Feasibility of the Maturity Concept for Strength Prediction in Geopolymer Based Materials

Abstract

The aim of this work is to investigate the effect of curing temperature and time on the development of compressive strength in geopolymer mortars produced using ground granulated blast-furnace slag (GGBFS) and fly ash (FA). Considering curing circumstances, both the activation energy and the reference temperature could be used properly to build a reliable anticipated model for predicting the compressive strength of geopolymer-based products (mortar and concrete) using maturity-based techniques. In this study, the compressive strength development of geopolymer mortar made from (FA) and (GGBFS) under varying curing conditions. The mortar was prepared using an alkali solution of sodium hydroxide (NaOH) and sodium silicate (Na₂SiO₃) in a 1:1 ratio, with NaOH molarity of 12. Specimens were cast following ASTM C109 standards, with a binder/sand ratio of 1:2.75, and compacted for full densification. FA-based mortar was cured at 40 °C, 80 °C, and 120 °C, while GGBFS-based mortar was cured at 5 °C, 15 °C, and 40 °C for durations of 0.5 to 32 days. Compressive strength was evaluated at each curing period, and data were analyzed using ASTM C1074 procedures alongside a computational model to determine the best-fit datum temperature and activation energy. The Nurse-Saul maturity method and Arrhenius equation were applied to estimate the equivalent age and maturity index of each mix. A predictive model was developed for geopolymer concrete prepared at an alkali-to-binder ratio of 0.45 and NaOH molarity of 12. The final equation demonstrated high accuracy, offering a reliable tool for predicting geopolymer strength under diverse curing conditions and providing valuable insights for optimizing geopolymer concrete formulations.

1. Introduction

The introduction of Portland cement concrete is the predominant building material; nevertheless, the use of Portland cement as a binding agent presents several obstacles to achieving sustainability standards [1]. Geopolymer concrete (GPC), a novel substitute for conventional Portland cement concrete, has received significant attention due to its potential to reduce greenhouse gas emissions and utilize industrial by-products such as fly ash (FA) and ground granulated blast-furnace slag (GGBFS) [2,3] The activation of aluminosilicate material with an alkali-activated solution yields higher mechanical qualities, enhanced chemical resistance, and a decreased carbon footprint compared to common Portland cement concrete [4]. Similar to conventional concrete, optimizing construction schedules and ensuring structural integrity rely on precise predictions of geopolymer concrete’s compressive strength [5]. It is usual to carry out labor-intensive and time-consuming experimental testing at certain intervals to assess the strength [6]. To overcome these issues, the maturity concept gives a practical and reliable method for estimating concrete strength based on its temperature history and curing duration [7,8]. A maturity function was modified by including the relative humidity component (γRH) with temperature features. The new model precisely predicted the enhancement in strength, and results from experiments indicated that both T and RH exhibited significant effects. The RH factor was precisely calculated using a modified hyperbola function [9]

On top of that, early studies conducted at National Bureau of Standards (NBS) verified that the maturity method may be used to predict how concrete’s compressive strength and other mechanical characteristics will change as a function of curing temperature [9,10]. The maturity-based strength development function from the fib Model Code 2010 has been utilized to improve the estimation of concrete compressive strength [11]. The equation is given as:

$$f_{cm}\left(t\right)=\mathit{exp}\left(s·\left(1-{\left({\displaystyle \frac{28}{t}}\right)}^{0.5}\right)\right)· f_{cm }\left(28\right)$$

(1)

where:

f_cm(t): is the mean compressive strength in N/mm² at age t in days,

f_cm, 28: is the mean compressive strength at an age of 28 days,

t: is the age of concrete in days,

s: is a coefficient depending on the cement strength class.

