Multiscale Dynamic Diffusion Model for Ions in Micro- and Nano-Porous Structures of Fly Ash: Mineralization Experimental Research

Abstract

The leaching concentration of alkaline ions plays a crucial role in the efficiency of the CO₂ mineralization reaction in fly ash. The multi-scale structural characteristics of micro–nano pores in fly ash are the primary factors that control the leaching and diffusion rate of alkaline ions. However, the existing theoretical models do not account for the multi-scale pore structure, leading to challenges in accurately describing the ion diffusion in fly ash and predicting the reaction rate and efficiency of CO₂ mineralization. To address this issue, a multi-scale dynamic diffusion model of ions was developed based on the micro–nano pore structure of fly ash. This model established the relationship between the ionic leaching rate and pore structure, as well as macroscopic changes over time, which were validated through experiments. Mineralization experiments with varying soaking times and uniaxial compression experiments on mineralized specimens were conducted to investigate the relationships among soaking time, ion leaching concentration, mineralization degree, and mechanical strength. The results elucidated the impact of alkaline ion concentration on the mineralization degree and mechanical strength of fly ash materials, offering theoretical insights to enhance mineralization and material properties.

1. Introduction

As global warming poses a significant threat to the earth’s life systems, there is a consensus among countries worldwide to strictly control CO₂ emissions in order to minimize greenhouse gas output [1]. The CO₂ capture, utilization, and storage technology (CCUS) serves as a crucial greenhouse gas emission-reduction technology, playing a significant role in expediting carbon emission reduction, ensuring energy security, fostering ecological civilization, and achieving sustainable development.

The CO₂ mineralization storage technology is widely regarded as one of the most promising and ideal methods, primarily due to its safety and stability. As a result, this technology finds extensive application in the ex situ mineralization and storage of CO₂ in industrial alkaline solid waste [2,3,4,5]. Due to its abundant raw materials, high reactivity, small particle size, low energy consumption, and lack of need for activation and mechanical grinding, fly ash has emerged as a promising alkaline material for mineralizing and sequestering CO₂ in recent years [6,7,8,9]. Nevertheless, studies have indicated that fly ash encounters challenges, including the multi-scale attenuation of the reaction rate and low mineralization efficiency, during the process of CO₂ mineralization and storage. Consequently, numerous domestic and international scholars have conducted extensive experimental research on the mineralization and storage of CO₂ using fly ash. Their studies have focused on investigating the evolution mechanism and factors that influence the mineralization reaction efficiency of fly ash and CO₂. These studies have also shed light on the occurrence of the multi-scale attenuation of the reaction rate and the low mineralization efficiency phenomenon, thereby offering valuable theoretical guidance for enhancing mineralization efficiency.

Existing research on the mineralization of CO₂ in fly ash primarily concentrates on investigating the impact of various external factors, including temperature, pressure, CO₂ flow rate, and water-to-solid ratio, on the efficiency of the process. Several scholars have examined the potential of utilizing fly ash mineralization as a means of storing CO₂ at ambient temperature and low-pressure conditions. Although this approach does not necessitate the use of extensive chemical reagents, it does require elevated reaction temperatures and pressures to enhance the reaction rate, resulting in significant energy consumption. Consequently, this method is currently insufficient to meet the requirements of industrial applications [10,11]. Hu et al. investigated the use of water as the reaction medium in the fly ash and CO₂ reaction system. They observed that CaO initially reacted with H₂O, resulting in the formation of Ca(OH)₂. Subsequently, Ca(OH)₂ reacted with CO₂ to form CaCO₃. The presence of H₂O significantly enhanced the mineralization reaction, effectively accelerating the reaction process and improving the kinetic characteristics of the reaction [12,13]. Water serves as the reaction medium between alkaline solid waste and CO₂, making the liquid-to-solid ratio (L/S) a crucial factor in determining mineralization efficiency. Previous experimental studies have highlighted that a low L/S ratio hinders the effective mixing of the slurry, leading to the inadequate transfer of calcium from the solid phase to the liquid phase. On the other hand, a high L/S ratio results in a low concentration of calcium ions in the solution, which is unfavorable for the mineralization reaction [14]. Increasing the temperature has a positive effect on the dissolution of CaO into the water phase and also enhances the reaction kinetics of the mineralization process. However, it is worth noting that the solubility of CO₂ in the liquid phase decreases at higher temperatures. Therefore, solely raising the temperature does not necessarily lead to an improvement in the reaction efficiency [15]. Relevant studies have shown that the pressure of CO₂ in the reaction system has a positive influence on the mineralization reaction. Increasing the pressure facilitates the diffusion of CO₂ in water and promotes its penetration into the pores of porous media for mineralization reactions. However, high pressure has a detrimental effect on carbon acid formation, which hinders mineralization efficiency [16,17,18].

