Thermodynamic Analysis in the Microscopic Structural Characterization of Civil Engineering Materials

The high strength and excellent durability of concrete make it indispensable in engineering projects that require high structural load-bearing capacity. Due to its high compressive strength, concrete can withstand huge loads and maintain the stability and safety of structures during long-term use. Whether in high-rise buildings, large bridges, tunnels, or major infrastructure projects such as water conservancy and dams, concrete always plays a crucial role in supporting structures. The plasticity of concrete allows it to be freely shaped under different construction conditions, adapting to various complex architectural design requirements. Whether in straight shapes or curved structures, concrete can be precisely formed through different pouring and molding techniques. This high degree of plasticity enables concrete to adapt to various structural designs and even find effective application in some innovative architectural designs.

As the most important foundational material in civil engineering, the mechanical properties and durability of concrete are directly influenced by its microstructure, especially the pore structure. Initially, research mainly focused on the porosity of concrete, with the belief that porosity had a direct relationship with the macroscopic mechanical properties of concrete. However, with the continuous deepening of research, scholars gradually realized that the mechanical properties of concrete are not only related to porosity but also closely linked to other key parameters of the pore structure, such as pore size distribution, pore morphology, and pore connectivity. These parameters determine the performance of concrete under stress, its durability, and its high-temperature resistance. As an important component of the concrete microstructure, the complexity of the pore structure makes traditional single-porosity models insufficient to fully reflect the mechanical behavior of concrete. The fractal characteristics of the pore structure are considered an effective way to describe the complexity and heterogeneity of the concrete pore structure, which is one of the reasons why many studies have attempted to establish concrete pore structure models using fractal theory in recent years.

Currently, fractal models based on pore structure testing are widely used in the analysis of concrete's microstructure, especially with the combination of mercury intrusion and optical methods. Although idealized geometric models, such as the Menger sponge model, space-filling models, and pore-axis fractal models, provide a rough description of the pore structure to some extent, the simplifying assumptions of these models lead to certain differences with the actual pore structure, thus affecting the accuracy and reliability of the models. Particularly in porous and irregular materials like concrete, idealized geometric assumptions often fail to adequately reflect the complex pore morphology and distribution characteristics. Therefore, establishing a fractal model that more closely aligns with the actual pore structure is crucial.

Thermodynamics-based methods for characterizing the microstructure of civil engineering materials provide a new approach for studying the pore structure of concrete. Through thermodynamic analysis, the pore structure of concrete can be combined with its thermodynamic properties, revealing the influence mechanism of the pore structure on the macroscopic mechanical properties of concrete. As a mathematical tool capable of quantitatively describing complex structures, the fractal model can effectively capture the self-similarity and multi-scale features of concrete's pores, thus more precisely simulating the microstructure of concrete. Thermodynamics-based fractal models not only consider the geometric features of concrete's pores but also integrate thermodynamic theory to further reveal the relationship between pore structure and the mechanical properties of concrete. Using this model, we can more accurately predict the mechanical behavior of concrete under different environmental conditions, particularly under high temperature or other extreme conditions.

Below, this paper provides a detailed introduction to the construction steps of the thermodynamics-based fractal model for concrete in civil engineering:

Step 1: The first step in constructing the thermodynamics-based fractal model is to measure the relationship between the pore volume and pore diameter of concrete using the mercury intrusion method. Mercury intrusion is a classic porosity testing method, the principle of which involves applying different pressures to gradually push mercury into the pores. Since mercury does not wet concrete, only sufficiently large pores can be completely filled by mercury. Therefore, by controlling the applied pressure and monitoring the amount of mercury intrusion N, multi-dimensional data about the concrete's pores can be obtained. These data are reflected in the pressure-intrusion curves, indicating the pore size distribution. In this process, the relationship between the applied pressure on mercury o and the mercury intrusion amount N is influenced by the pore structure, and the change in surface energy generated by the mercury entering the pores can be quantitatively described through thermodynamic analysis. Specifically, the applied pressure on the mercury is equal to the increase in the surface energy of the mercury inside the pores. Assuming the surface force of the mercury meniscus is represented by δ, the contact angle between the concrete surface and mercury is represented by ϕ, and the surface area of the concrete pores is represented by T, the equation is as follows:

$\int_0^N o d N=-\int_0^t \delta{COS} \varphi d T$       (1)

Through this thermodynamic relationship, we can obtain the structural information of the concrete pores, further revealing the geometric shape of the pores and their impact on the macro-mechanical properties of concrete.

