Quantitative Relationship Between Strength and Porosity of Nano-Silica-Modified Mortar Based on Fractal Theory

Abstract

Nano-silica (NS) is an ideal modifier for mortar materials, and exploring the evolution of the fractal dimension of the pore structure in NS-modified mortar is crucial for elucidating the mechanism by which NS enhances mortar strength. In this study, NS reinforced mortar was prepared using an NS sol solution, which inhibited the aggregation of NS particles. The relationship between the strength and pore structure of NS-modified mortar was quantitatively analyzed based on fractal dimension theory and gray correlation degree. The experimental system evaluated the mortar strength, pore structure distribution, and micro-morphology. Based on this evaluation, the fractal dimension of the mortar pore volume was calculated in detail. Subsequently, models for mortar strength and NS content were further established using grey analysis. The results indicate that NS significantly enhances the strength of mortar while also increasing its porosity due to reduced fluidity. NS can improve the compressive strength of mortar by up to 35%. The curve fitting of volume fractal dimension and box dimension is effective and can accurately reflect the complexity of the pore structure. The calculation of the grey correlation analysis model shows that the impact of varying silica content on the mechanical properties of mortar specimens is not linear; the distribution and quantity of bubbles are the main factors affecting the strength of the specimen.

1. Introduction

Cement-based materials are widely used in construction fields, such as high-rise buildings, bridges, and dams [1,2]. Additionally, concrete ranks as the world’s second-largest consumer product, trailing only water resources [3]. However, it exhibits certain drawbacks, such as low strength and poor toughness in the initial stages. A crucial step in enhancing the practicality of concrete structures involves improving their mechanical properties and durability [4]. With ongoing industrialization, the defects of concrete, such as long-term damage under external environmental and load conditions, readily impact the engineering structure and its service life. Traditional concrete structures no longer meet the escalating demands for usage conditions and environmental protection [5]. In practical engineering applications, many scholars focus on enhancing the cement base’s strength and toughness, mitigating concrete shortcomings. During the destruction and screening of construction waste in China, fine particles smaller than 0.16 mm are produced [6]. These particles are termed Recycled Concrete Powder. Unlike recycled powder, nanomaterials significantly bolster concrete performance. Nano-CaCO₃, nano-SiO₂, and carbon nanotubes are among the prevalent nanomaterials used [7].

Nano-SiO₂ (abbreviated as NS) represents a novel nano-scale high-performance material characterized by a vast specific surface area, small particle sizes, and potential volcanic ash performance [8]. Nanoparticles can fill the micropores in concrete, reducing their quantity and decreasing the material’s porosity, thus rendering the concrete denser. Incorporating a small quantity of NS minimizes slurry flow and shortens setting times, significantly enhancing concrete’s early strength. NS enhances the interface transition zone (ITZ) between an aggregate and cement at the microscale, thereby improving the bond strength of mortar structures. On a macroscopic scale, NS improves a concrete structures’ mechanical properties, durability, and operational performance [9,10,11,12]. NS added to mortar reacts with the hydration product Ca(OH)₂ in the volcanic ash, not only consuming Ca(OH)₂ but also speeding up the hydration rate of cement, thus improving the concrete’s compressive strength [13,14,15].

The primary component of NS in cement-based materials is approximately 3%, which can significantly enhance the early strength and tensile properties of cement mortar [16,17,18]. Prakasam et al. [19] indicated that tensile strength increases by 17% to 24% when the content ranges from 1% to 2% compared with the control group. Ma [20] studied the maximum tensile strength of mortar containing NS, revealing that the optimal strength is achieved at a content of 1.2%, representing a 17.42% increase over the reference group. However, exceeding a certain threshold leads to decreased compressive strength due to NS agglomeration [21]. Xu et al. [22] used an NS sol to prepare NS-reinforced mortar, which significantly modified the agglomeration of NS in NS.

