Hydration and Fractal Analysis on Low-Heat Portland Cement Pastes Using Thermodynamics-Based Methods

Abstract

Low-heat Portland (LHP) cement is a kind of high-belite cement, which has the characteristic of low hydration heat. Currently, it is extensively used in the temperature control of mass concrete. Based on the thermodynamic database of OPC-based materials, the thermodynamic software GEM-Selektor (noted as GEMS) is used for simulating the hydration products of the LHP cement paste. Then, according to the GEMS thermodynamic simulation results, MATLAB is used to visualize the initial and ultimate stages of LHP cement pastes; the effects of curing temperature and water to cement (w/c) ratio on hydration products are addressed; and the porosity, fractal dimension, and tortuosity of different pastes are calculated. It is found that an appropriately high curing temperature is important for reducing porosity, especially in the early hydration stage. Hydration time also has a significant impact on the hydration of LHP cement paste; long hydration time may reduce the impact of temperature on hydration products. The w/c ratio is another important consideration regarding the hydration degree and porosity of LHP paste, and under different curing temperatures, hydration times, and w/c ratios, the porosity varies from 5.91–32.91%. The fractal dimension of this work agrees with the previous findings. From tortuosity analysis, it can be concluded that the high curing temperature may cause significant tortuosity, further affecting the effective diffusivity of LHP cement paste. For cement pastes with low w/c ratio, this high curing temperature effect is mainly reflected in the early hydration stage, for ones with high w/c ratio, it is in turn evident under long-term curing.

1. Introduction

Low-heat Portland (LHP) cement presents low energy consumption, low harmful gas emissions, low production cost, and incomparable advantages over ordinary Portland cement [1,2,3]. It is a new type of cement with a wide range of applications and superior performance. For example, LHP is commonly used in high-strength and ultra-high-strength concrete [4] due to its strong ability of temperature control. Cracks caused by temperature drop and shrinkage of mass concrete during the process of concrete pouring and curing are reduced significantly [5]. Unlike ordinary Portland cement (OPC), the low hydration heat of LHP benefits from its high dicalcium silicate (C₂S) content and low tricalcium silicate (C₃S) content [6]. Compared to C₃S, C₂S has less reaction heat released by its complete hydration, as listed in

Table 1. The difference between properties of C₃S and C₂S.

Raw Clinker	Density (g/cm3) [10]	Content CaO (% Weight) [10]	Heat Evolved on Complete Hydration (cal/g) [11]
C3S	3.15	1.28	120
$\beta$-C2S	3.23	0.02	62

. However, the hydration degree of C₂S in Portland cement is lower compared to that of C₃S [7], perhaps leading to early compressive strength reduction of LHP-cement-based materials [6,8,9]. Therefore, it is worth exploring the early-stage hydration properties of LHP cement paste.

Thermodynamics is an important method for understanding chemical reactions of cement-based materials [12,13]. Lothenbach and Wieland [14] simulated the changes in the ion concentration of the pore solution of the Portland cement paste and the various substances in the phase dissolution and precipitation process by establishing a Portland cement thermodynamic database after evaluating the thermodynamic data of the Portland mineral phase. Additionally, a complete calculation model [15] was given, including the reactions that occurred in the process of hydration and the effects of temperature on raw clinkers hydration and for the thermodynamic calculation of cement-based materials, hydration reaction equations, and thermodynamic parameters of substances and ions. Sabine et al. [16] explored the stable ion concentration of the AFm phase in the solution and the thermodynamic parameters of new hydration products formed after the AFm phase combined with toxic substances and studied their ability to solidify toxic substances. With the help of the above thermodynamic databases, significant progress has been made in the study of the hydration properties of cement-based materials [17].

At present, research on the microstructure of LHP-cement-based materials primarily relies on experiments and can benefit from further exploration. Additionally, the application of LHP-cement-based materials in the hydraulic engineering is increasing, but there is no systematic research associated with the influence of curing temperature and water to cement (w/c) ratio on the reaction mechanism of LHP-cement-based materials, and there is also a lack of computer modeling research on LHP-cement-based materials. Therefore, further investigation of the hydration mechanism in LHP cement-based materials is needed. Computer simulations of such mechanism can potentially serve as a complement to experimental results of LHP-cement-based materials. Solid hydration products and the pore structure of LHP cement pastes with different w/c ratios under different curing temperature are addressed. Through this study, on one hand, we hope to improve the systematic summary of the hydration properties of LHP cement-based materials. On the other hand, we can verify the effectiveness of our computer model, providing guidance for the future application of LHP cement in practical work and even providing computer research models for other types of cement-based materials to improve research efficiency.