The concentration of alkaline ions plays a crucial role in the mineralization reaction between fly ash and CO₂, determining the efficiency of the process. Researchers both domestically and internationally have explored various methods to enhance the leaching concentration of alkaline ions in fly ash, aiming to improve the mineralization efficiency. Among these methods, the wet process has emerged as the preferred choice due to its advantages of low energy consumption, cost-effectiveness, and high leaching rate. The primary principle behind this process is to utilize the calcium-based compounds present in fly ash to react with a calcium ion-rich leachate in the presence of CO₂, resulting in the formation of calcium carbonate [19,20,21,22]. The wet leaching process typically involves using an acid solution or an ammonium salt solution as the leaching solvent. The external conditions, such as high temperature, high pressure, or microwaves, are often modified during the leaching process to enhance the leaching rate. Subsequently, the pH of the leaching solution is adjusted to remove impurities and obtain calcium. The enriched solution of calcium elements is then subjected to CO₂ treatment to facilitate a complete reaction with calcium ions. Finally, the solution is filtered to obtain calcium carbonate. Currently, the main leaching methods include deionized water leaching, acid leaching, and ammonia leaching. Among them, deionized water leaching is one of the earliest wet leaching methods [23]. The ionized water leaching method involves using deionized water as the leaching liquid. This method takes advantage of the solubility of alkaline oxides in fly ash in water. By using only deionized water and fly ash in the reaction system, the occurrence of oxidation reactions caused by the introduction of other impurities can be avoided. Currently, researchers focus on studying various factors that influence the leaching process, such as particle size, leaching time, liquid-to-solid ratio, and reaction temperature. It has been found that the optimal conditions for leaching a fly ash aqueous solution involve using fly ash that has passed through a 280-mesh sieve, maintaining a temperature of 60 °C, maintaining a liquid-to-solid ratio of 1000, and allowing a reaction time of 72 h. Under these conditions, the leaching rate of calcium can reach 23.81% [24,25]. In order to enhance the leaching rate of Ca²⁺ and expedite mineralization, some researchers have explored the use of ultrasonic waves and microwave radiation during the leaching process [26].

Di Huajuan [27] opted for tap water as the leaching solution and incorporated ultrasonic effects to enhance the leaching rate. By conducting a comparative analysis of the calcium ion leaching concentration and leaching rate under ultrasonic and stirring conditions, it was observed that the calcium ion concentration in the leachate after ultrasonic treatment was significantly higher compared to that after the corresponding stirring action. This indicates that ultrasonic waves can effectively promote the leaching of calcium ions. Furthermore, when the calcium ion leaching rate is the same, the ultrasonic treatment time can be significantly shorter than the stirring treatment time. Said et al. [28] discovered that the presence of silica and other common impurities in solid waste can create a porous layer on the surface of the calcium release material, which hampers the release of calcium. However, this porous layer can be effectively eliminated by utilizing ultrasonic waves. Additionally, ultrasonic waves can also strip off the surface layer of steel slag particles, resulting in a significant increase in the leaching rate of Ca²⁺, from 65% to 96%. In a similar vein, Tong et al. [29] incorporated microwave radiation technology into the Ca²⁺ leaching and mineralization process. Although the leaching effect of Ca²⁺ in the microwave irradiation system was not remarkable, the microwave irradiation technique accelerated the mineralization process and reduced the overall time required for mineralization from 60 min to 50 min.

According to the kinetics of mineralization reactions, the number of alkaline substances in fly ash corresponds to the consumption of alkaline ions. The leaching rate of alkaline ions in the solution directly determines the rate of the mineralization reaction. However, there is currently a lack of studies on the evolution of alkaline-ion leaching rates. This paper aims to address this gap by establishing an ion leaching external diffusion model that takes into account the multi-scale structural characteristics of fly ash micro–nano pores. This model effectively predicts the evolution of the alkaline-ion leaching rate in fly ash and provides a reasonable description of the mineralization reaction mechanism. The findings of this study offer theoretical guidance for improving the rate and efficiency of the fly ash mineralization reaction.