Concrete durability depends on permeability, which is associated with internal porosity. C-S-H and NS fill concrete pores, enhancing the compactness, mechanical properties, and durability of mortar materials [23,24]. Shen et al. [25], through a microstructural analysis, recorded the changes in pore diameter and number of mortar pores with 3% NS content. The addition of NS significantly reduced crack size and number, enhanced the internal structure density of mortar, and decreased the permeability of chloride ions.

As an advanced building material, NS-modified mortar has been extensively studied and recognized for its potential to enhance mortar performance. While the addition of NS can significantly improve the internal phase and pore network structure of polymer-modified cement mortars, the interactions between NS and different polymers vary, leading to inconsistent effects on the performance of different polymer-modified cement mortars. Economically, although NS modification can greatly enhance mortar performance, cost is a critical factor. In pursuing high performance, we must also consider the impact of material costs on the overall project economics. Globally, the promotion and use of NS-modified mortar may positively affect environmental protection and resource utilization. NS-modified mortar has demonstrated substantial potential in improving material properties, but its limitations, economic costs, and potential environmental and economic impacts must be considered when applying it.

Existing research on the pore structure of NS-reinforced cement mortar primarily relies on pore size distribution indices and corresponding pore content. However, these metrics do not adequately capture the underlying mechanisms of NS-reinforced cement mortar. In this study, fractal theory was employed to investigate the relationship between the pore structure fractal dimension and the strength of NS-modified mortar, aiming to elucidate the correlation between the macro and microstructure of the mortar. Fractal theory has been demonstrated to quantitatively characterize the complexity and irregularities of the pore structure in cement-based materials [26], thereby linking the pore structure characteristics to the macro-properties of these materials. This paper addresses the complex distribution characteristics of pore structure and its evolution on the mechanical properties of NS-reinforced mortar. Using various fractal theoretical methods, it deduces the fractal dimension of the pore structure and establishes a correlation between the fractal dimension of the pore structure and the strength of the mortar based on gray correlation analysis. This research has significant theoretical implications for understanding how NS enhances mortar properties and promotes the broader application of NS in various projects.

2. Experimental Design

2.1. Experimental Materials and Mix Ratio

NS sol, standard sand, reference cement, and superplasticizer were utilized as raw materials. Moreover, the excellent dispersion of NS sol prevents the agglomeration of NS particles in the mortar [20]. The NS sol is shown in Figure 1a and the SEM image of the NS particles is shown in Figure 1b, and the specific composition of the NS sol is presented in

Table 1. Properties of NS sol.

Appearance	Particle Size (nm)	Content	PH	Solvent	Specific Gravity	Specific Surface Area (cm2/g)	Viscosity (mpa.s)
Translucent liquid	30	30%	7.9	Water	1.202	250 ± 30	3.39

. Tetraethyl orthosilicate (TEOS) was used as a precursor and mixed uniformly with ethanol and deionized water. Under alkaline conditions, TEOS undergoes simultaneous hydrolysis and condensation reactions to form an NS sol system. The mortar’s fine aggregate consists of sand with an apparent density of 2744 kg/m³ and a fineness modulus of 2.94.

2.2. Mortar Configuration and Maintenance

Mortars containing 1.5% and 3% NS (S1 and S2) and mortars without NS were set as the control group (S0). These samples maintained a water/cement ratio of 0.5, with consistent proportions of sand, cement, and PS. The formulation of the various samples is presented in

Table 2. Mix proportions.

Specimen	Cement (g)	Sand (g)	Water(g)	Admixture
				NS	Water Reducer
S0	450	1350	225	0	1%
S1	450	1350	225	1.5%	1%
S2	450	1350	225	3%	1%

. The sizes of the mortar specimens made by the prescribed method were 40 × 40 × 40 mm and 40 × 40 × 160 mm, respectively, and mortar fluidity was assessed prior to casting the specimens. The experimental specimens were prepared by using a triple test mode, and three groups of specimens were set up to compare with each other. The preparation and strength testing of the cement mortar samples were carried out in accordance with GB/T 17671-2021 (ISO 679) standards [27]. The number settings of the experimental groups and the ratio of raw materials are shown in Table 2.