2. Methods and Algorithms

In this work, a new model for LHP cement paste hydration is proposed through thermodynamic simulation. With respect to the hydration products and porosity of LHP cement pastes, this work is carried out as follows: firstly, GEMS is used to simulate the change of reactant concentrations and hydration products for different cases of w/c ratios and curing temperature. Secondly, the initial and final stages of LHP cement pastes at a micro level are visualized by MATLAB. Finally, a porosity comparison, tortuosity analysis, and fractal analysis on the microstructure of different LHP cement pastes are presented.

2.1. Simulation Method

The paste notation and mix proportion of LHP cement pastes are shown in

Table 2. Paste notation and mix proportion of LHP cement pastes.

Paste Notation	w/c Ratio	Temperature (°C)
WC0.4T5	0.4	5
WC0.4T15	0.4	15
WC0.4T20	0.4	20
WC0.4T30	0.4	30
WC0.5T5	0.5	5
WC0.5T15	0.5	15
WC0.5T20	0.5	20
WC0.5T30	0.5	30

. The chemical and mineral compositions of LHP raw clinkers are analyzed using the Bogue method and the X-ray diffraction method combined with Rietveld refinement [18,19], and the result is listed in

Table 3. Chemical and mineral compositions of LHP raw clinkers.

Chemical Oxide Constituents (wt.%)	Percentage (%)	Mineral Composition (wt.%)	Percentage (%)
CaO	61.86	C3S	28.7 b (30.7) c
SiO2	23.98	C2S	47.0 b (44.3) c
Fe2O3	4.22	C3A	4.1 b (3.6) c
Al2O3	4.23	C4AF	12.8 b (14.1) c
MgO	2.89	Gypsum	3.9
SO3	2.31
R2O a	0.31
Loss on ignition (wt.%)	0.45

Notes: ^a Alkali content (R₂O) = Na₂O + 0.658K₂O; ^b from Bogue analysis; ^c from X-ray analysis.

. Due to the limitation of the Bogue method, there may be some discrepancies with reality. In this work, the LHP cement paste hydration process is simulated using the thermodynamic software GEM-Selektor (noted as GEMS). GEMS is a program package with Gibbs energy minimization, typically used for modeling the heterogeneous aquatic (geo) chemical systems based on the interactive thermodynamic, particularly in solid–aqueous solution equilibrium, sorption/ion exchange, metastability and dispersity of mineral phases, etc. [20,21,22,23] In this work, the hydration kinetics are not considered for simplicity. Modeling with new chemical and phase-extended thermodynamic databases or modifying certain thermodynamic properties is easily implemented in GEMS [20,21,22,23]. In the process of GEMS modeling, the circumstances of high ionic strength and complex ion reaction in LHP cement paste can be handled by the CEMDATA18 database [24].

In this simulation, a staged computer model is established to investigate the microstructure of LHP cement pastes. The actual particle contains multiple phases, each of which reacts with different rates, and it is difficult to determine these rates. Based on the thermodynamic calculation, we obtain reaction rates of the main phases in LHP cement paste. These rates are homogenized in this model for efficiency. In the model, it is assumed that there are nine phases in the hydration process: pore or water, raw clinker particles (C₃S, C₂S, C₃A, and C₄AF), gypsum, hydrated crystals, outer and inner hydrated products. In the model, the inner hydration products are defined as product layers with a thickness equal to or smaller than the raw clinker radius, on the contrary, they are outer hydration products. The molar mass and volume of reactants and hydrated products need to be set first, and then other parameters—w/c ratio and hydration degree—need to be input. The assumption of the growing of spheres is used in this model [25], the reaction starts from the boundary of the particles and progress toward inner layer. The initial stage of LHP cement pastes is simulated by randomly allocating raw clinker particles and other reactant particles into a representative elementary volume (REV), which is a 100 × 100 × 100 μm³ cube with the periodic boundary condition. The raw clinker particles with sizes smaller than 0.5 μm and larger than 45 μm are ignored for efficiency. The REV has 500 × 500 × 500 cubic voxels with a resolution of 0.2 μm³.

In the ultimate hydration stage, particles are reacted according to the degree of hydration obtained from GEMS simulation, while the hydrated products form layer by layer on the periphery of the original raw clinker particles until the desired volume is achieved at a given stage. Additionally, the algorithm called the “Disk Filling Method” is used in calculating porosity at the given stage of LHP cement pastes. This algorithm uses a disk to fill the pores in the paste. As the reaction progresses, the radius of the disk continuously increases until the pores are completely filled. Consequently, the cumulative porosity of the LHP cement paste can be determined by these disks.