The coefficient “ s” is defined in accordance with the cement strength classes specified in EN 197-1 [12] and its values as given in the fib Model Code are:

Cement class 32.5 N: s = 0.38

Cement class 32.5 R and 42.5 N: s = 0.25

Cement class 42.5 R, 52.5 N and 52.5 R: s = 0.20,

These values are applicable to concrete cured in water at 20 °C. The fib Model Code currently does not provide specific guidance on the “s” values for cement types other than CEM I. However, studies have shown that the “s” value is influenced by both the water-to-binder ratio and the supplementary cementitious material (SCM)-to-binder ratio [13]. Additionally fib model code in the case of maturity concept or time -temperature effect, the time (age of concrete) would be adjusted based on the Arrhenius function with E_a = 33,256 J/mol, R = 8.314 J/(K∙mol) and specified reference temperature T_s = 293 K, the equation is given as:

$$t_T={\sum }_{i=1}^{n}exp(13.65+{\displaystyle \frac{4000}{273+T_a.∆ti}})∆ti$$

(2)

where: t_T is the temperature-adjusted concrete age

(Δt_i) is the number of days where a temperature T_α prevails

T_α(Δt_i) is the average temperature of concrete during time interval Δt_i (°C),

The coefficients “s” and “E” have been resolved using the equivalent mortar approach [14]. To estimate concrete strength using maturity models, researchers looked at how activation energy and datum temperature interact with one another. The effect of temperature sensitivity on both parameters (activation energy and datum temperature) has been investigated with variations in concrete mixture [15]. The effect of GGBFS and siliceous fly ash (sFA) on the strength of concrete over time was studied [16]. A thorough literature database was established, which showed a relationship between s-values and the w/b + SCM/b ratios. Also, the impact of various curing temperature histories on the strength development of high-strength concrete has been analyzed [17]. An effort has been made to refine the conventional maturity function for enhanced predictive accuracy. The method precisely predicted the high-strength concrete (HSC) strength evolution by accounting for apparent activation energy, setting behavior, and rate constant, hence enhancing the reliability of maturity-based strength assessment for nuclear structure applications. Medrano et al. [18] established that the maturity index of accelerator-enhanced concrete corresponds with its compressive strength. The maturity index was calculated using an automated adiabatic method, and the mix design was created utilizing supplier-specific additives.

As a result of the development of an alternative activation energy model in response to increasing interest, the maturity model was formulated. Subsequently, this model was tested to evaluate its effectiveness in estimating the early-age strength of concrete. It was compared to experimental data from mixtures that included three different kinds of cement with FA. Because of its importance in predicting early-age strength and finding the best time for formwork removal or applying prestressing, modeling concrete maturity has become a primary area of research. An experimentally verified closed-form equation based on chemical kinetics was constructed to provide the best match for strength predictions. Multiple maturity calculation approaches, such as direct calculation and equivalent age calculation, have been analyzed [19] to assess their advantages and disadvantages, as well as to evaluate their prediction capabilities for compressive strength and elastic modulus against published test results. A concise overview of technical monitoring systems, including wired, wireless, and fully automated real-time systems, as well as a variety of maturity functions and strength-maturity connections, has been described [8]. The study showed that the maturity approach is reliable for assuring structural safety during construction by comparing it with standard concrete cylinder testing. A new strength function was developed by Sun and Lee [20] that integrates the Arrhenius equation. The findings show that the obtained E values provide more accurate predictions of FA concrete strength. The correlation between on-site strength and equivalent age, as assessed using a reference strength curve at 20 °C, has been thoroughly researched. While extensively applied to Portland cement concrete, the adaptation of the maturity concept to geopolymer concrete presents unique challenges due to the distinct chemical and thermal behaviors of GPC. This research applies the maturity concept to predict the compressive strength of geopolymer concrete. Maturity indices that assess the equivalent age rather than the actual age correlated with experimentally determined compressive strength. As per ASTM C1074 [14], the temperature-time factor at a specific age “t” is ascertained from the area beneath the curve of temperature and the reference temperature. The reference temperature is defined as the temperature below which cement hydration ceases and strength growth stops. The temperature was recorded as 10 °C. The time-temperature factor, referred to as the maturity index, could be calculated using the Nurse-Saul maturity function Equation (1).

$$M=\sum _0^t(T-T_o)·∆\mathrm{t},$$

(3)

whereas:

M = maturity index (°C-hours or °C-days); T = average concrete temperature (°C), during the time interval ∆t; T_o = datum temperature (usually taken to be −10 °C); t = elapsed time (hours or days); and ∆t = time interval (hours or days).