2. Mineralization Test of CO₂ and Fly Ash

2.1. Experimental Equipment

The CO₂ and fly ash mineralization reaction experiment was conducted using a self-developed dynamic–static reaction test device for CO₂ mineralization under multi-field coupling. The device comprises five main parts: a gas-injection unit, a vacuum-pumping unit, a static-mineralization unit, a dynamic-mineralization unit, a thermal-air-mixing unit, and a steam unit.

Figure 1 shows the schematic diagram of the experimental device. The gas-injection unit is composed of a gas storage tank, pressure gauges 1 and 2, valves V1 and V2, a booster pump, air compressor pressure gauge 3, and valve V3. The hot-mixing unit includes valves V4 and V5, the CO₂ tank, a N₂ tank, temperature sensors 1 and 2, pressure sensors 1 and 2, one-way valves 1 and 2, pressure sensor 3, temperature sensor 3, valves V6 and V7, pressure gauge 4, and a gas-mixing tank. The static-mineralization unit includes valves V8 and V9, flow meter 1, temperature sensor 4, pressure sensor 4, vent 1, pressure gauge 5, a static reactor, a condenser, valve V10, a discharge port, and a flue gas analyzer. The vacuum-extraction unit consists of a vacuum pump, pressure gauge 6, vent 2, valves V11 and V12. The dynamic-mineralization unit includes valves V13 and V14, vent 3, an ultrasonic stirred tank, temperature sensor 5, pressure sensor 5, a stirring tank, and a magnetic stirrer. The steam unit comprises a steam generator, a water-injection pump, a steam-flow meter, a discharge valve, and valve V15.

2.2. Experimental Method

The experimental material used in this study was fly ash obtained from a coal-fired power plant located in Shijiazhuang, Hebei Province. The composition analysis results of the fly ash are presented in

Table 1. Composition of experimental material.

Component	Mass Fraction
SiO2	88.61
Al2O3	5.72
Fe2O3	1.58
MgO	1.43
CaO	1.24
K2O	1.21

. A total of 200 g of fly ash with a particle size of 80 μm was selected and dried in a drying oven at 105 °C for 2 h. Subsequently, the dried experimental sample was placed in a reaction kettle, and 600 g of deionized water was added, following a water–fly ash ratio of 3:1. The sample was then soaked for different durations (0, 12, 24, 48, and 72 h), and the pH value was measured. A 2 MPa high-purity CO₂ gas was injected into the high-temperature kettle through the CO₂ gas storage tank. The mineralization reaction lasted for 3 h, during which the pressure data was recorded in real time.

The suspension after the mineralization reaction was filtered through filter paper. Then, the filtered product was placed in a cylindrical mold with an inner diameter of 25 mm and a height of 50 mm. The mold was vibrated until the surface became slurry, eliminating any internal bubbles, and the surface was scraped. The mold was allowed to stand for 24 h before unmolding. Subsequently, the specimen was transferred to a constant temperature curing box with a temperature of 60 °C. Then, it was kept in the curing box for 48 h. Once the specimen is fully cured, it can undergo a uniaxial compression test. The AG-I precision material testing machine was used in this experiment, performing the uniaxial compression at a displacement loading rate of 0.01 mm/min. Each group was tested with three specimens, and the average value was taken as the experimental result. Additionally, a portion of the product was collected and dried at 105 °C for 24 h. The dried product was then sampled and measured using a thermogravimetric analyzer (TGA) and a scanning electron microscope (SEM) to obtain the sample’s experimental parameters.

2.3. Data Handling

During the mineralization reaction process between fly ash and CO₂, the efficiency of CO₂ mineralization is commonly calculated using the following empirical formula [30]:

η = (actual amount of CO₂/theoretical mineralization amount of CO₂) × 100%

(1)

The ideal mineralization amount of CO₂ is calculated using Formula (2):

CO_2Theo = 0.785CaO% + 1.09MgO% + 0.93K₂O%

(2)

The actual mineralization amount of CO₂ is determined as the percentage of the decomposed mass of CaCO₃ and the initial dry fly ash mass, according to Formula (3):

CO_2real = (Δ_{m500–m850 °C}[g]/m_105°C[g]) × 100%

(3)

The formula for calculating compressive strength is as follows:

$${\sigma }_C=P/A$$

(4)

In Formula (4), ${\sigma }_C$ represents the uniaxial compressive strength (UCS), the unit of ${\sigma }_C$ is MPa. P represents the maximum axial pressure in the event of specimen failure; the unit of P is N. A represents the cross-sectional area of the specimen; the unit of A is mm².

3. Model Building

3.1. Mathematical Model of Ion Leaching and Diffusion in Fly Ash Micro–Nano Pores

As the medium for CO₂ mineralization reaction, fly ash primarily consists of tiny particles emitted after coal combustion. These fly ash particles are mostly spherical in shape, with internal pores and cracks.