2.2. Fractal Analysis of Pore Structure in LHP Cement Paste

Fractal theory is significant for analyzing the pore structure of cement paste. On the basis of fractal theory, the following relations exist between the quantity of pores whose diameters exceed $d$ ($\mathrm{N}\left(\mathrm{d}\right)$), the fractal dimension ($D_f$) of the pore size, and the probability function ($f\left(d\right)$) of the pore size distribution [26]:

$$\mathrm{N}\left(\mathrm{d}\right)={\left(d_{\max }∕d\right)}^{D_f}$$

(1)

$$f\left(d\right)=D_f·d_{min}^{D_f}·d^{-D_f-1}$$

(2)

where $d_{\max }$ and $d_{\min }$ represent the maximum and the minimum pore sizes, respectively. According to the image theory applied to cross section of paste, $D_f$ can be calculated using the box-counting method [27,28]. Ten cross sections are taken for each REV to calculate values of $D_f$, which is opposite to the slope of the paradigmatic logarithmic plot of cumulative pore volume and box size. Finally, the $D_f$ value is taken as the average of ten cross sections.

2.3. Tortuosity Analysis of Pore Structure in LHP Cement Paste

Tortuosity is an important index of diffusion of cement paste. In this work, the random walker algorithm is employed in the simulated microstructure to explore the tortuosity of LHP cement paste [29]. In the random walker algorithm, an ion is allowed to move in six directions (+x, −x, +y, −y, +z, −z) with a three-dimensional random walk—that is to say, the statistics of ions random motion is utilized to simulate diffusion. During a time interval of t, the square of the averaged distance of all the random walk ions is recorded as $\langle l^2\rangle$; $\langle l^2\left(t\right)/t\rangle$ is an indicator of tortuosity, which can be obtained by inputting the REV of LHP cement paste through the random walker model. $D^{self}$ represents the self-diffusion factor of ions, and $D_0^{self}$ represents the self-diffusion coefficient of walkers in the discretized open space, where $N_W$ is the sum of walk steps in a simulation, for the 500 × 500 × 500 cubic voxels REV. $N_W$= 50,000 is enough to make $\langle l^2\left(t\right)/t\rangle$ approach an effective value. The relationship among them is as follows [29]:

$$D^{self}=1/6\ast \langle l^2 \left(t\right)/t\rangle$$

(3)

When $D^{self}$ is determined, the diffusion tortuosity (${\tau }_D$) of the voxel can be calculated as [29]:

$${\tau }_D=D_0^{self}/D^{self}=\langle l^2 \left(t\right)/t{\rangle }^{-1}$$

(4)

The self-diffusion coefficient $F$, the effective porosity ϕ, and the diffusion tortuosity ${\tau }_D$ of a cement paste satisfy the following relationship [29]:

$$F={\tau }_D/\mathit{\varphi }$$

(5)

3. Results and Analysis

3.1. Influence of Temperature on Hydration Products

The hydration result of WC0.4T20 can be transformed into the volume accumulation diagram of reactants and products of each phase over time, as shown in Figure 1. The volume of each phase in LHP cement pastes is marked by a different color. The vertical axis in Figure 1 represents the volume of each hydration product, and the horizontal one stands for the hydration time of LHP cement paste. The hydration degree of reactants and volume proportion of some products of WC0.4T20 are shown in

Table 4. Hydration degree of reactants and volume proportion of some products of WC0.4T5-WC0.4T30.

	WC0.4T5			WC0.4T15			WC0.4T20			WC0.4T30
	1 Day	3 Days	27 Days	1 Day	3 Days	27 Days	1 Day	3 Days	27 Days	1 Day	3 Days	27 Days
Reactant
C3S	0.261	0.457	0.771	0.377	0.567	0.825	0.432	0.614	0.846	0.534	0.692	0.879
C2S	0.231	0.373	0.614	0.273	0.415	0.621	0.293	0.434	0.622	0.331	0.469	0.625
C3A	0.206	0.381	0.742	0.338	0.538	0.820	0.408	0.605	0.849	0.550	0.711	0.891
C4AF	0.091	0.267	0.634	0.170	0.362	0.682	0.213	0.407	0.700	0.297	0.489	0.729
Other unhydrated raw clinker	0.124	0.331	0.525	0.274	0.415	0.568	0.328	0.454	0.584	0.418	0.515	0.609
Hydrated products
C-S-H	0.177	0.263	0.377	0.211	0.288	0.376	0.226	0.298	0.374	0.250	0.311	0.371
CH	0.206	0.284	0.399	0.255	0.322	0.399	0.272	0.334	0.400	0.303	0.349	0.398

. The hydration stages of WC0.4T20 are shown in Figure 2. Different colors in Figure 2 stand for different phases of the paste, as illustrated in

Table 5. Different colors demonstrated for different phases.

Raw Clinkers	Color	Phases	Color
C3S	Red	Pore or water	Dark blue
C2S	Pink	Crystals a	Purple
C3A	Orange	Outer hydration product	Green
C4AF	Deep yellow	Inner hydration product	Light yellow
Other unhydrated raw clinker	Grey

Notes: ^a crystals include C-S-H, CH, AFt, AFm, hydrotalcite, etc.

.

