGBS content, activator ratio, molarity, MK content, and alkali-to-binder ratio on compressive strength

Compressive strength was evaluated across GGBS contents ranging from 30% to 70%. Within the well-replicated 40%–60% range (31-32 observations per level, pooled across multiple preliminary trial batches on the binary FA-GGBS system conducted at varying molarity and activator-ratio combinations, rather than independent replicates of a single fixed composition), strength increased steadily from 33.71 MPa to 45.69 MPa, supporting selection of 50% and 60% as the final levels. Among alkaline activator ratios, 1:1 produced the highest mean strength (43.71 MPa) compared with 1:1.5 (38.49 MPa) and 1:2 (38.24 MPa), supporting selection of 1:1 and 1:1.5. Molarity trials showed peak mean strength at 6M (49.61 MPa); 8M was retained as the second level to capture the higher end of the practical molarity range used in this study. For MK content, trials at fixed FA:GGBS (50:50) and A/B ratio (30%) showed strength peaking between 5% and 15% MK before declining sharply beyond 20%, supporting selection of 10% and 15%. For the alkali-to-binder ratio, trials at fixed MK content (10%) and FA:GGBS ratio (55:35) showed strength peaking at 25%-30% before declining at higher ratios, supporting selection of 30% and 35%.These results provided the empirical basis for selecting the final two-level ranges used in the subsequent L16 multifactor design.

3.2 Orthogonal experiments

3.2.1 ANOVA for the PIs of geopolymer paste

The results of the experiments on the property indices of geopolymer paste (Table 6) reveal a distinct correlation between compressive strength (CS), flowability (FL), and material cost. A maximum compressive strength of 115.87 MPa was attained in mix M15, consistent with the influence of molarity, GGBS content, and activator ratio on strength development, as discussed further below. This behavior is attributed to enhanced geopolymerization and the formation of dense C–A–S–H and N–A–S–H gels, which improve the mechanical characteristics of geopolymer systems (Chiranjeevi et al., 2023; Paruthi et al., 2024).

This combination yielded the highest cost (₹28.52/kg), indicating a trade-off between performance and economic viability. Recent studies have reported similar results regarding increased costs associated with higher slag content and activator concentration (Paruthi et al., 2024). Conversely, the most economical mixture, M14 (₹21.43/kg), exhibited moderate strength, suggesting that budget-friendly blends may negatively affect mechanical performance.

Flowability varied across all combinations, with higher flowability in M2 and the lowest flowability in M6, indicating that the liquid-to-solid ratio and activator content significantly affect workability. This conclusion is consistent with previous findings that increased liquid content enhances workability but may adversely affect stability and strength. The data indicate that an optimal geopolymer formulation requires a balance of strength, flowability, and cost-effectiveness. The L16 orthogonal design employed in this study is a fractional factorial array in which main effects A (GGBS content), D (alkaline activator ratio), and E (alkali-to-binder ratio) are aliased with each other’s two-factor interactions (A≡D × E, D≡A × E, E≡A × D); their individual contributions therefore cannot be statistically separated using the present dataset (Section 2.4). Accordingly, the error term in Tables 7, 8, 9 reflects the residual variance not attributable to the twelve distinguishable effects, and material cost (Table 10), being a deterministic quantity calculated directly from constituent unit prices rather than an experimentally measured response, is analysed using range analysis. Because A, D, and E are aliased with each other’s two-factor interactions, the contribution rates (rho%) reported for these three factors in Table 7 cannot be interpreted as independent main-effect contributions; each such rho% value reflects the combined variance of a main effect and its aliased interaction, not the main effect in isolation. For the same reason, the optimal levels (OL) listed for A, D, and E in Table 7 should be read as the level favoured by the combined main effect and its aliased interaction, not as evidence of the main effect’s direction alone; the mixture ultimately selected in this study (R2) is based on the multi-objective synthetic score (Section 3.2.2), not on Table 7’s individual OL values. Factor D (activator ratio) shows the largest relative contribution to compressive strength (17.92%), consistent with its role in enhanced dissolution and polycondensation of aluminosilicate precursors, though this effect cannot be fully distinguished from the A × E interaction. None of the twelve identifiable effects reached statistical significance at α = 0.05 for compressive strength (Table 7), reflecting the comparatively higher residual variance associated with this property index relative to flowability and the synthetic score. Documented findings indicate that alkaline activators and alkali-to-binder ratio strongly influence geopolymer strength (Amar et al., 2023; Reddy et al., 2024). Flowability is predominantly governed by Factor E (the alkali-to-binder ratio, 72.81% contribution), making it the principal predictor of workability. Material cost is most strongly influenced by Factor B (metakaolin content, range = 3.35 ₹/kg between levels), reflecting the higher unit cost of metakaolin relative to Fly Ash and GGBS. These findings align with those of other studies that highlight the influence of mix design parameters on both fresh and hardened properties (Reddy et al., 2024).

TABLE 7. ANOVA table of compressive strength (CS).

(DF = degree of freedom; SS = sum of squares; rho = contribution rate (%); OL = optimal level; “ns” = not significant (α = 0.05). Factors A, D, and E are mutually aliased with each other’s two-factor interactions in the employed L16 design (Section 2.4). The error term reflects the residual variance not attributable to the twelve distinguishable effects).

TABLE 8. ANOVA table of flow table spread (FL).