Figure 2 showcases the microstructure of fly ash, captured using a scanning electron microscope (SEM). The black part in the image represents the pores, while the white part represents the fly ash particles. Fly ash, being a porous medium material, contains matrix, pore, and crack structures of varying sizes, exhibiting a multi-scale characteristic in space. It is assumed that the fly ash particles are isotropic spheres. When a cross-section is taken through the center of the sphere, the spherical fly ash reveals a multi-scale structure, with larger circles encompassed by smaller circles, as illustrated in Figure 3.

Alkali metal oxides, such as Ca and Mg, present in fly ash, dissolve in water, resulting in the generation of alkaline metal ions that accumulate on the surface of fly ash particles. When CO₂ dissolves in water, it ionizes into CO₃²⁻. These two substances react to form carbonates, with all dissolved CO₂ participating in the reaction during the process. CO₂ actively participates in the reaction, while the alkaline ions diffuse from the interior to the exterior. Moreover, CO₂ reacts with alkaline ions on the surface of the fly ash particles, leading to the formation of carbonate crystals. Based on the aforementioned analysis, a mathematical model was developed to describe the dissolution and diffusion process of fly ash particles. The model incorporates the following assumptions: (1) Fly ash is considered a continuous medium. (2) Fly ash particles are assumed to be spherical and isotropic. (3) The leaching and diffusion of alkaline ions adhere to Fick’s second law of diffusion and the law of conservation of mass. (4) During the leaching process, alkaline ions dissolve and diffuse from the interior to the exterior of the pore surface. (5) All leached alkaline ions participate in the reaction.

The continuity mass conservation equation for particle ion dissolution and diffusion is presented in Equation (5).

$$\left\{\begin{array}{c}\frac{\partial C}{\partial t}=D(\frac{{\partial }^{2}C}{\partial r^2}+\frac{2}{r}\frac{\partial C}{\partial r})\hfill \\ \frac{\partial C}{\partial r}{|}_{r=0}=0,(r=0,t\ge 0)\hfill \\ C{|}_{r=r_0}=C_a=0,(r=r_0,t\ge 0)\hfill \\ C{|}_{t=0}=C_0,(t=0,0\le r\le r_0)\hfill \end{array}\right.$$

(5)

In the formula, C represents the dissolution and diffusion mass concentration of alkaline ions, which is a function of time (t) and distance (r), denoted as C = C(r, t). The unit of C is g/cm³. The variable t represents time and is measured in seconds. The variable r represents the diffusion path and is measured in centimeters. D represents the diffusion coefficient, with a unit of cm²/s. The variable r represents the particle radius, measured in centimeters. $C_a$ denotes the ion mass concentration at the particle surface during the diffusion process, with a unit of g/cm³, which serves as the boundary condition. $C_0$ represents the initial mass concentration of alkaline ions, with a unit of g/cm³, and it represents the initial condition.

Substituting the variables in Equation (5), supposing $u=\left(C-C_a\right)r$. Then, Equation (5) can be rewritten as follows:

$$\left\{\begin{array}{c}\frac{\partial u}{\partial t}=D\frac{{\partial }^{2}u}{\partial r^2}\hfill \\ u{|}_{r=0}=0,\left(r=0,t\ge 0\right)\hfill \\ u{|}_{r=r_0}=0,\left(x=L,t\ge 0\right)\hfill \\ u{|}_{t=0}=\left(C-C_a\right)r,\left(t=0,0\le r\le r_0\right)\hfill \end{array}\right.$$

(6)

Using the separation of variables method,

$$u=R\left(r\right)T\left(t\right)$$

(7)

Substituting Equation (7) into the first equation in Equation (6),

$$R\left(r\right)T^{\prime }\left(t\right)=DT\left(t\right)R^{″}\left(r\right)$$

(8)

Separating the variables in Equation (8),

$$\frac{T^{\prime }\left(t\right)}{DT\left(t\right)}=\frac{R^{″}\left(r\right)}{R\left(r\right)}=-{\lambda }^2$$

(9)

According to Equation (9),

$$R^{″}\left(r\right)+{\lambda }^{2}R\left(r\right)=0$$

(10)

$$T^{\prime }\left(t\right)+{\lambda }^{2}DT\left(t\right)=0$$

(11)

By solving Equation (10) and taking into account the two boundary conditions in Equation (6), Equation (12) can be derived. Similarly, by solving Equation (11), Equation (13) can be derived.