In this work, the ultimate stage is considered at 27 days. It can be seen that small C₃S and C₂S particles are almost completely hydrated, while the degree of hydration of those large particles is remarkably dominated by the solution diffusion. The inner hydration products are gradually formed around the C₃S and C₂S particles in the paste, and C₄AF, C₃A, gypsum, and other reactants do not adsorb inner hydration products, which is in accordance with the previous study [30]. Additionally, it is noted in Figure 2 that the crystal phases are gradually formed around the crystal nuclei, while for C₃A, C₄AF, and other particles, the surface is mainly covered by outer hydration products. Therefore, once the covering hydration product ruptures, the internally unhydrated raw clinker may rehydrate immediately.

In Figure 2b, there is a small amount of hydration products randomly scattered on the boundary of raw clinker particles, and the interior of the particles remains unreacted after 1 day of hydration. At this hydration stage, the crystal phase, mostly the AFt phase, has already been generated in the pastes in a large amount, while the outer hydration product forms around the unreacted particles and crystal particles. From Figure 2, it can be found that after 1 day, the hydration rate in the paste comes to a significant leap. When the hydration time is 3 days (Figure 2c), water invades from the capillary pores of initially formed C-S-H to the particle surface. At the same time, a large amount of CH is generated during the reaction. Outer hydration products are gradually filled in the pores among the particles, which results in a dense structure compared to Figure 2a. During the hydration process from 3 to 27 days, the quantity of the C-S-H phase significantly increases, and the inner hydration products are formed in a large quantity. After the long-term hydration, the outer hydration products interconnect with crystals, and an originally cohesive paste is clearly established. Additionally, from Figure 2d, C₃S and C₃A particles in the paste are almost completely hydrated, and the large particles of C₂S and C₄AF particles are covered with thick layers of inner and outer hydration products. In addition, during the hydration process, the incipient pores of LHP cement pastes are filled by hydration products, and a solid skeleton of the paste evolves, contributing to the increase of paste strength [31].

Figure 3, Figure 4 and Figure 5 illustrate the influence of curing temperature on hydration behavior and volume of different phases in WC0.4T5, WC0.4T15, and WC0.4T30. The hydration degree of these pastes with different hydration days simulated using GEMS is also shown in Table 4. Compared to Figure 1, Figure 3, Figure 4 and Figure 5, the change of curing temperature does not significantly affect the change of total volume during the hydration, however, the hydration degree of each raw clinker particles seems to be sensitive to the curing temperature. The most obvious change in Figure 1, Figure 3, Figure 4 and Figure 5 is the volume of AFm. The volume in the hydration reaction has a negative relation with curing temperature. The increase in curing temperature from 5 °C to 30 °C causes the decomposition of AFm [32]. Shirani et al. [33] pointed out that when the curing temperature was below 40 °C, the AFm and AFt amounts of Belite cement paste both reduced with the increase in curing temperature. However, it is evident that the high curing temperature has a larger impact on AFm content, and it can be speculated that the high curing temperature may cause an obstacle in the transition from AFt to AFm. In the study by Xuan and Wang [34], they pointed out that high C₃A could promote the transformation of AFt to AFm. The content of C₃A in LHP cement pastes is originally small. As the curing temperature increases, the hydration degree of C₃A increases and its content decreases significantly, which plays a certain role in inhibiting the conversion of AFt to AFm [35]. Therefore, it can be inferred that AFt is more stable in the LHP cement pastes than AFm [36].

Expectedly, the curing temperature increasing from 5 °C to 30 °C is helpful to shorten reaction the time of the reactants [37,38,39]. For example, when the temperature is 5 °C, the complete reaction time of gypsum is 1.156 days; while curing temperature is 30 °C, the time becomes 0.277 days. The high curing temperature is beneficial to the hydration of four main raw clinkers in a given curing temperature range, which is confirmed by findings from the literature [40,41,42]. Interestingly, it is observed that the calcite does not seem to participate in the whole reaction, and when the temperature is 5 °C, the calcite volume decreases slightly with the progress of the reaction. However, as the curing temperature increases, the residual volume in the reaction gradually increases slightly, and the influence of curing temperature on calcite is negatively correlated, corresponding to the findings by Baltrusaitis et al. [43]. A low curing temperature makes calcite convert into other phases in the reaction process [44], while a high temperature keeps the original stage of calcite.

Furthermore, it can be found in Figure 1, Figure 3, Figure 4 and Figure 5 that the effect of temperature on the overall hydration process of LHP cement paste is mainly prominent in the initial hydration stage. In the given temperature range (5–30 °C), the high curing temperature at the initial hydration stage can cause the high hydration rate of raw clinkers and the high hydration degree of cement paste. For example, when the hydration time is 1 day, the hydration degree of WC0.4T30 is much higher than that of WC0.4T5. However, when the hydration time is about 7 days, the difference between WC0.4T5 and WC0.4T30 becomes small. At the same time, large amounts of C-S-H and CH begin to be generated in the cement paste, and their amounts are also positively interrelated with curing temperature. During the curing process, high temperature is in favor of the production of C-S-H and CH within the same hydration time. It can be found that the difference of hydration degree between the WC0.4T5 and WC0.4T30 is even smaller when the hydration time is about 27 days. From Table 4, it can be obviously seen that the paste under low temperature curing has low hydration degree at early stage. It can be considered that curing temperature is an important factor affecting the hydration degree in the early stage. The hydration degree is closely related to the pore evolution of cement paste. Accordingly, it can be inferred that the low curing temperature and the large porosity of pastes in the initial stage may cause the slow the strength growth [45,46,47].