(DF = degree of freedom; SS = sum of squares; rho = contribution rate (%); OL = optimal level; “*” = highly significant; “” = significant. Ns = not significant).

TABLE 9. ANOVA table of the synthetic weighted scores for property indexes.

Synthetic weight matrix W = [0.309, 0.332, 0.359]^T for CS, FL, cost, respectively (Subjective T = [0.4, 0.25, 0.35]^T, Objective U = [0.218, 0.414, 0.368]^T). “*” = highly significant (α = 0.01); “” = significant (α = 0.05); “ns” = not significant.

TABLE 10. Range analysis of material cost (Cost).

The ranking analysis (Table 11) confirms that Factor D ranks first for compressive strength, Factor E ranks first for flowability, and Factor B ranks first for both material cost and the synthetic weighted score, highlighting their substantial impact on the relevant properties. The B × E and C × E interactions also rank highly for compressive strength, while Factor C (molarity) consistently ranks lowest across all property indexes, indicating its comparatively minor influence within the tested range. This ranking clearly demonstrates that no single factor prevails across all performance metrics, underscoring the need for a multi-objective Optimization technique to achieve a balanced geopolymer mix design, as previously emphasised in recent Optimization studies (Talaat et al., 2021; Fereidouni et al., 2025; Khodadadi et al., 2025).

TABLE 11. Ranking of factors for all property indexes (PIs).

(Numbers indicate the rank of each source within each PI column (1 = most significant). Blank = factor pooled into error (not significant).

3.2.2 Standardized estimation matrix

To reveal the optimal pattern of affecting elements for the property indexes, an enhanced synthetic weighted multi-index scoring technique was established and applied, accounting for interaction effects on the PIs. The five two-level factors, GGBS dosage (A), MK dosage (B), NaOH molarity (C), the alkaline activator ratio (D), and the alkaline-to-binder ratio (E), form the parameter space. The approach is outlined below. In the orthogonal multi-index test, the factors are represented as H = 1 , 2 , … , l, and n = 16 experimental schemes (referred to as the Set of Experimental Schemes, SES) are marked as I = 1 , 2 , … , n , = m = 3 performance assessment indicators (designated as the Set of PIs, SPI), as shown in J = 1 , 2 , 3. The value of PI j for experimental scheme i is represented as x_ij (where i = 1 , 2 , … , 16; j = 1 , 2 , 3). The matrix X = x i j 16 × 3 serves as the estimation matrix for the SES on the SPI, as shown in Table 6.

Owing to the distinct Optimization requirements of each PI, the 16 × 3 matrix X = (x_ij) necessitates standardization. Let K1 = {PI where “higher is better”} and K2 = {PI where “lower is better”}, with K1 ∪ K2 = K and Ks ∩ Kt = ∅ (s ≠ t; s, t = 1, 2). In this study, compressive strength (CS) and flow table spread (FL) are evaluated as attributes of K1 (where greater values are preferable), whereas material cost (Cost) is categorised under K2 (where lower values are preferable). The present investigation reveals an absence of expected-value PI (K3 = ∅). The matrix Y = (yij)16 × 3 is derived from Table 6 in accordance with Equations 1a,b, :

y i j = x i j , j ∈ K 1 h i g h e r i s b e t t e r : C S , F L (1a)

y i j = − x i j , j ∈ K 2 l o w e r i s b e t t e r : C o s t (1b)

The standardized estimation matrix Z = (z_*ij*) 16 × 3 is then computed by normalizing each column to the range [0, 100] using Equation 2:Where y_*j,min* = min {y_*ij* |i = 1, 2, … , 16} and y_*j* , _*max* = max {y_*ij* |i = 1, 2, … , 16}. The resulting standardized matrix Z, together with the synthetic weighted scores, is presented in Table 12. It should be noted that the standardized scores z ∼ ij ∼ are computed relative to the minimum and maximum values observed within this specific set of sixteen mixtures; the resulting normalized performance index therefore reflects each mixture’s relative standing within the tested range of this study and should not be interpreted as an absolute or transferable material property.

z i j = 100 × y i j − y j , ⁡ min y j , ⁡ max − y j , ⁡ min , i = 1 , 2 , … , 16 j = 1 , 2 , 3 (2)

TABLE 12. Synthetic weighted scores.

z_i1, z_i2, and z_i3 are the standardized scores (0–100) for CS, FL, and Cost, respectively. Score fi = W₁z_i1 + W₂z_i2 + W₃z_i3. A higher score indicates better overall performance. “*” = best mix G2 (fi structural = 78.9849); (fi SCC = 80.6247) “#” = worst mix G16 (fi structural = 17.2089); (fi SCC = 15.3742).

3.2.3 Synthetic weight of PIs

This method comprehensively considers both the subjective and objective weights of each PI to guarantee the assessment’s usefulness and precision.

3.2.3.1 Subjective weight

The subjective weight matrix is T = [T1, T2, T3]^T, where Σ T_j = 1 and T_j ≥ 0 for j = 1, 2, 3. In the context of geopolymer paste for structural applications, compressive strength serves as the principal mechanical performance metric and has the utmost subjective significance. The cost of materials directly influences the economic feasibility of the mixture and is considered equally significant. The flow table spread, which regulates workability and casting quality, is assigned modest significance. The subjective weight matrix utilized in this investigation is as follows:

T = 0.4 , 0.25 , 0.35 T for CS , FL , Cost respectively

A sensitivity analysis (Table 13) was performed to evaluate the robustness of this weighting scheme, testing the synthetic score ranking under three alternative subjective weight scenarios: equal weighting (T = [0.333, 0.333, 0.334]^T), cost-prioritized weighting (T = [0.30, 0.20, 0.50]^T), and strength-prioritized weighting (T = [0.50, 0.20, 0.30]^T). Mix M2 remained the top-ranked mixture and M16 the lowest-ranked mixture across all four scenarios, confirming that the identified optimum is not an artifact of the specific subjective weights selected.