After the completion of specimen fabrication, they were initially cured in a standard curing box for 48 h, followed by immersion in water for further curing after mold removal. Subsequently, the specimens were subjected to standard room temperature curing at 25 °C for durations of 3, 7, 14, and 28 days, respectively. After curing, the specimens were removed from water and subjected to compressive and flexural strength evaluation. Subsequently, their pore structures were observed and analyzed using a SEM.

2.3. Mortar Performance Test

2.3.1. Slurry Fluidity Test

The flowability test of the mortar complies with the “Test Methods for Flowability of Cement Mortar” (GB/T 2419-2005) [28]. The freshly mixed mortar is divided into two layers and compacted in a conical mold. Subsequently, the mold is removed, and any excess mortar is scraped off. Finally, the mold is lifted vertically upwards, and the mortar flowability test apparatus is initiated. Upon completion of compaction, both the maximum diffusion diameter and the vertical diameter of the bottom surface are measured using calipers, with the average value calculated to a precision of 1 mm. This procedure evaluates the fluidity of the mortar.

2.3.2. Mortar Strength Test

After removing the 40 × 40 × 160 mm mortar specimens, they were placed on a specimen bending loading frame. The loading device was then initiated, employing a three-point bending loading method with a loading rate set at 0.36 mm/h and a span of 120 mm. Following the bending failure of the specimens, the affected 40 × 40 mm samples were placed on compression loading fixtures and compressed at a rate of 0.2 mm/min. A total of 12 specimens were tested for strength, with 3 specimens tested at each maturity stage. The strain rates for compressive and flexural tests were 5 × 10⁻⁵ s⁻¹ and 2.5 × 10⁻⁵ s⁻¹, respectively.

2.3.3. Pore Structure Test

After curing for 28 days, the mortar was tested for porosity structure using the linear traversal method specified in the “Code for Testing of Hydraulic Concrete” (Chinese National Standard SL352-2020) [29]. After the bending test, slices measuring 30 × 30 mm were selected. These slices were approximately 15–20 mm thick. Initially, three central and three edge slices were considered for measurement from each group. The slices were ground and polished using a grinder rotating at a speed of 50–60 revolutions per minute for 30 min. The samples were then dried at 45 °C for three hours to dry the polished surfaces. After compression, any excess powder was removed. Subsequently, the processed slices were introduced into an automated pore structure analyzer for the examination and computation of pore distribution and porosity. The pore size analysis range of the cement mortar obtained by a pore structure analyzer was 3 to 1500 μm. Figure 2 provides an overview of the pore structure test.

2.3.4. SEM Test

The analysis used a SEM to examine the microstructure of mortar specimens with varying amounts of NS. Subsequently, bending tests were conducted; cube particles measuring 1 cm³ were extracted from the fractured specimens at 3, 7, 14, and 28 days, respectively, and preserved in ethanol to prevent hydration. They were then analyzed using a scanning electron microscope to investigate the distribution of NS particles in the mortar and their effects on porosity and hydration products.

3. Fractal Model and Calculation Method

Nanomaterials possess characteristics that can modify the pore structure of mortar and its overall mechanical properties. Observing the internal pore structure of nanomaterials requires the utilization of fractal dimensions for representation. There are several methods for characterizing the fractal dimension of pore structures, with commonly used techniques including the Mengler sponge model [30], box-counting method, and size method.