This equation relies on the assumption that the initial rate of strength growth during the acceleratory phase following setup is a linear function of temperature [21]. The linear approximation may be invalid when curing temperatures fluctuate significantly. In 1977, Freiesleben Hansen and Pedersen introduced a novel function to calculate a maturity index based on the measured temperature history of concrete [21], which illustrates the influence of temperature on the rate of a chemical process. The new function (2) facilitated the calculation of the concrete’s equivalent age as follows:

$$t_e=\sum _0^t{e}^{-{\displaystyle \frac{E}{R}}\ast \left({\displaystyle \frac{1}{T}}-{\displaystyle \frac{1}{T_o}}\right)}·∆t$$

(4)

whereas;

$t_e$ = the equivalent age at the reference temperature; E = apparent activation energy, J/mol; R = universal gas constant, 8.314 J/mol-K; T = average absolute temperature of the concrete during interval ∆t, Kelvin; and $T_o$ = absolute reference temperature, Kelvin.

The true age of concrete is converted into its equivalent age via Equation (2). The curing temperature has a non-linear relationship with the initial rate of strength development. Equation (2) depends on a critical parameter, the “activation energy”, which illustrates the influence of temperature on the rate of strength development. Then, a method for ascertaining the “activation energy” of a cementitious mixture was developed. The approach depends on determining the rate constant for strength development and its variation with curing temperature. By applying an appropriate equation to the strength vs. age data collected during isothermal curing, one may ascertain the rate constant, which correlates with the curing time required to achieve a designated fraction of the long-term strength. The “activation energy” for concrete with a water-cement ratio (w/c) of 0.45 ranged from 30 to 64 kJ/mol, whereas for w/c = 0.60, it ranged from 31 to 56 kJ/mol, depending on the kind of cementitious materials and admixtures used. Van Deventer et al. [22] stated that calcium facilitates the alkaline activator’s participation in chemical processes, leading to the development of calcium silicate hydrate (C-S-H) phases alongside geopolymeric gels. This dual reaction approach enables GGBFS-based AABs to achieve significant early-age strength without the necessity for thermal activation [23]. Another alumino-silicate source material for geopolymerization that has been extensively explored is metakaolin, which is made by heating kaolinite. The amorphous form and high alumina concentration of metakaolin make it naturally reactive, unlike FA and GGBFS. Geopolymers based on metakaolin can become strong at room temperature, although they are commonly enhanced in microstructure and mechanical characteristics after light heat curing (e.g., 40–60 °C).

Despite the benefits and the distinct chemical and thermal characteristics of GPC, it is difficult to forecast its compressive strength. Conventional techniques for assessing strength frequently depend on experimental testing, which is labor-intensive, time-consuming, and unsuitable for real-time applications. A feasible method for evaluating the compressive strength of GPC is the maturity concept, which is frequently used to forecast the strength of Portland cement concrete. However, the maturity concept must be modified to meet the unique needs of GPC due to its unique curing methods and material qualities. Furthermore, because FA and GGBFS have differing activation energies and temperature-dependent strength development characteristics, their combined usage as alumina-silica sources in GPC adds even more complication. The maturity concept has been extended to alkali-activated and geopolymer materials to assess their strength development as a function of time and temperature. Maturity indices are calculated using effective age models that incorporate activation energy, reflecting the cumulative temperature-time effect on geopolymerization. However, the reliance on temperature activation for some precursors, such as FA, requires adjustments to traditional maturity models to account for these unique reaction mechanisms.

The fundamental goal of this study is to develop an accurate prediction model that is specific to GPC and its characteristics. Such a model has the potential to considerably enhance quality control by providing real-time strength estimates during construction and facilitating faster decision-making. Given these challenges, the present work aims to establish a dependable predictive model for geopolymer concrete based on the maturity concept. The objective of this study is to predict the compressive strength of GPC under various curing conditions. Initially, the reference temperature and activation energy for FA and GGBFS would be determined, followed by the formulation of an effective age equation. This research is essential due to the lack of a predictive model, which limits the extensive optimization and use of GPC in practical construction situations. The specific objectives are to investigate the strength development behavior of geopolymer mortar made with FA and GGBFS as alumina-silica sources under varying curing temperatures and durations. Determine the activation energy and reference temperature for both FA-based and GGBFS-based geopolymer mortar. Formulate the equivalent age equation tailored to the unique curing characteristics of GPC. Propose a reliable and accurate strength prediction equation based on the maturity concept, incorporating experimental data and best-fit analysis. Finally, this study may promote more widespread usage of sustainable construction approaches. Furthermore, the study intends to use the maturity concept to link laboratory results to real-world geopolymer concrete applications.