TABLE 13. Sensitivity analysis of synthetic weighted score under alternative subjective weightings.

3.2.3.2 Flow priority weight

The subjective weight matrix is T = [T1, T2, T3]^T, where ∑T_j = 1 and Tj > 0 for j = 1,2,3. In the realm of geopolymer paste for self-compacting applications, flowability is the foremost performance criterion and thus has the utmost subjective significance. The compressive strength determines the load-bearing capacity and overall mechanical performance of the material and is considered equally important. The cost of materials affects the economic viability of the mixture and is considered relatively less significant. The subjective weight matrix utilized in this investigation is as follows:

T = 0.4 , 0.4 , 0.2 T representing CS , FL and Cost , respectively

This weighting scenario anticipates the flow-oriented performance criteria relevant to future self-compacting concrete applications explored later in this research program; self-compacting concrete-specific acceptance criteria (slump-flow, T500, V-funnel) were not evaluated in this paste-scale study.

3.2.3.3 Objective weight (entropy method)

The final objective weight matrix U = [U1, U2, U3]^T is derived using the entropy evaluation process, where Σ U_j = 1 and U_j ≥ 0. The entropy of PIj is as follows:where p_*ij* = z_*ij*/Σi z_*ij* and p_*ij* ln p_*ij* = 0 when p_*ij* = 0. The objective weight of PI j is as follows:

h j = − ln n − 1 . Σ i p i j . ⁡ ln p i j (3a)

U j = 1 − h j Σ k 1 − h k . (3b)

The computed entropy values are as follows: h₁ (CS) = 0.9596, h₂ (FL) = 0.9232, and h₃ (Cost) = 0.9319. The resulting objective weight matrix is as follows:

U = 0.218 , 0.4144 , 0.3677 T for CS , FL , Cost respectively

The considerable objective weight of FL (U2 = 0.4144) signifies increased variability in flowability across the 16 blends, suggesting that this performance metric possesses superior discriminative capability. Similarly, cost has a significant objective weight (U₃ = 0.3677) owing to its wide range, driven by the elevated unit costs of metakaolin and the NaOH activator. To fully utilize insights from both subjective preferences and objective entropy weighting and to reconcile subjective and objective assessments, the synthetic weight W = [w1, w2, w3]^T is obtained by minimising the decision-making objective function F(w) as specified in Equation 4, where μ (0 ≤ μ ≤ 1) is the preference coefficient established at μ = 0.5 A value of μ = 0.5 was adopted to weight subjective and objective assessments equally, avoiding bias toward either engineering judgment or purely data-driven weighting; this choice, together with the subjective weight matrix, is evaluated in the sensitivity analysis presented in Table 13.

min F w = Σ _ { i } Σ _ { j } { µ [ w j − T j z i j 2 + 1 − µ [ w j − U j z i j 2 (4)

When Σᵢ z² ij > 0 (j = 1, 2, 3), the solution to Equation 4 is as follows:

W = µ T 1 + 1 − µ U 1 , µ T 2 + 1 − µ U 2 , µ T 3 + 1 − µ U 3 T (5)

Substituting T = [0.4, 0.25, 0.35]^T, U = [0.218, 0.4144, 0.3677]^T and μ = 0.5, the synthetic weight matrix is as follows:

W structural = 0.309 , 0.3322 , 0.3589 T for CS , FL , Cost , respectively (6a)

Similarly, for SCC, when T = [0.4, 0.4, 0.2]^T, U = [0.218, 0.4144, 0.3677]^T and μ = 0.5; the synthetic weight matrix is as follows:

W SCC = 0.309 , 0.4072 , 0.2838 T for CS , FL , Cost respectively (6b)

The synthetic weighted score of each experimental scheme i is then calculated using Equation 7, based on the entropy weighting method (Equations 3a,b), the weight-optimisation solution (Equation 5), and the substituted structural and SCC weight matrices (Equations 6a,b), as follows:

f i = Σ j w j . z i j = W i . z i 1 + W 2 . z i 2 + W 3 . z i 3 , i = 1 , 2 , … , 16 (7)

Table 12 displays the synthetic weighted scores for all 16 geopolymer paste formulations. The highest score is achieved by mix M2 (Score fi (structural) = 78.9849) and (Score fi (SCC) = 80.6247), corresponding to the parameter combination: GGBS 50%, MK 10%, Molarity 6 M, AA ratio 1:1.5, and A/B ratio 35%. The minimum score is attained by M16 (Score fi (structural) = 17.2126) and (Score fi (SCC) = 15.3742), which is associated with a combination of higher MK (15%), high molarity (8 M), high AA ratio (1:1.5), and diminished GGBS (60%), yielding the lowest compressive strength and the highest cost simultaneously. The essential mechanical and economic characteristics of geopolymer paste improve with increasing synthetic weight score. This demonstrates the effectiveness of the modified multi-index scoring method in evaluating and optimizing ternary FA–GGBS–MK geopolymer paste systems and SCC applications.