The change in curing temperature may lead to a change in reaction rate, but more importantly, it affects the equilibrium constants of different reactions and usually leads to the change of hydration products. It can be noticed from Figure 6b,c that layers of inner and outer hydration products at low curing temperatures are thin and that the number of crystals generated is also significantly low, but in Figure 7b,c and Figure 8b,c, the layers of inner and outer hydration products seem to be thick. With the increase in hydration time up to 27 days, from Figure 6d, Figure 7d and Figure 8d, the characteristics of hydration products at different curing temperatures are close. This phenomenon shows that the final reaction rates tend to be the same at different curing temperatures during the long-term hydration reaction. However, the amount of hydration products in the paste at a low curing temperature is still smaller than that at a high temperature; the difference in the amount of hydration products is principally owing to the change in AFt phase, while the C-S-H phase is not affected significantly. This potentially shows that although the eventual hydration products are different in amount, the influence of temperature on LHP cement hydration is mainly reflected in the early stage of the hydration process. As the hydration time increases, the influence of temperature on hydration products becomes weak.

3.2. Influence of w/c Ratio on Hydration Products

Figure 9, Figure 10, Figure 11 and Figure 12 illustrate the effect of curing temperature on hydration behavior and volume of solid assemblages in WC0.5T5, WC0.5T15, WC0.5T20, and WC0.5T30. It can be found that when w/c ratios are 0.4 and 0.5, the influence of curing temperature on reactants and products in the pastes are similar. Additionally, compared to WC0.4T5-WC0.4T30 and WC0.5T5-WC0.5T30, a high w/c ratio causes the amount of clinker particles in each unit volume of REV to decrease, and the pore amount increases significantly, indicating that the change of w/c ratio has a significant impact on the porosity.

When the hydration time is 1, 3, and 27 days, for the case of WC0.4T5-WC0.4T30 and WC0.5T5-WC0.5T30, the hydration rate of reactants and the volume change of phases in the early stage of hydration are very similar. However, when the hydration time reaches 27 days or more, obvious differences begin to appear between WC0.4T5-WC0.4T30 and WC0.5T5-WC0.5T30. When the hydration time is long enough, it can be considered that the paste is fully hydrated. At such a stage, WC0.5T5-WC0.5T30 produces more hydration products than WC0.4T5-WC0.4T30. This phenomenon shows that under a long enough hydration time, the same number of raw clinkers produces many hydration products under the condition of a high w/c ratio; that is to say, the condition of high w/c ratio is conducive to the full hydration of cement paste. It can be observed that the residual volume of reactants in WC0.5T5-WC0.5T30 is low after 27 days or more of hydration. In particular, the content of C₂S is significantly lower than that of WC0.4T5-WC0.4T30. A similar result can be obtained from Figure 13, Figure 14, Figure 15 and Figure 16. The overall volume proportion of the products in WC0.5T5-WC0.5T30 is larger than that in WC0.4T5-WC0.4T30. In particular, the contents of C-S-H and CH are significantly higher than those of WC0.4T5-WC0.4T30. The above phenomena show that high w/c ratio can not only increase the hydration degree of raw clinkers when the time is long enough but also promote the partial hydration of C₂S [48]. When the w/c ratio is high, the rate of C₂S hydration is higher than that of other raw clinkers, and the proportion of C-S-H and CH in the products is higher than that in low w/c ratio paste. It is noteworthy that the change in C-S-H content after 14 days may be mainly affected by C₂S content [49].

In addition, from Figure 2, Figure 6, Figure 7, Figure 8, Figure 17, Figure 18, Figure 19 and Figure 20, it is obvious that the different w/c ratios have a large impact on the hydration behavior of cement pastes. When hydration time is 1 day, the degree of hydration of raw clinkers is close at different w/c ratios. It can also be obtained from Figure 8b and Figure 16b that at the beginning of the hydration reaction, the hydration characteristics of raw clinkers in the figures are very similar. When the hydration time reaches to 27 days, in WC0.5T5-WC0.5T30, some small C₃S particles come to complete hydration, while WC0.4T5-WC0.4T30 rarely presents such a phenomenon. Additionally, the hydration degrees of C₂S, C₄AF, and other particles increase with the increase in w/c ratio. Even though the hydration degree at a high w/c ratio is slightly larger than that of the low one after curing 27 days, the formation rate of products is lower when the w/c ratio is 0.4 and the number of crystals is significantly lower.