The Mengler sponge model calculates the volume fractal dimension and pore structure fractal dimension of mortar. This model classifies the pore structure based on the size of bubbles, allowing for the determination of different grades’ quantities and the total porosity of the specimen. The description of the Mengler sponge model is as follows:

The initial element is defined as a regular hexahedron with an edge length of R. It is then divided into m³ small cubes, each with an edge length m/R. By randomly removing n small cubes, the remaining cubes are N = m³ − n. This process is iterated continuously, removing cubes of different sizes each time, which represent the pores in the mortar material. As the iterations progress, the size of the remaining cubes decreases, while the number of cubes increases. After k iterations, the size r_k and quantity N_k of the remaining cubes are represented by the following formula:

r_k = R/m^k

(1)

N_k = N₁^k = (m³ − n)^k

(2)

Accessibility:

k = lg(R/r_k)/lgm

(3)

Substitute (3) into (2):

N_k = (r_k/R)^−D

(4)

The fractal dimension is denoted by

D = lg(m³ − n)/lgm

(5)

The volume of the remaining cube can be expressed as shown above:

V_k = r_k³N₁^k = r_k^3−D/R^−D

(6)

The pore volume is

Vφ = R³ − V_k

(7)

Porosity can be defined as follows:

φ = Vφ/R³ = 1 − (r_k/R)^3−D

(8)

The box is described as follows: define F as any nonempty bounded subset of R_n and let δ be the box size. Then, N(δ) represents the number of boxes of size a needed to cover F:

N_δ(F)∝δ^−D

(9)

As the size of the box δ approaches zero, a constant D (referred to as the box dimension of F) becomes associated with a positive number K:

$$\underset{\delta \to 0}{\lim }{\displaystyle \frac{N_{\delta }\left(F\right)}{1/{\delta }^D}}=K$$

(10)

$$D=\underset{\delta \to 0}{\lim }{\displaystyle \frac{\lg K-\lg N_{\delta }\left(F\right)}{\lg \delta }}=\underset{\delta \to 0}{\lim }{\displaystyle \frac{\lg N_{\delta }\left(F\right)}{\lg 1/\delta }}$$

(11)

The explanation of the box-counting dimension involves covering an object with boxes of a specific size δ, which may be square or round. These boxes must cover the objects individually without overlapping. As the box size δ decreases, the ratio between the logarithm of the box count and the logarithm of the reciprocal of the box size determines the box dimension. Linear regression is used with double logarithmic coordinates for the bubble diameter and the total number of converted bubbles. The slope of the regression line represents the box dimension, which can be mathematically defined as Equation (12). Based on bubble characteristic parameters, this model yields the box dimension that represents the bubble distribution. In this study, the model is referred to as the bubble distribution fractal model, with the associated fractal dimension termed as the bubble distribution fractal dimension.

$$\lg N_d=-D\lg d+C$$

(12)

4. Experimental Result

4.1. Mortar Fluidity

Mortar fluidity is shown in Figure 3. When the NScontents were 0, 1.5%, and 3%, the average values of mortar fluidity were 160 mm, 143 mm, and 102 mm, respectively. Compared to S0, the fluidity of the NSmodified mortar was reduced. This is because NS particles have a high specific surface area and strong surface attraction, enabling them to adsorb water molecules from the mixing water. Consequently, the reduced availability of free water in the fresh cement mortar leads to decreased mortar fluidity [13,31,32].

The negative impact of NS on the workability of cementitious materials has been recognized. It is due to the fact that the surfaces of NS particles have a powerful attraction and high specific surface area, which allow them to adsorb water molecules in the mixing water on their surfaces. Thus, the reduced free water in the fresh cement mortar leads to reduced mortar fluidity.

4.2. Strength of Mortar

The strengths of three sets of mortar specimens are shown in Figure 4. NS significantly enhances the initial mortar strength, with the strengthening effect gradually diminishing with the specimen’s cure. Compared to the control mortar without NS, the compressive strength enhancement is over 60% at 3 days for 3% NS, and over 45% for 1.5% NS. In addition, the compressive strength of control groups S1 and S2 increases by 47.4% and 60.3% on the 7th day, 32.8% and 48.9% on the 14th day, and 21.6% and 35.0% on the 28th day. The flexural strength of the mortar increases after adding NS sol, with S1 showing an increase of 22.6% at 3 days compared to S0. S2 demonstrates an increase of 32.0% on day 3, 27.7% on day 7, 15.9% on day 14, and 10.8% on day 28. Similarly, the increase in flexural strength becomes more pronounced with a higher dosage of the NS sol.