The relationship between the flow spread and compressive strength (CS) for the 16 orthogonal mixes is shown in Figure 7, with the bubble size and colour representing the synthetic weighted score (f). Overall, mixtures with moderate-to-high flow spread (∼17–22 cm) tend to achieve higher compressive strength, but strength alone does not affect performance. For example, M15 and M10 exhibit high strength, whereas M2 shows a balanced combination of good strength, high flow, and the highest synthetic score, indicated by the largest and darkest bubble. In contrast, mixes such as M16 and M8 have both low strength and low scores, reflecting poor overall performance. This confirms that the synthetic score effectively captures multiperformance Optimization, where the best mix is not necessarily the one with the highest strength but the one with the best balance of properties.

3.2.4 Validation of the assessment method

The PIs of this study are the compressive strength (CS), flow table spread (FL), and material cost of the ternary FA–GGBS–MK geopolymer paste. The effectiveness of the updated synthetic weighted multi-index scoring system is confirmed by repeated validation tests conducted on five selected mix combinations, as presented in Table 14. The mixture with the lowest score among the 16 orthogonal test mixtures is M16 (f = 17.2089), characterized by GGBS = 60%, MK = 15%, NaOH molarity = 8 M, AA ratio = 1:1.5, and A/B ratio = 30%. This mixture demonstrated the lowest compressive strength (CS = 50.04 MPa) among the 16 groups, along with insufficient flowability (FL = 13 cm) and a modest material cost of ₹25.92/kg, confirming its overall worse performance across all three metrics.

TABLE 14. Optimization combinations and repeat test results of geopolymer paste PIs.

(Bold values in each PI column represent the best value among the five repeated test groups. The material cost is a deterministic calculated value based on fixed mix proportions and unit material prices; it is identical between the original and repeated tests for each mix. Optimal reference values from the 16-mix study: CS = 115.87 MPa (M15), FL = 22 cm (M2), Cost = ₹21.43/kg (M14).

The group achieving the highest score in the L16 orthogonal array is M2 (f = 78.9849), defined by 50% GGBS, 10% MK, 6 M molarity, an AA ratio of 1:1.5, and an A/B ratio of 35%. Relative to M16, the compressive strength (CS) increases from 50.04 to 87.81 MPa, the flow spread improves from 13 to 22 cm, and the material cost decreases from ₹25.92 to ₹22.98/kg, demonstrating that all three performance indicators (PIs) consistently increase as the synthetic weighted score increases. M15 achieves a peak individual compressive strength of 115.87 MPa (z₁ = 100) but is positioned 10th overall (f = 54.3971) because of its high material cost of ₹28.52/kg. G14 displays the lowest material cost (₹21.43/kg, z₃ = 100) but scores sixth overall, illustrating the inverse relationship between strength/flowability and cost within this framework.

Table 9 presents the ANOVA results, providing a quantitative evaluation of the influence of each factor and interaction on the synthetic weighted score for the PI system (SPI). Factor B (MK dosage) is the predominant variable, with the highest contribution rate to the SPI, followed by Factor E (A/B ratio) and Factor A (GGBS dosage). The regression relationship between the relevant elements and the composite weighted score is expressed as follows:

S = 55.05 + 5.97 A + 12.30 B + 4.92 C + 3.84 D + − 6.48 E (8)

The regression model includes only the five main effect terms (A–E); the interaction terms A × D, A × E, and D × E were removed, as they are aliased with E, D, and A, respectively, in the employed design (Section 2.4) and do not represent additional independent information. Consequently, the coefficients for A, D, and E should be interpreted as main-effect estimates under the assumption that their aliased two-factor interactions are negligible, rather than as independently identifiable effects; this assumption is a limitation of both the regression model and the grid search prediction derived from it. The model correlation coefficient (R ² = 89.97%) signifies that the regression model effectively accounts for the variability in the synthetic weighted score. However, when the same model is evaluated by leave-one-out cross-validation, in which each of the sixteen mixtures is held out in turn and its score predicted from a model fitted to the remaining fifteen, the adjusted R ² is 84.95% and the cross-validated R ² (Q²) is 74.3%, both appreciably lower than the in-sample R ² of 89.97%. This gap indicates that the model generalises to untested compositions less reliably than the fitted R ² alone would suggest, and the regression and grid-search results reported below should accordingly be read as identifying a plausible predicted optimum within the tested design space, rather than as a fully validated predictive optimization capability. A grid search across all 32 potential factor combinations (2⁵) within the test domain {A [50%, 60%]; B [10%, 15%]; C [6M, 8M]; D [1:1, 1:1.5]; E [30%, 35%]} resulted in a maximum predicted score of 88.5534 for the combination: GGBS = 50%, MK = 10%, molarity = 6M, AA ratio = 1:1, A/B ratio = 35% (designated R_opt). This composition differs from R2 (the multi-objective scoring optimum) specifically in the alkaline activator ratio (1:1 for R_opt versus 1:1.5 for R2); all other factor levels coincide.