$$\left\{\begin{array}{c}R\left(r\right)=A\mathit{sin}\left(\lambda r\right)+B\mathit{cos}\left(\lambda r\right)\hfill \\ R^{\prime }\left(0\right)=R\left(r_0\right)=0\hfill \end{array}\right.$$

(12)

$$T\left(t\right)=\gamma \mathit{exp}\left({\lambda }^{2}Dt\right)$$

(13)

In Equations (12) and (13), A, B, γ, and λ are undetermined constants. By substituting the second boundary condition from Equation (12) into the first equation of Equation (12), Equation (14) can be derived.

$$\left\{\begin{array}{c}{R}^{\prime }\left(0\right)=\lambda A\mathit{sin}\left(0\right)+\lambda B\mathit{cos}\left(0\right)=0\hfill \\ R\left(r_0\right)=A\mathit{sin}\left(\lambda r_0\right)+B\mathit{cos}\left(\lambda r_0\right)=0\hfill \end{array}\right.$$

(14)

According to Equation (14),

$$\left\{\begin{array}{c}A\ne 0,B=0\hfill \\ \mathit{sin}\left(\lambda r_0\right)=0\Rightarrow \lambda =\frac{n\pi }{r_0},n=1,2,3\dots \infty \hfill \end{array}\right.$$

(15)

By substituting B and λ from Equation (15) into (7), the nth special solution Equation (16) can be derived.

$$u_n=c_n\mathit{sin}\left(\frac{n\pi }{r_0}r\right)\mathit{exp}\left({\left(\frac{n\pi }{r_0}\right)}^{2}Dt\right)$$

(16)

In the equation, $c_n=B_n{\gamma }_n$, the general solution of Equation (6) is obtained by applying the superposition principle.

$$u={\sum }_{n=1}^{\infty }{c}_n\mathit{sin}\left(\frac{n\pi }{r_0}r\right)\mathit{exp}\left({\left(\frac{n\pi }{r_0}\right)}^{2}Dt\right)$$

(17)

By substituting the initial conditions from Equation (6) into Equation (17),

$$((C_0-C_a)r={\sum }_{n=1}^{\infty }{c}_n\mathit{sin}\left(\frac{n\pi }{r_0}r\right)\mathit{exp}\left({\left(\frac{n\pi }{r_0}\right)}^{2}Dt\right)$$

(18)

Equation (18) is obtained by multiplying both ends of the equation by $\mathit{sin}\left(\frac{n\pi }{r_0}r\right)$, using the orthogonality of trigonometric functions. This results in the integral from 0 to r, as given by Equation (19).

$$c_n=\frac{\left(C_0-C_a\right){\int }_0^{r_0}\mathit{rsin}\left(\frac{n\pi }{r_0}r\right)dr}{\mathit{exp}\left({\left(\frac{n\pi }{r_0}\right)}^{2}Dt\right){\int }_0^{r_0}{\mathit{sin}}^2\left(\frac{n\pi }{r_0}r\right)dr}=\frac{-2r_0{(-1)}^n\left(C_0-C_a\right)}{n\pi \mathit{exp}\left({\left(\frac{n\pi }{r_0}\right)}^{2}Dt\right)},n=1,2,3\dots \infty$$

(19)

By substituting Equation (19) back into Equation (17), the final solution for u = u(r,t) can be derived.

$$u\left(r,t\right)=\frac{-2r_0\left(C_0-C_a\right)}{\pi }{\sum }_{n=1}^{\infty }\frac{{(-1)}^n}{n}\mathit{sin}\left(\frac{n\pi }{r_0}r\right)\mathit{exp}\left(-\frac{n^2{\pi }^{2}Dt}{{r_0}^2}\right)$$

(20)

$$C\left(x,t\right)=\frac{-2r_0\left(C_0-C_a\right)}{\pi r}{\sum }_{n=1}^{\infty }\frac{{(-1)}^n}{n}\mathit{sin}\left(\frac{n\pi }{r_0}r\right)\mathit{exp}\left(-\frac{n^2{\pi }^{2}Dt}{{r_0}^2}\right)+C_a$$

(21)

Equation (21) can be rewritten as

$$\frac{C-C_a}{C_0-C_a}=\frac{-2r_0\left(C_0-C_a\right)}{\pi r}{\sum }_{n=1}^{\infty }\frac{{(-1)}^n}{n}\mathit{sin}\left(\frac{n\pi }{r_0}r\right)\mathit{exp}\left(-\frac{n^2{\pi }^{2}Dt}{{r_0}^2}\right)$$