The mechanical properties test results of the mortar indicate that adding NS dispersion improves mortar strength. The enhancement is more noticeable at 3 days and 7 days but is weaker in bending strength compared to compressive strength. NS exhibits higher pozzolanic activity than silica fume, enhancing the hydration process in the cement matrix. The nanoscale silica particles fill micropores in the cementing material, enhancing the microstructure of the cement matrix [33].

4.3. Pore Structure

The pore size distributions are shown as Figure 5a,b. The porosity fluctuates within the pore radius range of 50 to 300 μm and reaches its peak in the range of 300 to 1000 μm. The S2 edge slices exhibit the highest peak porosity at 2.9%. Additionally, the pore size in the S0 and S2 slices rarely exceed 1000 μm. Based on the pore structure distribution test results of the mortar specimens taken from two positions, the average porosity of S0, S1, and S2 is calculated to be 6.21%, 8.42%, and 11.91%, respectively.

The addition of NS enhances the mechanical properties of cement mortar and densifies its microstructure, thereby reducing the penetration rate of corrosive substances, such as sulfate ions, into the mortar. This significantly improves the durability of the mortar. NS can enhance the freeze–thaw resistance and ion erosion resistance of mortar, which will remain a key focus in future research. We will continue to conduct more in-depth studies on the porosity of NS-modified mortar during freeze–thaw cycles, among other aspects.

4.4. SEM Results

Figure 6 illustrates the microdistribution of nanomaterials in mortars with varying amounts of NS, where white areas represent internal pores of the mortar and black areas represent internal materials. The internal structure of the mortar specimens is consistent in pore distribution and shape between the edges and the center, with the air volume decreasing as the nano material content increases. Furthermore, there are significant differences in the microstructures of C-S-H in the three groups of mortar specimens, indicating variations in pore size and the content of nanomaterials within the mortar [34]. These differences reflect the quality of the mortar’s densification properties [35,36].

In the control mortar, the hydrated cement slurry, as illustrated in Figure 6a, comprises an accumulation of angular particles. The reference sample exhibits a relatively porous microstructure, as highlighted by the yellow markings in the image, characterized by numerous large voids.

In the microstructure of the image, spherical particles sized between 100 and 200 nm are distributed on the hardened cement slurry, comprising compact layered formations. In Figure 6b, hardened cement paste contains spherical particles of different diameters, known as the hydration product C-S-H. Figure 6c reveals that the hardened cement paste is predominantly composed of densely packed layers, which are stacked layer-by-layer to form a network structure. It is evident that a high content of C-S-H significantly affects the microstructure of mortar, reducing the porosity of the mortar, which is crucial for enhancing its strength. These findings align with the studies by Fen [32] and Hou [37,38]. Moreover, due to NS’s high activity and nucleation properties, NS concrete shows shortened setting times, reduced slump, and increased shrinkage rates.

5. Pore Structure Fractal Dimension

Based on test results, the porosities of the central slices S0, S1, and S2 are 6.45%, 8.22%, and 11.13%, respectively. The introduction of 0.15% NS results in a porosity exceeding twice that of ordinary mortar, while 0.3% NS doubles the porosity compared to ordinary mortar. Traditional parameters cannot fully capture the complexity of pore structure. Fractal dimension is used to analyze changes in the pore structure of mortar, and we present the calculation results.

5.1. Fractal Dimension of Volume

The following formula can be written based on Equation (5):

V_k∝r_k^3−D

(13)

Both sides of the equation change logarithmically:

lgV_k = (3 − D)lgr_k

(14)

where V_k and r_k are solid-phase volume and aperture, respectively. Based on the data obtained from the pore structure test, the volume of the solid phase and its corresponding pore diameter are computed, and then, logarithmic curves are drawn, respectively. The slope of the curve indicates the fractal dimension of the pore volume, designated as D.