2. Materials and Methods

The experimental work consisted of creating geopolymer mortar samples by using fly ash (FA) and ground granulated blast-furnace slag (GGBFS) as sources of alumina and silica and testing their performance under various curing conditions.

2.1. Raw Materials

2.1.1. Fly Ash

FA used in this study was collected from India and was used as the primary alumina-silica source material. The FA consists of silica (Si) and alumina (Al) oxides with a Si/Al ratio of 2.22; the chemical composition is shown in

Table 1. Chemical composition of FA and GGBFS.

Oxides (%)	SiO2	Al2O3	Fe2O3	CaO	MgO	MnO	SO3	LOI
FA	62.71	28.31	3.39	0.84	0.40	--	0.38	0.46
GGBFS	38	12	0.8	34	11	0.7	1.5	0.7

.

2.1.2. GGBFS

GGBFS is a fine powder made by grinding a vitrified material made by the rapid cooling of a slag melt obtained by melting iron ore in a blast furnace. It is considered to be at least two-thirds glassy slag by mass and possesses hydraulic properties when suitably activated. The specific surface area measured by the Blain method has the range 5800 to 6100 cm²/gm, the activity index at 7 days is between 50 and 60%, and the activity index at 28 days is between 75 and 85%.

2.1.3. Alkali Solution

The alkali activator solution was prepared by combining sodium hydroxide (NaOH) and sodium silicate (Na₂SiO₃) in a 1:1 ratio. The sodium hydroxide solution has a molarity of 12 M, whereas the sodium silicate solution consists of Na₂O (17.98%) and SiO₂ (36.14%), with the remainder being water. During synthesis, the ratio of sodium silicate (Na₂SiO₃) to sodium hydroxide (NaOH) in the alkaline activator solution determines the mechanical characteristics and microstructure of geopolymer materials. The final geopolymer product’s curing time, workability, and compressive strength are all heavily influenced by this ratio. The ideal ratio of Na₂SiO₃ to NaOH might range from 1 to 3, according to several research studies [24,25]. The exact value depends on the materials used and the purpose of the mixture.

According to research carried out in the area, the optimal ratio of Na₂SiO₃ to NaOH for producing lightweight geopolymer concrete using FA as the main alumino-silicate ingredient was determined to be 1:1 [26]. This indicates that for specific geopolymer uses, including FA, lower silicate-to-hydroxide ratios may provide superior results, maybe as a result of enhanced polymerization and geopolymer gel formation. The ideal ratio of Na₂SiO₃ to NaOH (1:1) also produced the greatest compressive strength in another study that concentrated on geopolymer pervious concrete using GGBFS as the principal geopolymer binder [27].

2.1.4. Aggregate

The fine aggregate in the combination consisted of fine silica sand with rounded, uniform-sized particles in a saturated surface dry state. The sand was passed through an 850-micron aperture and subsequently collected on a 600-micron sieve. This standard sand was employed for making geopolymer mortar cube specimens of size 50 mm. Nevertheless, for the casting of concrete cubes, locally available natural sand as fine aggregate with a specific gravity of 2.75, water absorption of 1.023%, and a fineness modulus of 3.1 was employed, while the coarse aggregate was categorized under grade (9.5 to 2.36) mm, with a specific gravity of 2.77, and demonstrates 0.8% water absorption. The sample complies with ASTM C33/03. Figure 1 presents the grading curves for coarse and fine aggregates. The natural river sand used as fine aggregate has a well-graded distribution with particle sizes ranging from approximately 0.15 mm to 4.75 mm. Only 10% of the sand was retained on sieve 4.75 mm. The smooth and continuous grading curve demonstrates a wide variety of particle sizes, which aids in packing and workability. This sand is classified as coarse sand, based on its fineness modulus (FM) measurement of 3.1. In contrast, coarse aggregate typically contained particles ranging in size from 4.75 mm to 12.5 mm, with a maximum particle size of 12.5 mm. Coarse aggregate has a narrow size distribution and seems to be graded uniformly, as seen by the steeply rising grading curve.