Step 2: After obtaining the pore structure data from the mercury intrusion test, dimensional analysis is used to reveal the geometric characteristics of the pore structure and correlate them with the intrusion volume N, pore diameter e, and other physical quantities. Dimensional analysis is a mathematical method that reveals the inherent relationships of physical quantities by deriving their dimensions. For porous materials like concrete, there is a complex dependency between the pore surface area T, pore diameter e, and the intrusion volume V. Through dimensional analysis, these physical quantities can be normalized to obtain a series of dimensionless quantities, which can then be used to derive their quantitative relationships. Let the average pressure during the u-th intrusion be denoted as o^-_u, the intrusion volume during the u-th intrusion be denoted as ΔN_u, the number of pressure intervals during intrusion be denoted as v, the pore diameter during the v-th intrusion be denoted as e_v, and the cumulative intrusion volume be denoted as N_v. The coefficient is denoted as Z, and the fractal dimension calculated is denoted as F_t. For the intrusion operation, the relationship between o and N can be discretized as follows:

$\sum_{u=1}^v \overline{o_u} \Delta N_u=Z e_v^{2-F_t} N_v^{F_t / 3}$           (2)

This fractal model quantitatively describes the complexity of the concrete pore structure through the fractal dimension Ft. The fractal dimension is a dimensionless parameter that represents the self-similarity and complexity of the pore surface or volume. Specifically, the larger the fractal dimension, the higher the complexity of the pore structure, and the ratio of pore surface area to pore volume also increases. Further, the above equation can be reorganized as:

$\sum_{u=1}^v \overline{o_u} \Delta N_u=Z e_v^2\left(\frac{N_v^{1 / 3}}{e_v}\right)^{F_t}$        (3)

Let $Q_v=\sum^v{ }_{u=1} O_u^{-} \Delta N_u, W_v=N^{1 / 3} v / e_v$, then:

$\lg \left(\frac{Q_v}{e_v^2}\right)=F_t \lg W_v+\lg C$        (4)

The above steps quantify the complexity of the pore structure and describe its scale self-similarity through mathematical formulas. By analyzing the relationship between the applied pressure and the intrusion volume during the mercury intrusion process, a series of parameters related to the pore structure can be derived. These parameters not only reflect the pore diameter distribution characteristics but also reveal the morphology and distribution patterns of concrete pores.

Step 3: Based on the dimensional analysis and fractal model established, further calculate key physical quantities such as Q_v/e²_v and W_v, and plot them as curves. Q_v represents the surface area of the pores, while e²_v and W_v reflect the geometric shape and distribution characteristics of the pores, respectively. By calculating these physical quantities and fitting the curves, the fractal dimension of the concrete pores can be further determined. The slope of the curve is directly related to the fractal dimension, and the shape of the curve reflects the distribution pattern and variation law of the pore structure. Through this process, we can quantitatively analyze the self-similarity of concrete pores and reveal the evolution characteristics of its microstructure, especially under environmental factors like carbonation and freeze-thaw, where the changes in pore structure directly affect its mechanical properties and durability.

Step 4: Finally, by verifying the pore structure data of actual concrete samples, the fractal model based on the thermodynamic relationship can be further optimized. When processing the pore data of concrete after 0 days of carbonation, a scatter plot of lgQ_v/e²_v-lgW_v can be drawn, and the fractal dimension can be determined through regression analysis. Figure 1 shows an example of the scatter plot of the thermodynamic relationship lgQ_v/e²_v-lgW_v. The data points in the scatter plot present a certain linear relationship, and its slope is the fractal dimension of the concrete pore structure. Through the analysis of the scatter plot, the accuracy and effectiveness of the established fractal model can be further validated.

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Figure 1. Example of the scatter plot of the thermodynamic relationship lgQ_v/e²_v-lgW_v

This study primarily focuses on the relationship between the microstructure and mechanical properties of concrete, especially the evolution of pore structure under high-temperature conditions and its effect on mechanical properties. Table 1 shows the experimental data for three different concrete mix proportions, where the control group concrete and the concretes with fly ash or slag replacing cement were compared for their effects on the performance of concrete. Table 2 lists the chemical composition of various concrete raw materials, showing the major chemical components of control cement, fly ash, and slag. In the cement, components like CaO, SiO₂, and Fe₂O₃ dominate, while fly ash and slag contain higher amounts of SiO₂, Al₂O₃, and MgO. This chemical composition combination is closely related to the mechanical properties and pore structure of concrete, providing support for predicting the mechanical properties of concrete under high-temperature conditions in thermodynamic models.