The CS values recorded in all five repeat test groups are consistently lower than the corresponding original L16 mix results. R16 exhibited a strength of 33.54 MPa, a decrease of 32.97% from the original M16 value of 50.04 MPa; R2 demonstrated 73.45 MPa, reflecting a 16.35% reduction from the original M2 value of 87.81 MPa; R15 recorded 102.76 MPa, indicating an 11.31% decrease from 115.87 MPa; and R14 showed 75.48 MPa, a reduction of 12.32% from 86.08 MPa. The consistent direction of decline observed across all groups—though variable in magnitude, ranging from 11.31% (R15) to 32.97% (R16) — is more plausibly attributable to a shared cause of variance than to random dispersion. Repeat tests were performed utilizing a new production batch of sodium silicate solutions. Sodium silicate is recognized as a batch-sensitive component in geopolymer systems, with interbatch discrepancies in the SiO₂/Na₂O molar ratio, total solids content, and solution viscosity capable of affecting the rate and extent of polycondensation, and thereby mechanical strength, even when mixing ratios and curing conditions remain constant. The strength reduction observed across all repeat groups is consistent with this explanation, although the chemical properties of the two sodium silicate batches were not directly measured, and batch variability should be regarded as the most plausible rather than a confirmed cause. The reduction in compressive strength is greater in low-strength mixes (R16: 32.97%) than in higher-strength mixes (R14: 12.32%). This finding suggests that mixtures with elevated dosages of GGBS and MK may exhibit reduced sensitivity to variations in sodium silicate batches, potentially reflecting the enhanced influence of slag hydration and pozzolanic reactions on strength development, though this mechanism was not directly investigated in the present study.

The flow table spread values in the repeated tests slightly differ from the initial measurements: R14 remains constant at 15 cm (0.00%), M2 deviates by 4.55% (22 → 21 cm), R15 deviates by 5.26% (19 → 18 cm), and R16 deviates by 7.69% (13 → 14 cm). The minor variations fall within the expected range of experimental reproducibility for fresh-paste workability assessments and do not affect the rankings or results.

The material cost is a deterministic figure derived from the fixed mix proportions and the unit pricing of the constituent materials (FA, GGBS, MK, NaOH, and sodium silicate). Given that the mix proportions are consistent between the original and repeated tests for each group and that the unit material costs are stable, the cost values are precisely replicated in all five repeat groups (R16: ₹25.92/kg, R2: ₹22.98/kg, R15: ₹28.52/kg, R14: ₹21.43/kg, R_opt: ₹23.85/kg). Consequently, cost is not influenced by batch-to-batch experimental variability, necessitating no evaluation of repeatability.

The computed synthetic weighted scores for the five repeat groups employing the original 16-mix normalisation ranges are as follows: R2 (f = 60.0750) > R_opt (f = 59.8386) > R15 (f = 56.7092) > R14 (f = 45.9892) > R16 (f = 10.6676). The ranking is fully consistent with the method’s predictions: R16 is the lowest performer, whereas R2 and R_opt occupy the top two positions. The synthetic weighted ratings of R2 and R_opt differ by only 0.39%, indicating that the two mixtures achieve closely comparable overall performance under the proposed multi-objective scoring framework. It is worth noting, however, that this realised score (59.8386) is substantially lower than the score of 88.5534 that the regression model predicted for this same composition in Section 3.2.4; this 32% overprediction, observed on the model’s own designated optimum, is consistent with the limited out-of-sample predictive accuracy indicated by the cross-validated R ² (Q² = 74.3%) reported above, and reinforces that the regression model is better suited to identifying a promising region of the design space than to precisely forecasting the performance of an untested composition. The closely comparable scores between the scoring-method optimal (R2) and the regression-model optimal (R_opt) suggest that the two Optimization approaches identify a similar region of the design space, although this should be interpreted as convergence between methods rather than independent validation of either.

R15, although attaining the highest individual compressive strength (CS) among the repeat groups at 102.76 MPa, ranks third overall because of its material cost of ₹28.52/kg the highest in the dataset resulting in a cost-normalized score (z₃) of zero, hence negating any cost contribution to the composite score. R14, with the lowest material cost (₹21.43/kg, z₃ = 100), ranks fourth because of its relatively low compressive strength (75.48 MPa) and flow spread (15 cm). The results unequivocally demonstrate the adversarial relationship between mechanical/workability performance and material economy within this ternary binder system: increased dosages of GGBS and MK that increase compressive strength and flowability concurrently increase material costs because of the elevated unit prices of MK (₹70/kg) and NaOH (₹130/kg), rendering it fundamentally unfeasible for any single mix combination to simultaneously attain all three individual performance indices.

The outcomes of the repeated verification experiments indicate that the relative rankings produced by the modified synthetic weighted multi-index scoring technique were broadly preserved under repeat testing, supporting its usefulness for the comprehensive assessment and Optimization of ternary FA-GGBS-MK geopolymer paste systems. The method accurately identifies optimal and suboptimal mix combinations, and the regression model’s predicted optimum is within 0.39% of the scoring method’s optimum, indicating that both tools converge in the same region of the design space. The systematic reduction in CS due to variations in sodium silicate batches does not alter the relative ranking of the mixes or the validation conclusions, since the scoring technique assesses relative performance within a normalized framework rather than absolute strength values.