2.2. Mix Design and Testing Methods

For the preparation of mortar specimens that were employed to determine the activation energy and reference temperature, the ratio of geopolymer binder to sand was established at 1:2.75, and the alkali solution-to-binder (A/B) ratio was fixed at 0.7 to produce a flowable mortar. Despite utilizing an alkali-to-binder (A/B) ratio of 0.7 to create the geopolymer mortar, the results were excessively sticky. To increase workability, a polycarboxylate-based superplasticizer (Flowcrete SP 33) was added at a concentration of 0.5% by weight of the binder. This change provided the target degree of workability by raising flowability to 115%, as measured by an ASTM C1437 [28] compliant flow table test for hydraulic cement mortar.

The prepared mortar was poured into 50 mm cube molds and compacted with a mechanical vibrator to remove air voids. To achieve a good compromise between compressive strength and workability, geopolymer mixes typically use a 12 M NaOH solution. The compressive strength can be improved with a higher NaOH molarity, although the workability might be negatively impacted. A 12 M concentration seems to be the ideal value for this equilibrium [29]. The compressive strength of the mortar specimens was determined using standard 50 mm cube molds in accordance with ASTM C109 [30]. Testing was performed with an automatic concrete compression machine (BESMAK) equipped with a SEMATRON Touch Series controller, having a capacity of 2000 kN and a measurement accuracy of 1 N. Since various precursors display variable setting and strength development properties, the curing parameters for geopolymer specimens were selected according to the binder type. To improve the geopolymerization process and attain optimal strength development, geopolymers based on FA usually need to be cured at elevated temperatures [29,31]. As shown in Figure 2, the geopolymer mortar that was made using FA in this work was cured at temperatures of 40 °C, 80 °C, and 120 °C. Alternatively, geopolymers derived from GGBFS often set and harden under room temperature because the high calcium content of these materials speeds up the reaction even at lower temperatures [31]. So, to test how well the GGBFS-based mortar specimens held up at various temperatures, we cured them at 5, 15, and 40 °C. Figure 2 shows the cement mortar specimens cured in a JEOTEST drying oven, 250 Liter, capable of heating from ambient to 250 °C with a precision of 1 °C. A four-channel thermocouple temperature data logger (Tekneka Model T540) was utilized to record specimen temperatures with a measurement accuracy of ±(0.3% t + 0.4) °C at designated time intervals. The apparatus, produced by Tekneka, was acquired from Ontario, Canada.

The curing period varied from 0.5 to 32 days. The temperatures are recorded twice daily. Three specimens of geopolymer mortar were tested for compressive strength according to ASTM C109 for hydraulic cement mortar. The specimens were each made as a 50 mm cube and subjected to varying curing times. Both geopolymer mortar and geopolymer concrete curing temperatures and durations are detailed in

Table 2. Mix Proportions, Curing Temperature, and Curing Duration of Geopolymer Mortar and Concrete.

Mix Designation	Source Material	B:S:G	A/B	Na2SiO3/NaOH	Molarity of NaOH	Curing Temp. °C	Curing Time
M1	GGBFS	1: 2.75:0	0.7	1	12	5, 15, 40	0.5, 1, 2, 4, 8 and 32 days
M2	FA	1:2.75:0	0.7	1	12	40, 80, 120
C1	GGBFS	1:2:2.86	0.45	1	12	15, 40, 80	2, 4, 8 days
C2	FA	1:2:2.86	0.45	1	12	15, 40, 80

.