(22)

By multiplying both sides of Equation (22) by −1 and adding 1,

$$\frac{C_0-C}{C_0-C_a}=1+\frac{-2r_0\left(C_0-C_a\right)}{\pi r}{\sum }_{n=1}^{\infty }\frac{{(-1)}^n}{n}\mathit{sin}\left(\frac{n\pi }{r_0}r\right)\mathit{exp}\left(-\frac{n^2{\pi }^{2}Dt}{{r_0}^2}\right)$$

(23)

The cumulative mass of alkaline ions diffusing on the particle surface is denoted as $M_t$(g).

$$\begin{array}{c}{M}_t={\iiint }_V\left(C_0-C\right)dV={\iiint }_V\left(C_0-C\right)r^2\sin \phi d\theta d\phi dr\hspace{0.33em}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} ={\int }_0^{2\pi }d\theta {\int }_0^{\pi }\mathit{sin}\phi d\phi {\int }_0^{r_0}\left(C_0-C\right)r^{2}dr\hfill \end{array}$$

$$M_t=\frac{4\pi r_0^3}{3}(C_0-C_a)\left[1-\frac{6}{{\pi }^2}{\sum }_{n=1}^{\infty }\frac{1}{n^2}\mathit{exp}\left(-\frac{n^2{\pi }^{2}Dt}{r_0^2}\right)\right]$$

(24)

When the time is long enough (t→∞), the limit diffusion amount is $M_{\infty }$.

$$M_{\infty }=\frac{4\pi {r_0}^3}{3}\left(C_0-C_a\right)$$

(25)

The ratio of ion diffusion amount to limit diffusion amount at time t is given by

$$\frac{M_t}{M_{\infty }}=1-\frac{6}{{\pi }^2}{\sum }_{n=1}^{\infty }\frac{1}{n^2}\mathit{exp}\left(-\frac{n^2{\pi }^{2}Dt}{r_0^2}\right)$$

(26)

3.2. Model Validation

During the experiment, real-time detection of the pH value of the liquid environment was conducted. By establishing the conversion relationship between the pH and ion concentration, the relationship between the concentration of alkaline ions in the solution and the soaking time was determined (Figure 4a). Each data point in the figure represents the ion concentration at different time intervals during the experiment. Figure 4b,c display the fitting curves of the model at both multi-scale and single-scale levels, respectively. Both curves indicate that the dissolution and diffusion rate of fly ash decrease over time. However, the single-scale model shows a relatively slower change in the ion concentration as time progresses, while the multi-scale model demonstrates a rapid increase in the ion concentration followed by fast dissolution and diffusion in the early stage of the reaction. Figure 4d illustrates that, under the same conditions, the results calculated by the model align with the experimental findings, confirming that the model effectively predicts the evolution of alkaline ion concentration during the dissolution and diffusion process of fly ash. A comparison and verification were performed among the single-scale model, the multi-scale model, and the experimental data. The results show that the single-scale model fails to accurately match the experimental results, indicating its inability to properly calculate and describe the evolution of ion concentration in the dissolution and diffusion process of fly ash.

4. Results and Discussion

According to the kinetics of mineralization reactions, the concentration of alkaline ions that are leached and dissolved in fly ash corresponds to the mineralization reaction product. As a result, the leaching and dissolution rate of alkaline ions directly impact the chemical reaction rate of mineralization. Section 3 reveals that the multi-scale effect of micro–nano pores in fly ash is the primary factor that restricts the leaching and diffusion rate of alkaline ions. However, the relationship between alkaline ions and micro–nano pore size cannot be characterized at a macro level. Instead, the leaching rate of alkaline ions in fly ash is influenced by pore size at the micro level, exhibiting dynamic changes over time at the macro level. Therefore, this research now focuses on discussing the relationship between alkaline-ion leaching concentration and time, and subsequently, the relationship between mineralization efficiency and time. Changes in the concentration of alkaline ions not only affect mineralization efficiency but also impact the structure and mechanical properties of the material. Consequently, this article conducted further analysis to examine the relationship between the ion leaching concentration and the material structure, as well as its mechanical strength.