3.3. Pore Structure of LHP Cement Paste

It is noteworthy that different curing temperatures and w/c ratios have a great impact on the formation of the pore structure of LHP cement paste. It is considered that the porosity of cement-based materials has a clear correlation with their mechanical properties and durability [50]. Hence, the porosity is utilized to evaluate the property of LHP cement pastes. The porosities of LHP cement paste under different temperatures and w/c ratios are calculated using MATLAB. A comparison of the cumulative porosities of LHP cement pastes under different curing temperatures and w/c ratios is presented in Figure 21 and Figure 22. Typically, the pore size of the LHP cement paste is concentrated from 100 nm to 5000 nm. From Figure 21 and Figure 22, it can be seen that the cumulative porosity vs. pore diameter shows an S-shaped trend. It is attributed that the REV model may not cover all large particles, resulting in the neglect of a relatively large proportion of large and small pores. The trend of the cumulative porosity curve is resemble to that of the high-belite cement determined via mercury intrusion porosimetry [33]. This trend is explained in the study by Thomas et al. [51]: the accumulation of hydration products or the collapse of macropores may cause the pore refinement of cement paste.

The porosities of WC0.4T5-WC0.4T30 are shown in Figure 21. In the early stage, when hydration time is 1 day, the lower the temperature, the higher the cumulative porosity of the LHP cement paste. In the 1-day hydration of WC0.4T5-WC0.4T30, the porosity of LHP cement paste decreases from 23.3% to 14.0% with the increase in curing temperature from 5 °C to 30 °C. Obviously, the high temperature has a positive relation with the hydration reaction rate in a given curing temperature range. When the hydration age is 3 days, the porosities of WC0.4T5 and WC0.4T30 still have a 4.1% difference. But for 27 days, the cumulative porosity under various curing temperature conditions tends to a certain value with only small difference. As demonstrated in Figure 21, when the hydration time reaches 27 days, the cumulative porosity under curing temperatures from 5 °C to 30 °C is around 6.0% different. Due to the hydration heat released by the LHP cement paste itself, the difference in the reaction degree gradually decreases and the cement paste is basically completely hydrated after 27 days, so the temperature has little impact on the ultimate stage of LHP cement paste. The result of simulation in this work has good agreement with the work by Wang et al. [52] and Xie et al. [53]; they considered that the hydration process of cement paste at 0 °C and 5 °C was very slow at the beginning but that it would progress rapidly after 7 days. Other researches also showed that the high curing temperature could lead to the rapid reaction of Portland cement in the preliminary stage of hydration and have high early compressive strength [54], which potentially corresponded to the low porosity [55].

Figure 22 shows the cumulative porosity of WC0.5T5-WC0.5T30. The influencing trend of curing temperature on cumulative porosity is similar to the case of w/c = 0.4 above, but the overall porosity is larger than that of pastes with a w/ c= 0.4 at the given temperature, which is identical to the previous illustration. From the comparison of the cumulative porosity of LHP cement paste under different w/c ratios during hydration at 20 °C, the difference in the cumulative porosity between WC0.4T20 and WC0.5T20 is about 9.0% after 1 day, and with long-term hydration, it gradually decreases to 4.0% at the ultimate stage. This phenomenon can be explained as the differences in water content significantly affecting the molarity in the solution and thus the hydration degree. Moreover, it can be found from Figure 21 and Figure 22 that a high w/c ratio accounts for the large rate of porosity reduction with the increase in hydration time. This phenomenon accords with the findings by Li et al. [56] that a high w/c ratio would reduce the quantity of unfilled voids in the previous concrete. In general, the porosity increases by increasing the w/c ratio, which is followed by the time required for cement paste hydration to reach a certain porosity also increases. A high w/c ratio is responsible for low cement strength according to relevant findings [57].

3.4. Fractal Analysis of Pore Structure in LHP Cement Paste

Based on Equations (1) and (2), a fractal analysis of pore structure in LHP cement paste was performed. The calculated $D_f$ in LHP cement paste and $D_f$ from other researchers [58,59,60] are shown in Figure 23. The difference between the simulated and experimental data may be caused by the investigated cement paste or pore size distribution. In Figure 23 $D_f$ in the LHP cement paste varies between 1.5 and 1.9. In addition, $D_f$ value in LHP cement paste is closely related to its porosity. $D_f$ value decreases as the porosity decreases, but this is not a strictly linear relationship—the larger the porosity, the smaller the changes in $D_f$ value, which is consistent with the previous findings [61,62]. In this work, the $D_f$ of WC0.5T5-WC0.5T30 is generally larger than the $D_f$ of WC0.4T5-WC0.4T30 because the porosity is sensitive to the w/c ratio, and under the same hydration time, the higher the curing temperature is, the lower $D_f$ value is, which is consistent with their porosity because the small $D_f$ attributes to the small porosity in the stage.