4 XRD and SEM analysis of the selected geopolymer paste mixtures

Based on the multi-objective optimization study (Section 3.2), the geopolymer paste mixtures R2 and R15 were selected for detailed microstructural characterization. R2 was identified as the optimum formulation under the proposed multi-objective weighting scheme, balancing compressive strength, flowability, and material cost (73.45 MPa, 21 cm, ₹22.98/kg). R15 achieved the highest individual compressive strength among all sixteen mixtures (102.76 MPa) at a comparatively higher cost and without a workability advantage. Comparing the two formulations microstructurally illustrates why single-metric optimization based on compressive strength alone would favour R15, whereas the proposed multi-objective framework identifies R2 as the more balanced, practically deployable formulation. To understand the fundamental mechanisms responsible for their enhanced performance, X-ray diffraction (XRD) and scanning electron microscopy (SEM) analyses were performed. XRD was employed to identify the crystalline and amorphous phases formed during geopolymerization and to evaluate the extent of reaction, while SEM was used to examine the morphology, matrix densification, pore distribution, and the interfacial bonding between the precursor particles and the geopolymer gel. The combined XRD and SEM analyses provide valuable insights into the phase evolution and microstructural characteristics that govern the observed strength difference between the R2 and R15 geopolymer paste mixtures.

4.1 X-ray diffraction (XRD) analysis of geopolymer pastes

Figure 8 presents the X-ray diffraction (XRD) patterns of the R2 and R15 geopolymer paste mixes recorded over a 2θ range of 10°–90°. This scanning range is widely adopted for alkali-activated binders because it encompasses both the broad amorphous halo associated with geopolymer gel formation and the principal crystalline phases inherited from the precursor materials. Both diffractograms exhibit similar crystalline phases owing to the use of the same raw materials (Fly Ash, GGBS, and metakaolin), while the principal differences arise from the degree of geopolymerization, as reflected by the development of the amorphous phase and the relative intensity of the diffraction peaks.

The dominant crystalline phase identified in both mixes is quartz (SiO₂), with the principal diffraction peak located at approximately 26.6° (2θ), corresponding to the (101) crystallographic plane (ICDD PDF No. 46–1,045) (Kim et al., 2025; Gasca-Tirado et al., 2026). Quartz is a stable crystalline constituent of Fly Ash and metakaolin that remains largely inert during alkaline activation and therefore persists after geopolymerization. A secondary crystalline phase corresponding to calcite (CaCO₃) is identified near 29.4° (2θ), indexed to the (104) reflection (ICDD PDF No. 05-0586) (Kalinkin et al., 2021). The formation or persistence of calcite may be attributed to residual limestone impurities in the precursor materials and partial carbonation of calcium-bearing reaction products during curing and exposure to atmospheric CO₂. Additional weak quartz reflections observed near 50.1° and 60.0° (2θ) further confirm the presence of residual crystalline silica within the geopolymer matrix. The diffraction pattern of the R2 mix exhibits a relatively weak and poorly defined amorphous hump extending from approximately 20°–35° (2θ), indicating a limited degree of dissolution of the aluminosilicate precursor phases. These observations are qualitative; precise quantification of the amorphous fraction would require Rietveld refinement or an internal-standard method, which was outside the scope of the present XRD analysis. The low background intensity and subdued amorphous halo suggest that only a small fraction of the reactive silica and alumina participated in the dissolution–polycondensation process, resulting in incomplete geopolymer gel formation. Furthermore, the quartz peak at 26.6° (2θ) remains comparatively less integrated with the amorphous background, implying that a considerable proportion of the precursor material remained unreacted. The calcite reflection at 29.4° (2θ) is also relatively weak, indicating limited participation of calcium-bearing species in the reaction. Overall, the XRD pattern of R2 reflects a comparatively lower degree of aluminosilicate dissolution relative to R15, consistent with its lower individual compressive strength; this lower reaction extent is offset in the multi-objective ranking by R2’s substantially better flowability and material cost (Section 3.2).

In contrast, the R15 mix exhibits a significantly more pronounced amorphous hump within the 20°–35° (2θ) region, demonstrating a substantially greater degree of geopolymerization. The increased diffuse background intensity is characteristic of a larger volume of amorphous aluminosilicate gel formed during alkaline activation. The principal quartz peak at 26.6° (2θ) remains visible because crystalline quartz is chemically stable; however, its appearance superimposed on a stronger amorphous background indicates that the surrounding reactive glassy phases have undergone extensive dissolution and subsequent polymerization. Likewise, the calcite reflection at 29.4° (2θ) is more clearly resolved, consistent with, though not direct proof of, calcium contribution from GGBS to calcium-rich reaction products, which is further examined via SEM in (Section 4.2). The improved definition of the secondary quartz reflections around 50.1° and 60.0° (2θ) further indicates a more stable and homogeneous geopolymer matrix. The differences between the R2 and R15 diffractograms clearly demonstrate that the phase assemblage remains unchanged, whereas the degree of reaction differs considerably. Since both mixes were synthesized from the same FA–GGBS–MK precursor system, the improvement in performance cannot be attributed to the formation of new crystalline phases but rather to the increased transformation of the amorphous aluminosilicate fraction into geopolymeric binding gel. The stronger amorphous halo observed for R15 indicates enhanced dissolution of reactive SiO₂, Al₂O₃, and CaO species, leading to more extensive gel formation and a higher reaction extent. Conversely, the weaker amorphous hump observed for R2 signifies incomplete geopolymerization, leaving a greater proportion of the precursor material unreacted. Overall, the XRD analysis confirms that R15 exhibits the highest degree of aluminosilicate dissolution and gel formation among the mixtures examined, consistent with its highest individual compressive strength, while R2 exhibits a comparatively lower reaction extent. This microstructural comparison illustrates precisely why single-metric optimization targeting maximum compressive strength alone would favour R15 — a valid but practically inferior engineering choice given its higher cost and lower workability—while the proposed multi-objective framework correctly identifies R2 as the more balanced formulation. The XRD results therefore provide mineralogical support for the mechanistic basis of the observed strength difference between R2 and R15, rather than validating the multi-objective optimization methodology itself, which is separately supported by the statistical ranking presented in (Section 3.2).