4.1. Effect of Leaching Time on Mineralization Degree

Figure 5 illustrates the relationship between the concentration of alkaline ions in the reaction environment and the soaking time obtained from the experiment. It demonstrates an initial increase followed by a decrease. Initially, the leaching rate of alkaline ions in fly ash is higher than the rate of formation and consumption of calcium silicate. As time passes, the leaching ion concentration increases. However, in the later stage of the reaction, the leaching rate becomes lower than the rate of calcium silicate formation and consumption. Consequently, the ion concentration gradually decreases over time. Figure 6 presents the mineralization efficiency of the reaction under different soaking times. Among the experimental samples, the mineralization degree without soaking treatment was 28.35%. With a soaking time of 12 h, the mineralization degree increased to 30% approximately. The maximum mineralization degree of 43.85% was achieved with a soaking time of 24 h. Beyond 24 h, the degree of mineralization slightly decreased, reaching 42.88% and 38.98% after soaking for 48 h and 72 h, respectively. The overall trend of mineralization degree with soaking time exhibits an initial increase, followed by a slight decrease, and eventually reaching a plateau. This trend aligns with the analysis of changes in the ion concentration. After soaking the fly ash, the concentration of alkaline ions in the liquid reaction environment gradually increases over time. In the early stage of soaking, the alkaline ions in the fly ash rapidly dissolve and diffuse, leading to a rapid increase in the ion concentration and reaction with CO₂. In the later stage of soaking, the ionic concentration in the liquid environment stabilizes, and the number of generated products remains relatively constant.

Figure 7 shows the mineralization efficiency of the mineralization reaction with supercritical CO₂ under different soaking times. Among the experimental samples, the trend is similar to that of non-supercritical CO₂ in Figure 6. Comparing the two figures, the degree of mineralization has increased by about 10% with supercritical CO₂.

4.2. Effect of Leaching Time on Product Composition and Micromorphology

Figure 8 and Figure 9 present SEM photos and the energy spectra of samples with different soaking times, illustrating the carbonization products. The original fly ash particles exhibit a relatively smooth surface with a regular spherical shape, along with a small number of attached tiny particles composed mostly of soluble substances and quartz. After 24 h of soaking, the particle surface gradually changes from smooth to rough, with the appearance of aggregated precipitates, indicating the dissolution of a significant number of alkaline substances. After 48 h of soaking, an increased presence of silicate (CSH) can be observed, potentially an intermediate product of fly ash hydration, with continuous formation and precipitation of disordered precipitates on the particle surface. The EDS analysis revealed that the main components of the precipitate were Si, Mg, Al, and Ca. After 72 h of soaking, the original fly ash particles become covered by a dense layer of hydration products, and the fly ash spheres are filled with sediment. As the hydration reaction products continue to generate and cover the particle surface, a layered hydration product layer forms, enclosing the products. The longer the soaking time, the denser the generated carbonate becomes, promoting the improvement of compressive strength. Additionally, more silicates (CSH) can be observed, which may indicate the intermediate products of fly ash hydration. During the subsequent curing process, the carbonate of the mineralized specimen gradually becomes enclosed and cemented by CSH, effectively filling the product’s pores.

4.3. Effect of Leaching Time on Compressive Strength

Figure 10 illustrates the compressive strength of the specimen after a mineralization reaction for 3 h under different soaking times. The specimen without any soaking treatment exhibited the lowest compressive strength of 0.099 MPa. However, as the soaking time exceeded 12 h, the compressive strength significantly increased. Notably, the specimens soaked for 24 and 48 h displayed the highest strengths of 0.658 MPa and 0.592 MPa, respectively, which is nearly six times higher than the strength of the specimen without soaking treatment. The compressive strength of specimens subjected to soaking for 12 to 72 h is higher compared to those without soaking treatment. The trend in compressive strength with soaking time indicates an initial sharp increase, followed by a slight decrease. This increase in strength is attributed to the soaking of fly ash, leading to the continued dissolution and diffusion of the alkaline metal ions into the liquid environment. This process promotes mineralization reactions, enhancing the degree of mineralization. Prolonged soaking results in higher concentrations of alkaline ions in the liquid environment, leading to an increased generation of carbonate and amorphous calcium silicate hydrate (CSH) through the mineralization reactions. This densifies the specimen, consequently increasing its strength. Consequently, the growth rate of compressive strength gradually diminishes. For instance, compressive strength increases by 565% when soaking time rises from 0 to 24 h, but the growth rate slows down as the carbonization time extends from 24 to 72 h.