Table 3 provides detailed data on the fractal dimension of concrete as temperature changes under high-temperature conditions. As the temperature rises from room temperature (20℃) to 200℃, 400℃, 600℃, and 800℃, the fractal dimension of concrete changes. For example, at 20℃, the fractal dimension is generally higher, around 1.856. As the temperature increases, the fractal dimension slightly decreases. At 200℃, the fractal dimension drops to 1.8125, at 400℃ and 600℃, it decreases to 1.8124 and 1.7985, and at 800℃, it slightly rebounds to 1.7954. Moreover, some data show fluctuations at high temperatures, demonstrating a certain nonlinear change. For instance, at 400℃, the fractal dimension rises to 1.8562, while at 800℃, there is a slight decrease. These changes suggest that high temperatures affect the self-similarity characteristics of the microstructure of concrete, particularly at higher temperatures, where the fractal dimension slightly decreases. This is related to the impact of high temperatures on the pore structure and material strength.

Table 1. Concrete mix proportions for civil engineering/kg/m³

Table 2. Chemical composition of raw materials for civil engineering concrete (%)

Table 3. Fractal dimension of concrete under high-temperature conditions

Figure 3 displays the change in the fractal dimension of concrete over time at different high temperatures. From the data in the table, it can be observed that the fractal dimension of concrete fluctuates with time at different temperatures. Under normal temperature conditions, the fractal dimension remains stable, with values of 1.838 at 7 days, 1.842 at 14 days, and 1.858 at 28 days, indicating that the microstructure of the concrete undergoes minimal changes and maintains a relatively consistent self-similarity feature at normal temperature. As the temperature rises, the impact of temperature on the fractal dimension of concrete gradually becomes evident. At 200℃, the fractal dimension slightly increases, reaching 1.868 at 28 days; however, when the temperature rises to 400℃, 600℃, and 800℃, the fractal dimension generally decreases. Especially at 600℃ and 800℃, the fractal dimension significantly decreases, reaching 1.828 and 1.776, respectively, indicating that the self-similarity of the microstructure of concrete is significantly affected by high temperatures. It is important to note that at higher temperatures, the change in the fractal dimension is influenced not only by the direct effect of temperature but also by changes in the hydration reaction of the cement, an increase in porosity, and material degradation.

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Figure 3. Relationship between fractal dimension of concrete and temperature

Table 4 shows the distribution of harmless and less harmful pore ratios in concrete under different temperature conditions. The proportion of harmless pores generally decreases as the temperature increases. At room temperature, the proportion of harmless pores is highest, particularly at 3 days, reaching 35.21%, and fluctuates slightly over time, decreasing to 37.85% at 28 days. However, when the temperature reaches 200℃, the proportion of harmless pores decreases to 31.44%, further reducing to 32.15% at 400℃. At 600℃, the proportion significantly increases to 43.21%, but drops sharply to only 5.14% at 800℃. The proportion of less harmful pores also changes significantly under high-temperature conditions, gradually decreasing as the temperature increases. At room temperature, the proportion of less harmful pores is 26.25% at 3 days, but under high-temperature conditions, particularly at 400℃ and 600℃, the proportion significantly decreases. For example, at 400℃, the proportion is 22.15%, while at 800℃, it decreases to 7.32%. This indicates that the evolution of the concrete's pore structure under high temperature has a significant impact on the distribution of harmless and less harmful pores, especially at high temperatures, where the pore structure of the concrete becomes rougher and more irregular.

Table 5 shows the variation trend of composite porosity of concrete under different temperature conditions over time. Composite porosity reflects the overall porosity of concrete, encompassing all types of pores. Under room temperature conditions, the composite porosity gradually decreases over time, from 58.265% at 3 days to 46.235% at 28 days, indicating that as the hydration reaction progresses, the porosity decreases and the structure becomes denser. As the temperature increases, the change in composite porosity exhibits significant fluctuations. At 200℃, the composite porosity decreases to 51.245%, and at 7 days, it drops to 42.361%. At 400℃, the composite porosity increases again to 48.124%, and remains high at 14 and 28 days. At 600℃, the composite porosity gradually increases to 54.147%, and then gradually decreases under higher temperatures. At 800℃, the composite porosity significantly increases, particularly at 7 days when it reaches 35.625%, and increases to 41.258% at 28 days. These changes indicate that the pore structure of concrete undergoes significant changes under high-temperature conditions. Especially at extremely high temperatures, the increase in porosity indicates significant internal damage to the concrete structure.