4.2 SEM Microstructural Analysis

Figures 9a–f presents the scanning electron microscopy (SEM) micrographs of the R2 and R15 geopolymer paste mixes, selected based on the multi-objective optimization study (Section 3.2) The microstructural features reveal significant differences in the extent of geopolymerization, pore refinement, and gel formation, which directly correlate with the measured compressive strengths of 73.45 MPa for R2 and 102.76 MPa for R15. The SEM images of the R2 mix (Figures 9a–c) reveal a comparatively heterogeneous and porous microstructure, indicating a relatively lower degree of geopolymerization. As shown in Figure 9a, numerous micropores are distributed throughout the matrix, suggesting incomplete filling of the interstitial spaces by geopolymeric reaction products. These micropores reduce the effective load-bearing area of the hardened paste and serve as preferential sites for crack initiation under compressive loading. Similar observations have been reported by Provis and van Deventer, who demonstrated that increased residual porosity adversely affects the mechanical properties of alkali-activated aluminosilicate binders.

Further evidence of incomplete geopolymerization is observed in Figure 9b, where several unreacted spherical Fly Ash particles remain embedded within the matrix together with microcracks extending across the geopolymeric binder. The persistence of intact Fly Ash particles indicates that only partial dissolution of the aluminosilicate precursor occurred during alkaline activation, thereby restricting the release of reactive Si and Al species required for the formation of continuous geopolymeric gels. Consequently, the precipitation of binding phases was limited, resulting in weak interparticle bonding and the development of microcracks during curing. Such microstructural defects are commonly associated with insufficient geopolymerization and inferior mechanical performance. The higher magnification micrograph shown in Figure 9c further reveals the presence of large interconnected open pores, indicating that the geopolymer gel was unable to completely occupy the available pore volume. These interconnected voids increase the overall porosity and permeability of the matrix while reducing its density, thereby facilitating crack propagation under compressive loading. The absence of a continuous and compact reaction product clearly demonstrates that the dissolution–polycondensation process remained incomplete in the R2 mix. Consequently, the combined presence of micropores, unreacted precursor particles, open pores, and microcracks explain the comparatively lower compressive strength of 73.45 MPa recorded for this mixture.

In contrast, the R15 mix (Figures 9d–f) exhibits a significantly denser and more homogeneous microstructure, reflecting a substantially higher degree of geopolymerization. As illustrated in Figure 9d, only a limited number of partially unreacted Fly Ash particles are observed within an otherwise compact geopolymer matrix. Although a few residual GGBS particles remain visible, their intimate embedding within the surrounding binder indicates that most reactive phases participated in the dissolution process, resulting in extensive gel formation. The reduced amount of unreacted precursor demonstrates improved reaction kinetics compared with the R2 specimen. Figure 9e shows the occurrence of several localized shrinkage-induced microcracks within the dense matrix. These cracks are considerably narrower than those observed in the R2 specimen and are primarily attributed to moisture loss during thermal curing rather than inadequate geopolymerization. Importantly, the surrounding matrix remains continuous and compact, suggesting that these localized cracks have only a minor influence on the overall structural integrity of the hardened paste. The most significant microstructural feature is observed in Figure 9f, where two morphologically distinct gel regions are clearly developed throughout the matrix, consistent with the coexistence of calcium-aluminosilicate-hydrate (C--A--S--H) and sodium-aluminosilicate-hydrate (N--A--S--H) gel phases reported in comparable FA-GGBS-MK systems (Chiranjeevi et al., 2023; Paruthi et al., 2024). As energy-dispersive X-ray spectroscopy (EDS) was not performed in this study, these gel-type assignments are based on morphological features consistent with the literature rather than direct compositional confirmation. The observed morphology is nonetheless consistent with the synergistic contribution of GGBS, Fly Ash, and metakaolin during alkaline activation, with calcium-rich GGBS expected to promote C--A--S--H-type gel and aluminosilicate-rich Fly Ash and metakaolin expected to promote N--A--S--H-type gel. The intergrowth of these gel phases effectively fills internal voids, enhances interparticle bonding, and produces a dense three-dimensional binding network. Such a compact microstructure significantly improves stress transfer throughout the matrix and minimizes crack propagation during compressive loading.