3.5. Tortuosity of LHP Cement Paste

The comparison of tortuosity of LHP cement pastes under different w/c ratios, curing temperature, and hydration time is presented in Figure 24, Figure 25, Figure 26, Figure 27, Figure 28 and Figure 29. It can be seen from the figures below that the index $\langle l^2\left(t\right)/t\rangle$ evaluating the tortuosity gradually stabilizes with the increase of the number of $N_W$. In addition, $\langle l^2\left(t\right)/t\rangle$ value increases and gradually approaches 1 in pace with the hydration time increasing. The large $\langle l^2\left(t\right)/t\rangle$ is represent the high tortuosity of cement paste, while the large tortuosity means that the cement paste has complex pore structure. In Figure 24, when curing temperature is low, such as 5 °C or 15 °C, the $\langle l^2\left(t\right)/t\rangle$ value is very close, and when the curing temperature becomes higher than 20 °C, the $\langle l^2\left(t\right)/t\rangle$ value increases with the increasing curing temperature. In Figure 25, the final $\langle l^2\left(t\right)/t\rangle$ value shows a positive relation with the curing temperature. With the increase in hydration time, the effect of curing temperature on the pore structure of LHP cement paste perhaps becomes profound. In Figure 26, when the hydration time approaches 27 days, $\langle l^2\left(t\right)/t\rangle$ values of WC0.4T5-WC0.4T30 are close. Remarkably, for such cases, the hydration time is a dominant influencing factor of tortuosity. The LHP cement paste after hydration for a long time is relatively dense, thus presenting a similar pore structure. $\langle l^2\left(t\right)/t\rangle$ values are similar; this similarity is consistent with that of porosity in Figure 21 and Figure 22. In Figure 27, Figure 28 and Figure 29, the final $\langle l^2\left(t\right)/t\rangle$ values of WC0.5T5-WC0.5T30 increase as the curing temperature increases, even if the hydration time reaches 27 days. Due to the high porosity at a w/c ratio of 0.5, the pores of the cement paste cannot be filled in at the hydration time of 27 days. Compared to WC0.4T5-WC0.4T30 and WC0.5T5-WC0.5T30, it is found that the tortuosity of WC0.5T5-WC0.5T30 is generally higher than that of WC0.4T5-WC0.4T30. The larger pores of cement pastes with high w/c ratios and the more complex distribution of the hydration products result in higher tortuosity. In addition, high-temperature curing may cause high tortuosity. For WC0.4T5-WC0.4T30, this effect is mainly reflected in the early stage of hydration, while for WC0.5T5-WC0.5T30, the influence of curing temperature on the tortuosity is also obvious under long-term curing. WC0.5T5-WC0.5T30 has large pores, which is conducive to the continuous hydration reaction of cement paste under high-temperature and long-term curing conditions. Apparently, the large of the tortuosity of cement paste is due to the accumulation of hydration products. However, accurate determination of tortuosity is a great challenge, and there is little literature on this aspect, particularly a lack of related research on LHP cement paste. Only Dr. Ma [29] and Guo et al. [63] have carried out investigations about cement paste, and our result has good agreement with their findings.

Furthermore, ${\tau }_D$ and $F$ are calculated for further diffusion analysis based on $\langle l^2\left(t\right)/t\rangle$ values from Equations (4) and (5). The results are shown in

Table 6. Random walker simulation and calculation results of LHP cement (pastes hydrated at 1 day).

Paste Notation	$\mathbf{Porosity} \mathit{\varphi }$(%)	$\mathbf{\langle }{\mathit{l}}^{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{/}\mathit{t}\mathbf{\rangle }$	${\mathit{\tau }}_{\mathit{D}}$	$\mathit{F}$	$1/\mathit{F}$
WC0.4T5	23.33	0.3936	2.541	10.890	0.092
WC0.4T15	18.45	0.3930	2.545	13.791	0.073
WC0.4T20	16.60	0.4424	2.260	13.617	0.073
WC0.4T30	13.97	0.4674	2.139	15.315	0.065
WC0.5T5	32.91	0.4238	2.360	7.170	0.139
WC0.5T15	28.13	0.4585	2.181	7.753	0.129
WC0.5T20	25.66	0.4737	2.111	8.227	0.122
WC0.5T30	22.23	0.5109	1.957	8.805	0.114

,

Table 7. Random walker simulation and calculation results of LHP cement (pastes hydrated at 3 days).