After determining the activation energy and reference temperature of the binder material from geopolymer mortar specimens, with a mix ratio of 1:2:2.86 (binder: sand: gravel) and an alkaline solution-to-binder (A/B) ratio of 0.45, geopolymer concrete was produced utilizing the same binder system, including ground granulated blast furnace slag (GGBFS) and fly ash (FA). The A/B ratio has a major impact on both the mechanical and microstructural performance of geopolymer concrete. With 0.45 usually found as excellent for encouraging good geopolymerization, most research implies that an acceptable A/B ratio for balancing workability, geopolymerization, and mechanical strength falls between 0.40 and 0.50 [32,33,34]. Lower A/B ratios (below 0.45) may improve compressive strength by lowering porosity, but they may also cause partial geopolymerization, leaving unreacted precursors and hence lowering long-term durability. Consistent with the mortar mixes, the alkaline activator included a 12 M sodium hydroxide solution with a sodium silicate-to-sodium hydroxide ratio equal to 1. Based on preliminary trials, a gravel-to-sand ratio of 1.43 was chosen to increase workability and attain a homogeneous mix. For the concrete, the last ideal mix ratio was therefore 1:2:2.86 (FA/GGBFS: sand: gravel).

3. Results

3.1. Compressive Strength of Geopolymer Mortars

Experimental methods are required to establish both the datum temperature and the Q-value, which is calculated by dividing the activation energy by the gas constant, by ASTM C1074 [14]. These experiments were carried out with the use of geopolymer mortar specimens, and the findings were subsequently utilized to predict the strength of geopolymer concrete. Figure 3 illustrates the compressive strength progression of geopolymer mortars derived from GGBFS, whereas Figure 4 depicts that of FA-based mortars. The error bars represent the standard deviation of the observed data. Figure 3 illustrates the influence of curing temperatures (5 °C, 15 °C, and 40 °C) on the strength development of GGBFS-based mortar. Strength gains were consistently rapid throughout the initial 3 to 7 days, thereafter decelerating. After 32 days at 40 °C, the compressive strength was around 35 MPa, whereas at 15 °C and 5 °C, it was about 27 MPa and 22 MPa, respectively. Compressive strength increased by 60–70% on average from day 1 to day 32 across all curing temperatures. Our findings indicate that within the examined temperature range, elevated curing temperatures accelerate strength development without markedly changing the overall strength-time profile, implying that GGBFS has consistent geopolymerization behavior. The selected temperatures accurately reflect the ambient curing conditions for geopolymers derived from GGBFS, which generally do not need heat activation for setting and hardening.

Conversely, as seen in Figure 4, geopolymer mortars derived from fly ash require elevated curing temperatures to attain their optimal strength development. Temperatures of 40, 80, and 120 degrees were employed to cure the specimens. During the initial 3 days, all participants exhibited notable enhancements in strength, after which progress ceased. Curing at 80 °C resulted in a marginally reduced but still consistent strength of around 30 MPa, whereas the maximum strength of about 35 MPa was consistently attained at 120 °C. The strength development of specimens subjected to treatment at 40 °C was delayed; after 32 days, they attained around 24 MPa. Notably, 50 percent of the ultimate strength was attained within the initial 2 to 3 days of curing, indicating that geopolymers develop early-age strength more rapidly than Portland cement concrete [22].

As shown in Figure 3 and Figure 4, the compressive strength values were averaged to reduce variability, and the experimental data were used to calculate the activation energy and reference temperature for both FA, and GGBFS-based geopolymer mortars. When compared to regular Portland cement concrete, geopolymers made of GGBFS and FA both show a quicker rate of early strength growth. Take FA-based geopolymers cured at high temperatures as an example; they may reach up to 90% of their 28-day strength in the first week, but Portland cement concrete takes around 7 days to obtain 70% of its strength. In the same manner, geopolymers derived from GGBFS show impressive early strength growth, reaching a large fraction of their 28-day strength in the first week alone.

3.2. Activation Energy and Datum Temperature

Geopolymer mortar specimens were cured in a controlled environment at different temperature levels, as shown in Figure 2. For each curing temperature, 21 mortar cubes were manufactured, and three cubes were tested for measuring the compressive strength at seven different ages: 0.5, 1, 2, 4, 8, 16, and 32 days. Firstly, datum temperature and activation energy were determined based on the procedure described in ASTM C1074. Figure 5 and Figure 6 show the relationship between the value of K versus temperature for GGBFS, and FA-based geopolymer mortars, respectively. It can be seen that a linear relation between K and temperature with a high coefficient of determination was obtained.