Table 4. Distribution of harmless and less harmful pore ratios of concrete used in civil engineering under high temperature conditions

Table 5. Composite porosity of concrete used in civil engineering under high-temperature conditions

Table 6. Pore parameters of concrete used in civil engineering under high-temperature conditions

Table 6 shows the variation trend of pore parameters of concrete under different temperature conditions. The data in the table includes pore size, pore volume, and pore distribution. At room temperature, the pore parameters of concrete are relatively stable. At 3 days, the pore parameter is 612.214, at 7 days it is 587.265, at 14 days and 28 days it is 589.244 and 468.235, respectively, showing that as the hydration reaction progresses, the porosity gradually decreases, and the concrete pore structure becomes denser. However, as the temperature increases, the pore parameters show significant fluctuations. At 200℃ and 400℃, the pore parameters slightly increase. For example, at 200℃, the pore parameter at 3 days is 641.235, and at 7 days it is 423.256, showing that the volatilization of water and the formation of micro-cracks caused by high temperatures lead to an increase in pore volume. At 600℃ and 800℃, the pore parameters increase dramatically, especially at 800℃, where the pore parameter at 3 days reaches 887.124, indicating that under extremely high temperatures, the pore structure of concrete is severely damaged, and the pore volume and pore size increase significantly, causing the structure to loosen and leading to a significant decrease in overall performance.

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(a) Differential thermal scanning curve

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(b) Thermogravimetric curve

Figure 4. Differential thermal scanning curve (a) and thermogravimetric curve (b) of concrete used in civil engineering

The analysis results from the differential thermal scanning curve shown in Figure 4(a) can be used to investigate the thermodynamic behavior of concrete at high temperatures, exploring the relationship between the microstructure of concrete and its mechanical properties. In the differential thermal scanning curve analysis, as the temperature increases, the exothermic and endothermic characteristics of concrete change significantly. In the low-temperature range, concrete exhibits a relatively smooth exothermic phenomenon. This phase mainly involves the evaporation of moisture and a small amount of hydration reaction, indicating that the microstructure of concrete gradually densifies during the hydration process, with a decrease in porosity. However, when the temperature exceeds 200℃, especially in the range of 400℃ to 600℃, the curve shows a distinct endothermic peak, indicating that within this temperature range, hydration products begin to decompose, and the degradation of the microstructure becomes evident, with the pore volume rapidly increasing. Beyond 600℃, especially at 800℃, the differential thermal scanning curve shows a strong endothermic phenomenon, indicating the intense decomposition of hydration products and minerals in concrete, leading to a significant increase in porosity and severe physical and chemical degradation of the concrete structure.

Through the analysis of the thermogravimetric curve in Figure 4(b), the thermal stability and mass changes of concrete under high temperatures can be further explored, reflecting the physical and chemical changes of concrete under different temperature conditions. According to the content of this study, the thermogravimetric curve shows some mass loss in the low-temperature range from 20℃ to 200℃, mainly caused by the evaporation of free and combined water in concrete. As the temperature further increases to 200℃ to 400℃, the mass loss accelerates, indicating the dehydration of hydration products (such as calcium hydroxide) and the gradual decomposition of hydration reactions. Especially between 400℃ and 600℃, the thermogravimetric curve shows a significant mass loss, indicating further decomposition of hydration products and the phase transition of silicate minerals, resulting in an accelerated mass loss of concrete. When the temperature exceeds 600℃, especially at 800℃, the mass loss in the thermogravimetric curve increases significantly, exceeding 50%. This stage is mainly caused by the decomposition of calcium silicate hydrates and other minerals in the concrete, and the loss of the hydrated structure causes the internal pores to expand and further exacerbates the propagation of cracks.

Combined with the study's findings and the analysis of the thermogravimetric curve, it can be concluded that high temperatures significantly affect the thermal stability and microstructural evolution of concrete. In the low-temperature range, the mass loss in concrete is mainly due to the evaporation of moisture, which corresponds to the decreasing trend of porosity in concrete during this phase. As the temperature increases, especially between 200℃ and 400℃, the accelerated mass loss indicates the decomposition of hydration products and changes in hydration reactions, leading to an increase in porosity. In the 400℃ to 600℃ range, the mass loss rate of concrete further accelerates, and the decomposition of hydrates and phase transitions of minerals causes the microstructure of concrete to degrade, resulting in increased pore volume and a significant decrease in mechanical properties. Especially when the temperature exceeds 600℃, the thermogravimetric curve shows significant mass loss, indicating that the concrete's structure is severely damaged. This phenomenon directly leads to a significant loss in the compressive strength and durability of concrete.