Figure 7 displays logarithmic curves representing the solid-phase volume and pore diameter of mortar samples S0, S1, and S2. The data indicate that the volume of the solid phase varies with changes in pore diameter, with varying rates of increase among the samples. Concrete with 1.5% NS has a higher air void content compared to concrete with 3% NS.

The graph shows that the overall variations of the three curves are quite similar, but the internal air bubble content in mortars with different NS contents varies significantly. Mortar with 3% NS content has the fewest bubbles, followed by mortar without NS and, finally, mortar with 1.5% NS content. The logarithmic curves exhibit a slow initial growth rate and a stable trend. As the bubble size increases, the amplitude of curve variation significantly increases, and the growth rate of each grade differs markedly. Therefore, fitting a single linear regression model does not align with the overall trend of logarithmic curves and fails to capture the complexity of the internal pore structure of mortar. The volume fractal dimensions of these two regions are calculated independently, and detailed findings are outlined in

Table 3. Volume fractal dimension.

Specimen	Region I	Region II
S0	2.9983	2.9722
S1	2.9495	2.9996
S2	2.8916	2.9972

.

The calculations show different fractal dimensions between Region I and Region II. Region I’s volume fractal dimension R² is less than 0.535, implying a limited reflection of structural intricacy for bubble chord lengths below 140 microns. On the other hand, Region II’s R² exceeds 0.844, showing the significance of fractal characteristics for bubble string lengths of 140 microns. The volume fractal model effectively depicts pore volume distribution and structure complexity. Thus, the volume fractal dimension of Region II quantifies changes in the concrete pore structure with varying NS content.

5.2. Box Dimension

The number of bubbles in each string length grade is specific. Assuming all bubbles are regular circles, by the principle of equal area, bubbles with a diameter greater than d are transformed into bubbles with a diameter of D. The logarithmic curve relating the aperture and the number of bubbles is then plotted. The logarithmic curve in Figure 8 illustrates the variation in the total number of transformed bubbles, N, in concrete doped with 0, 1.5%, and 3% NS for the pore size. The figure reveals that the logarithmic curves for S0, S1, and S2 are similar, exhibiting a trend where the total number of bubbles decreases as the aperture increases. Furthermore, the bubble distribution logarithm curve differs from the pore volume and surface area logarithm curves. The bubble logarithm curve is almost linear, indicating that the pore structure lacks multifractal characteristics, allowing for a linear regression fit.

The regression equation’s negative slope indicates the fractal dimension of bubble distribution: 2.0577 for S0, 2.1071 for S1, and 2.1518 for S2. The R² of the linear regression is greater than 0.99, indicating significant fractal characteristics in the concrete bubble distribution. Therefore, the fractal model provides a precise and quantitative representation of the intricacy within the pore structure, acting as a parameter to delineate the intricate alterations in concrete pore structure induced by fluctuations in NS content.

6. Grey Correlation Analysis

Grey Correlation Analysis (GRA) is an essential component of the grey system theory. This approach targets uncertain systems characterized by small samples and limited information. It examines the superficial and profound relationships among various factors within the system, identifies the primary factors from the numerous influencing elements, and, thus, captures the system’s main characteristics. This study explores concrete pore structure parameters such as pore volume fractal dimension, pore ratio fractal dimension, bubble distribution fractal dimension, and porosity. Limited experimental data exist for each parameter. Grey relational analysis assesses the impact of various factors on concrete strength. The computational process for Genetic Risk Assessment (GRA) is specified below.

Porosity, volume fractal dimension, porosity fractal dimension, and bubble distribution fractal dimension are designated as comparison sequences in concrete structural parameters. Concrete compressive strength data sequences with varying NS content serve as reference sequences. Normalizing the test data involves considering quantity and dimensions, utilizing Equations (15) and (16):

X_x^′(k) = X_i(k)/X_i(1)

(15)

Y′(k) = Y(k)/Y(1)

(16)

Then, the grey level is calculated using Equation (17):

$$r_{\mathrm{i}}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{r}_{\mathrm{i}}\left(k\right)$$

(17)

In these equations, the gray level ranges from 0 to 1, indicating the numerical correlation measure between the reference and comparison sequences. A higher degree of similarity between the two sequences corresponds to a gray value closer to 1.