From the intercept of the line with the temperature axis, the datum temperature was determined to be 2.73 °C for GGBFS-based mortars and 40.58 °C for FA-based mortar. The natural logarithm of the K values correlated with the reciprocal absolute temperature (in Kelvin), as shown in Figure 7 and Figure 8. The best-fit straight line to these three points demonstrated the negative slope of the line, which represents the Q value, which is the activation energy divided by the gas constant that is used to compute the effective age of geopolymer mortar and concrete specimens.

Table 3. The best-fit parameters determined for the effective age and strength development of geopolymer mortar samples.

Mixes	Curing Temperature	Parameters
	T, °C	${\mathit{S}}_{\mathit{u}}$, MPa	K	to	Q = E/R (k − 1)	To (°C)	Method of Analysis
Alkali-activated GGBFS-based Mortar	5	25.36	0.256	0	6895.7	2.7	Solver function in Excell
	15	25.60	1.82	0
	40	31.74	5.14	0
	5, 15, 40	25.84	0.564	0.293	7751.3	8.89	Computer program
Alkali-activated FA-based Mortar	40	33.25	0.099	0	4979	40.6	Solver function in Excell
	80	33.84	0.92	0
	120	34.77	2.43	0
	40, 80, 120	29.24	1.93	0	3608	46.67	Computer program

shows the best fit parameters (S_u, k), determined using the solver function required to estimate the equivalent age while t₀ was fixed equal to 0. It can be seen that the activation energy E determined ranged from 57.3 to 64.4 kJ/mol, and the minimum datum temperature was 2.7 °C for GGBFS-based GPM, with a coefficient of determination R² = 0.96 and a minimum mean squared error MSE = 29.0. In addition, for FA-based GPM, the activation energy E ranged from 30 to 41.4 kJ/mol, and the minimum datum temperature was 40.6 °C with the coefficient of determination R² = 0.98, and the minimum mean squared error MSE = 26.0 was obtained. When considering t_o as a variable, the solver function resulted in a low correlation coefficient. To improve the fitting process, an alternative approach was attempted by developing a computer program written with Python version 3.11.5, packaged by Anaconda, was designed to determine the optimal parameter for both the equivalent age (Q and T_o) and user-defined function (S_u, k, and t_o) that correlates strength with equivalent age, as follows.

$$CS=S_u. {\displaystyle \frac{k\left(t_e-t_o\right)}{1+k\left(t_e-t_o\right)}}$$

(5)

The program iterates over a range of t_o and Q values, computing the effective age for each, and fits the strength model “Equation (5)” using nonlinear regression. The best combination of t_o and Q, minimizes the mean squared error (MSE) and maximizes R-squared (R²), indicating a better fit to experimental strength data. The maximum coefficient of determination R² was 0.94, and the minimum MSE: 30.3 was obtained for GGBFS-based mortar, and R² was 0.98 with minimum MSE: 19.85 were obtained for FA-based mortar. the relationship between predicted and measured values of compressive strength is shown in Figure 9 and Figure 10 for GGBFS- and FA-based mortars, respectively. Ferreira et al. [35] applied the maturity concept to FA-based, alkali-activated mortar, revealing activation energy values between 51 and 92 kJ/mole across various mortar mixtures. The inclusion of cement in the mixtures did not substantially influence the values of E, and mixtures with a water-to-cement ratio of 0.45 necessitated higher energy values to initiate compared to those with a ratio of 0.5. The temperature datum for all series varied from 44.5 °C to 54.2 °C.