Paste Notation	$\mathbf{Porosity} \mathit{\varphi }$ (%)	$\mathbf{\langle }{\mathit{l}}^{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{/}\mathit{t}\mathbf{\rangle }$	${\mathit{\tau }}_{\mathit{D}}$	$\mathit{F}$	$1/\mathit{F}$
WC0.4T5	13.41	0.5124	1.952	14.553	0.069
WC0.4T15	10.87	0.5493	1.820	16.748	0.060
WC0.4T20	10.36	0.5829	1.716	16.559	0.060
WC0.4T30	9.32	0.6148	1.627	17.452	0.057
WC0.5T5	22.49	0.5145	1.944	8.642	0.116
WC0.5T15	19.34	0.5940	1.684	8.705	0.115
WC0.5T20	18.12	0.6180	1.618	8.930	0.112
WC0.5T30	16.11	0.6646	1.505	9.340	0.107

and

Table 8. Random walker simulation and calculation results of LHP cement (pastes hydrated at 27 days).

Paste Notation	Porosity $\mathit{\varphi }$ (%)	$\mathbf{\langle }{\mathit{l}}^{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{/}\mathit{t}\mathbf{\rangle }$	${\mathit{\tau }}_{\mathit{D}}$	$\mathit{F}$	$1/\mathit{F}$
WC0.4T5	6.00	0.8793	1.137	18.954	0.053
WC0.4T15	5.91	0.9016	1.109	18.767	0.053
WC0.4T20	5.94	0.9190	1.088	18.319	0.055
WC0.4T30	5.99	0.9111	1.098	18.323	0.055
WC0.5T5	11.52	0.9859	1.014	8.805	0.114
WC0.5T15	10.36	1.0802	0.926	8.936	0.112
WC0.5T20	9.90	1.1291	0.886	8.946	0.112
WC0.5T30	9.13	1.2217	0.819	8.965	0.112

. In general, the tortuosity indicator $\langle l^2\left(t\right)/t\rangle$ does not have strict linearity with porosity. This fits with the findings by Guo et al. [63]. The factor 1/F has been used to estimate the effective diffusivity of cement-based materials [29,64]. It can be found that high porosity can lead to a high 1/$F$, indicating that high porosity may be beneficial to effective diffusivity. For WC0.4T5-WC0.4T30 and WC0.5T5-WC0.5T30, the porosity varies from 5.91–32.91%, and the values of 1/$F$ are from 0.053–0.139, implying that the effective diffusivity of LHP cement paste is low.

4. Conclusions

• Curing temperature has a significant impact on the hydration process and porosity of LHP cement paste, of which we investigated the hydration characteristics in the temperature range of 5–30 °C. High curing temperature potentially leads to high early hydration degree of LHP cement paste. At the same time, long hydration time can alleviate the influence of temperature on hydration products to some extent.
• The w/c ratio significantly affects the hydration degree and porosity of LHP pastes. Two values of w/c ratio (0.4 and 0.5) are compared in this work. At the same curing temperature, a high w/c ratio leads to the high porosity of cement paste and hydration degree of raw clinkers. After curing for 27 days, the formation rate of products for a low w/c ratio paste is low and the number of crystals is significantly small. Remarkably, high w/c ratio is helpful to the compact hydration structure of LHP cement pastes.
• In this work, fractal analysis and tortuosity analysis are used to explore the pore structure characteristics of LHP cement paste. The $D_f$ value of LHP cement paste is closely related to porosity, and a high w/c ratio and low curing temperature can lead to a high $D_f$ value. From the tortuosity analysis, it can be concluded that the tortuosity of LHP cement paste increases with a decrease in its porosity. The tortuosity of the high-w/c pastes is generally higher than that of the low w/c pastes. This may indicate that the internal structures of cement pastes with larger pores are more complex, which is consistent with the conclusion of fractal analysis. In addition, high-temperature curing may cause high tortuosity. For the low-w/c-ratio pastes, this effect is mainly reflected in the early stage of hydration, and for high values, it is also obvious under long-term curing. Based on the tortuosity value, we calculated the 1/$F$ value of LHP cement paste, and the 1/$F$ value of LHP cement paste is low, which may mean that the effective diffusivity of LHP cement paste is low.

5. Prospects

• The modeling of LHP cement paste can effectively assist in future practical engineering research on cement paste, which also provides ideas for other types of cement research. Using computer simulation for pilot experiments can extremely improve experimental efficiency. However, this model also has certain limitations. This model only explores the hydration properties of LHP cement from a thermodynamic perspective without considering hydrodynamics. The hydration model of LHP cement paste is still not accurate enough, and future research can try to establish a more accurate cement paste hydration model combined with hydrodynamics to explore deeper cement hydration properties.
• In this research, the fractal dimension analysis and the tortuosity calculation of LHP cement paste mainly focus on the establishment of a calculation model. The results obtained are consistent with the relevant findings. After verification, the model is feasible. However, this research is limited to the establishment of models and does not conduct in-depth discussions on the nature and application of fractal dimension and tortuosity of LHP cement paste. The tortuosity of cement paste may be related to its permeability and strength to a certain extent. We hope this can provide ideas and basic data for further research by subsequent scholars.