Calculations in

Table 4. Concrete strength and parameter values.

NS Content	Volume Fractal Dimension	Fractal Dimension of Bubble Distribution	Porosity (%)	Compressive Strength (MPa)	Flexural Strength (MPa)
0	2.9722	2.2468	6.21	40.9	7.09
1.5%	2.9996	2.2022	8.42	47.5	7.55
3%	2.9972	2.4139	11.91	52.6	7.85

show that the volume fractal dimension and box-counting dimension effectively represent the complexity of pore structures in mortar. Each fractal dimension reflects specific aspects of a single pore structure parameter. Mortar under different mix proportions, curing conditions, and experimental environments exhibits varying pore structures, necessitating the selection of suitable fractal dimensions to depict the changes in pore structure. Given the variability in the mix ratio, curing conditions, and working environments, concrete exhibits diverse pore structures, necessitating the selection of an appropriate fractal dimension to depict pore structure changes. To investigate the primary factors influencing the strength variation of concrete with varying NS contents, we employed gray relational analysis to examine the influence of each parameter.

In

Table 5. Grey correlation analysis of fractal dimension of each rank.

Grey Coefficient	Volume Fractal Dimension	Fractal Dimension of Bubble Distribution	Porosity
ri(1)	1	1	1
ri(2)	0.8793	0.9237	0.8271
ri(3)	0.8041	0.8152	0.9251
ri	0.8944	0.9129	0.9174

, GRA is used to calculate the correlation between volume fractal dimension, box dimension, compressive strength, flexural strength, and porosity to explore the changes in the pore structure and strength of mortar under different NS contents and to evaluate the influence degree of each parameter.

The correlations between compressive strength and volume fractal dimension, surface integral fractal dimension, bubble distribution fractal dimension, and porosity were calculated and analyzed using the grey correlation analysis method. Table 5 reveals that compared to porosity, the relationships between compressive strength and fractal dimensions, volume, surface area, and bubble distribution are more pronounced, with the grey correlation degree reaching 0.70. The correlation hierarchy is as follows: bubble distribution fractal dimension > surface integral dimension > volume fractal dimension > porosity. These results show that while porosity is not the main factor affecting compressive strength, the fractal dimensions, particularly the bubble distribution fractal dimension, are significant pore structure parameters affecting strength change. The fractal dimension of bubble distribution shows the highest correlation with compressive strength.

The above calculation and analysis indicate that the bubble distribution’s shape dimension correlates most strongly with both compressive and flexural strengths. Bubble distribution significantly impacts the strength of concrete.

7. Conclusions

(1) The results indicate that NS significantly enhances the strength of mortar while also increasing its porosity due to reduced fluidity. The compressive strength of mortar containing 3% NS was enhanced by 35%, while the early flexural strength was increased by approximately 20%. Compared to the control mortar, the porosity increased by 26.5% and 72.3% with NS contents of 1.5% and 3%, respectively.

(2) The feasibility of using fractal dimension to evaluate the pore structure of mortar was explored. The volume fractal dimension, surface integral fractal dimension, and bubble distribution fractal dimension were calculated using linear regression analysis. The square R² values of the linear fitting correlation coefficients all exceeded 0.84, indicating that fractal dimensions can accurately depict the complexity of the pore structure. These dimensions are reliable parameters for quantitatively characterizing the dynamic changes in concrete pore structure as nano-silica content varies.

(3) The primary factors influencing the strength of mortar were investigated using the grey correlation analysis method. The study’s results indicated that fractal dimensions are the key parameters affecting strength variations relative to traditional porosity. Specifically, the fractal dimension of bubble distribution shows the strongest correlation with compressive and flexural strengths, highlighting that pore volume distribution is the critical factor affecting the strength of concrete.