From the activation energy and reference temperature determined for the geopolymer mortar. The Arrhenius equation was employed to determine the equivalent age of the geopolymer concrete specimens, which takes into consideration the impact of the curing temperature on the rate of strength development. Subsequently, a predictive strength model was established by establishing a correlation between the compressive strength values experimentally measured and the equivalent age. A multivariable regression model was developed by utilizing best-fit analysis to establish this correlation. The exponential function of the following form was used to characterize the compressive strength development of the geopolymer concrete:

$$CS=f_u\ast \left(1-\mathit{exp}\left(-k{t}_e\right)\right)$$

(6)

$$f_u={\displaystyle \frac{a0\ast {\left(R4\right)}^{a1}\ast {\left(R2\right)}^{a4}}{{\left(R3\right)}^{a2}\ast {\left(R1\right)}^{a3}}}$$

(7)

in which CS is the compressive strength (MPa), t_e_ is the equivalent age (hours or days), R1 is the sodium silicate to sodium hydroxide ratio (Na₂SiO₃/NaOH), R2 is the alkali solution to binder ratio, and R3 is a variable that accounts for the FA variety (SiO₂/Al₂O₃). R4 accounts for the calcium content (CaO). Regression coefficients such as a0, a1, a2, a3, a4, and k are determined through best-fit analysis.

Other mixture parameters, such as the NaOH molarity, the Na₂SiO₃/NaOH ratio, and the alkali solution to binder ratio, also influenced the activation energy used in the Arrhenius equation. The variety of FA was the primary determinant. Compressive strength results for geopolymer concrete mixtures produced with a variety of FA types were compiled from three distinct literature sources [36,37,38]. In total, 72 data points were collected. The mixtures encompassed a variety of Na₂SiO₃/NaOH ratios (1 to 3) and alkali solution to binder ratios (0.4 to 0.7), with a constant NaOH molarity of 12 M.

Figure 11 A total of 72 data points were collected from previously published studies to evaluate the proposed model. The relationship between the predicted and observed compressive strength of geopolymer concrete is presented. The model’s accuracy was assessed using statistical parameters, including the coefficient of determination (R²) and the sum of squared errors (SSE). For FA-based geopolymer products, the correlation between equivalent age and compressive strength resulted in an R² value of 0.77 and an SSE of 5535 with optimal parameters: a0 = 130, a1 = 0.18, a2 = 2, a3 = 0.09, a4 = 0.013 and k = 0.5, as illustrated in Figure 11.

This correlation demonstrates that the activation energy and reference temperature that were determined can be used to predict the strength of geopolymer concrete. However, modified activation energy and reference temperature are necessary for geopolymer concrete with varying compositions.

The effects of different curing temperatures (80, 100, and 120 °C) and durations (4, 6, and 20 h) on geopolymer mortars based on FA were studied by Adam and Horianto [36]. The researchers discovered that compressive strength was improved with increasing curing temperatures and durations. The maximum strength was reached after 20 h of curing at 120 °C. Hassan et al. [37] conducted a 26-h heat curing experiment at 75 °C against ambient curing. Strength growth was shown to be greatly enhanced by heat curing, as opposed to ambient-cured specimens, according to the study. Muhammad et al. [38] investigated curing temperatures between ambient temperature and 100 °C over 24 h. The results showed that raising the curing temperature had a positive effect on speeding up the development of strength, with the best results seen at higher temperatures. Ahmed et al. [39] formulated statistical methods to analyze the compressive strength of FA-based geopolymers, taking into account factors such as curing temperature and duration. Their study was based on using 247 experimental datasets. The evaluation indicated that the nonlinear regression model has given the best predictive results with an R² of 0.933. Their models indicated that curing regimes ranging from 40 °C to 100 °C for durations of 4 to 48 h created optimal conditions for geopolymer synthesis and strength development.

4. Conclusions

The maturity concept effectively predicts the rise of compressive strength in geopolymer concrete incorporating FA and GGBFS under diverse curing conditions, as demonstrated in this study. Precise strength estimations were achieved by incorporating the empirically derived activation energy and reference temperature into equivalent age models. These results demonstrate the significance of adjusting maturity parameters for different binder mixtures to get more precise forecasts.

The proposed maturity model must be evaluated across a broader range of geopolymer systems with diverse activator concentrations, curing protocols, and aluminosilicate sources to be deemed robust for future study. Enhanced real-time strength monitoring and broader use of geopolymer technology for sustainable construction can be accomplished by expanding these models.