The observed microstructural differences between the R2 and R15 specimens closely correlate with their compressive strength behaviour. The R2 mix, characterized by abundant micropores, interconnected voids, microcracks, and residual unreacted Fly Ash particles, exhibited a relatively low compressive strength of 73.45 MPa, indicating incomplete geopolymerization and limited matrix densification. Conversely, the R15 mix developed a dense and continuous geopolymer matrix with visibly fewer and smaller pores in the SEM micrographs, resulting in a substantially higher compressive strength of 102.76 MPa; precise quantification of pore volume would require image analysis or porosimetry, which was outside the scope of the present SEM examination. These SEM observations are fully consistent with the XRD results, where R15 exhibited a more pronounced amorphous hump corresponding to greater geopolymer gel formation, whereas R2 showed a weaker amorphous halo indicative of incomplete precursor dissolution. Therefore, the combined SEM and XRD analyses indicate that enhanced dissolution of the aluminosilicate precursors, extensive gel formation, and improved microstructural densification are the principal factors responsible for R15’s approximately 40% higher compressive strength relative to R2 (102.76 MPa versus 73.45 MPa). This microstructural evidence explains the mechanistic basis of the strength difference between the two formulations; it does not, on its own, validate the multi-objective optimization methodology, which selected R2 over R15 on the combined basis of strength, flowability, and cost (Section 3.2). Rather, the comparison illustrates the practical trade-off underlying the optimization: R15 represents the outcome of single-metric (strength-only) optimization, while R2 represents the more balanced, cost-effective, and workable formulation identified by the proposed multi-objective framework.

5 Conclusion and future aspects

The present study successfully established an integrated statistical and microstructural approach for optimizing ternary FA-GGBS-MK geopolymer paste through single-factor experiments, L16 orthogonal design, ANOVA, synthetic weighted multi-objective decision analysis, and experimental validation. The preliminary investigation confirmed that curing at 60 °C for 24 h provided sufficient thermal energy to accelerate aluminosilicate dissolution and polycondensation, consistent with near-complete geopolymerization within the range tested; this inference is based on compressive strength alone and was not independently confirmed via reaction-degree analysis.

Effect analysis (Section 3.2) indicated that the alkaline activator ratio showed the largest relative influence on compressive strength, and the alkali-to-binder ratio the largest relative influence on flowability, though these effects could not be fully separated from their aliased two-factor interactions within the L16 design employed (Section 2.4). Three distinct outcomes emerged from this study and are reported separately to avoid conflating them: M15 (60% GGBS, 15% MK, 8M NaOH, AA ratio 1:1, A/B ratio 0.35) achieved the highest individual compressive strength (115.87 MPa) among all sixteen mixtures; R2 (50% GGBS, 10% MK, 6M NaOH, AA ratio 1:1.5, A/B ratio 0.35) was identified as the optimum formulation under the proposed multi-objective weighting scheme, balancing strength, flowability, and cost; and R_opt, predicted by the regression model (Equation 8), differed from R2 only in alkaline activator ratio (1:1 versus 1:1.5) and achieved a synthetic score within 0.39% of R2’s, indicating that both optimization approaches converge on a closely related, though not identical, region of the design space. Repeat validation testing indicated that the relative ranking of the selected mixtures was broadly preserved, although compressive strength values in the repeat tests were consistently lower than the original results (ranging from 11.31% to 32.97% across the five validated mixtures), attributed to probable batch-to-batch variation in sodium silicate composition that was not directly measured in this study.

Furthermore, XRD and SEM analyses of R2 and R15 provided microstructural support for the observed strength difference between the two formulations, showing greater amorphous gel development and a visibly denser, less porous matrix in R15 relative to R2; this comparison illustrates the mechanistic basis of the strength-cost-workability trade-off underlying the multi-objective optimization, rather than serving as independent validation of the weighting methodology itself. The proposed optimization methodology represents a practical framework for designing high-performance, lower-carbon-precursor geopolymer binders, with the statistical and microstructural limitations noted above and detailed below taken into consideration.

Several limitations of this study should be acknowledged. The L16 orthogonal design employed resulted in Factors A, D, and E being mutually aliased with each other’s two-factor interactions, such that their individual effects could not be statistically separated within the present dataset (Section 2.4); a higher-resolution design would be required to resolve this in future work. The observed reduction in compressive strength between the original and repeat validation tests is attributed to probable sodium silicate batch variability, but the chemical composition of the two batches was not directly measured. Gel-phase identification (C-A-S-H, N-A-S-H) in the SEM analysis was based on morphological features rather than direct compositional confirmation, as energy-dispersive X-ray spectroscopy (EDS) was not performed. No durability testing (chloride ingress, sulfate attack, carbonation, freeze-thaw) or quantitative environmental impact assessment was conducted. Finally, each design factor was evaluated at only two levels, which may not capture non-linear effects across a broader compositional range.

Future work should prioritize resolving the limitations identified above. A higher-resolution fractional factorial or full factorial design would allow the individual contributions of activator ratio, GGBS content, and alkali-to-binder ratio to be statistically separated. Direct measurement of sodium silicate batch chemistry (SiO2/Na2O ratio, solids content) alongside repeat testing would confirm the batch-sensitivity mechanism proposed in this study, and energy-dispersive X-ray spectroscopy (EDS) alongside SEM would allow direct compositional confirmation of the C-A-S-H and N-A-S-H gel phases inferred here. The optimized mixtures should also be evaluated for durability performance (chloride ingress, sulfate attack, carbonation, freeze-thaw cycling) and assessed via quantitative life-cycle or embodied-carbon analysis to substantiate the lower-carbon-precursor framing of this study. These paste-scale findings are already being extended within this broader research program to mortar-scale rheology and mix design, and subsequently to ambient-cured structural concrete, providing a direct pathway from the present mechanistic understanding to field-relevant structural application.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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Footnotes

Edited by: Dumitru-Doru Burduhos-nergis, Technical University Gheorghe Asachi Iasi, Romania

Reviewed by: Pan Zhou, Sichuan University, China

Ruobin Su, Hohai University, China

Atila Demiröz, Konya Technical University, Türkiye