Figure 11 presents the mineral pressure strength of the specimen during mineralization reaction with supercritical CO₂ at different soaking times. The compressive strength of the specimen without any soaking treatment is the lowest, measuring 0.519 MPa. However, when the soaking time exceeds 12 h, there is a significant increase in compressive strength. Notably, the specimens soaked for 24 h and 48 h exhibit the highest strengths, measuring 1.324 MPa and 1.059 MPa, respectively, which is more than twice the strength of the specimen without soaking treatment. Furthermore, the compressive strength of the specimens with immersion times ranging from 12 to 72 h is higher than that of those without immersion treatment. The overall trend of compressive strength with immersion time indicates a sharp increase initially, followed by a slight decrease, and eventually stabilizing. Comparatively, compressive strength is doubled when compared to non-supercritical CO₂ reactions. This can be attributed to the similar density of supercritical CO₂ and water, requiring minimal critical energy to overcome surface tension. Consequently, it enhances the transport capacity of CO₂ in water, facilitating rapid diffusion into the liquid environment to provide the necessary reactive substances for mineralization.

Figure 12 illustrates a linear relationship between the degree of mineralization and the compressive strength in the experimental specimens, where higher mineralization correlates with greater strength. As soaking time increases, the rate of ion leaching decreases, reaching a maximum ion concentration and slowing down the reaction speed.

The fly ash specimens prepared by the reaction of fly ash and carbon dioxide mineralization have good uniaxial compressive strength and pore structure. The mineralization reaction conditions directly determine the degree of mineralization reaction between fly ash and carbon dioxide and the amount of mineralization reaction products, so the mineralization reaction conditions have an important influence on the internal pore structure and mechanical strength of the mineralized fly ash specimens. Experimental studies have shown that immersion treatment, hot steam treatment, ultrasonic treatment, and stirring treatment will have a positive effect on the internal structure and strength of mineralized fly ash specimens, with immersion treatment being the most significant. Both the improvement in the strength of the material and the optimization of the microstructure are similar to the results of other scholars, such as Di.

During the mineralization reaction between fly ash and carbon dioxide, the concentration of leached dissolved alkaline ions in fly ash is positively correlated with the mineralization reaction products, and the leaching and dissolution rate of alkaline ions directly affects the mineralization reaction rate. The multi-scale effect of micro- and nano-pores in fly ash is the main influencing factor restricting the diffusion rate of alkaline ions in fly ash, and the alkaline-ion leaching rate in fly ash is affected by pore size at the microscopic level, while showing a dynamic change with time at the macroscopic level. The change in the concentration of alkaline ions affects the mineralization efficiency and then the structure and mechanical properties of the material. Through experimental research, it has been found that because the alkaline-ion leaching rate in the early stage of fly ash is greater than the rate of calcium silicate formation and consumption, and the alkaline ion concentration increases first and then decreases with the soaking time, the leaching ion concentration increases with time, but the leaching rate in the later stage of the reaction is smaller than the calcium silicate formation and consumption rate, and the ion concentration gradually decreases with the increase in time. Combined with the analysis of ion concentration, the overall trend of mineralization degree with soaking time showed a trend of first increasing, then decreasing slightly, and finally flattening. As the mineralization reaction product continues to form and coat the surface of the original particles, a layered layer of hydration products is formed. The longer the immersion time, the denser the generated carbonate becomes, and the tightly packed carbonate fills the pores, improving the compressive strength of the specimen. After the soaking treatment, the structure of the mineralized fly ash specimen can be denser and more stable, the mechanical properties are better, and the strength of the specimen is nearly six times higher than that of the specimen without immersion treatment.

5. Conclusions

The dissolution and diffusion characteristics of alkaline ions in fly ash play a crucial role in the mineralization reaction of CO₂. To address this scientific issue, a multi-scale dynamic diffusion model of ions was developed based on the micro–nano pore structure of fly ash. This model established the correlation between the ion leaching rate and micro pore structure, as well as elucidated the dissolution and diffusion process of the alkaline ions in fly ash. The model also analyzed the change in medium mass concentration over time. The experimental validation of the model confirmed the relationship between the fly ash soaking time and the ion leaching concentration, the link between the ion leaching concentration and the mineralization degree, and the impact of the mineralization degree on the mechanical strength of specimens. This study revealed the influence of alkaline ion concentration on the mineralization degree and mechanical strength of fly ash materials. The main findings are as follows:

(1)

A mathematical model was developed to study ion leaching and diffusion in the micro–nano pores of fly ash. The model effectively predicts the evolution of the alkaline ion concentration during the dissolution and diffusion process.

(2)

Alkaline ions dissolve and diffuse rapidly, leading to a rapid increase in the ion concentration. As the time progresses, the ion concentration stabilizes.

(3)

The mineralization degree shows an initial increase, followed by a slight decrease, and eventually levels off with soaking time.

(4)

The compressive strength initially sharply increases with the mineralization degree, followed by a slight decrease.