The samples of cement paste and mortar are shown in Fig. 1. The cementitious materials used in this experiment consist of 30% Portland cement and 70% ground granulated blast furnace slag (GGBS). The basic physical parameters of cement are listed in Table 1. The chemical composition of the cementitious materials is presented in Table 2. The fine aggregate is made of river sand. The measured water content is 4.8% (mass fraction), the fineness modulus is 2.6 (belongs to medium sand), the mud content is less than 1% (mass fraction), and the apparent density is 2.54 g/cm³. The mix proportions of cement paste (P-0.3, P-0.4, P-0.5) and mortar (M-0.3, M-0.4, M-0.5) are provided in Table 3. The fresh cement pastes and mortars were loaded into ϕ50 mm × 100 mm molds. After 1 day, they were removed and placed in water for curing. After 90 days, hydration was completed and the pore structure was relatively stable. Samples were taken out and cut into 3–5 mm thick slices to shorten the time to reach water equilibrium during the isothermal adsorption–desorption experiment [19].

The samples were divided into two groups: the adsorption group and the desorption group. The desorption group was taken from the saturated samples, while the adsorption group was dried to a constant weight before the experiment. A drying temperature of above 100°C is capable of decomposing C-S-H gel; hence, the selected temperature should not be too high [8]. A temperature of 60°C was set such that the free water could evaporate slowly without damaging the microstructure of the matrix. The mass of a sample was measured using an analytical balance and the temperature and humidity of the experiment were controlled with a constant temperature and humidity box (see Fig. 2a). Temperature was maintained at 20°C; and RH was set as 20%, 40%, 60%, 80% and 100%. For each humidity level, three samples of cement pastes and mortars with different water-binder ratios were placed in the sealed chamber and weighed every 7 days until reaching constant mass. The entire adsorption–desorption experiment took 110 days. Given an RH value, the adsorption and desorption amounts of water can be calculated as: (1)Wa=13∑mi−midmid;Wd=13∑mis−mimidwhere Wa and Wd are the adsorption and desorption amounts of water for the given RH value. mi is the mass of sample i (i = 1, 2, 3) when the equilibrium state is reached. mid and mis denote the mass of a dried sample and a saturated sample, respectively. The saturation degree Sw after adsorption (desorption) balance can be calculated as: (2)Sw=13∑mi−midmis−mid

The isothermal adsorption-desorption curves that reflect the variation of water content with RH are shown in Fig. 3. The results indicate evident hysteresis between the desorption and adsorption processes.

The dried cement paste and mortar samples were subjected to MIP test by using an Autopore IV 9500 fully automatic mercury intrusion instrument (Fig. 2b). The cumulative pore size distribution curves of cement-based materials are presented in Fig. 4. The open porosities of P-0.3, P-0.4 and P-0.5 are 19.39%, 23.74% and 32.34%, respectively. Pore diameter ranges are 0.003–0.09 μm in P-0.3 and 0.003–0.2 μm in P-0.4 and P-0.5. With regard to the mortar samples, the open porosities of M-0.3, M-0.4 and M-0.5 are 10.91%, 17.77% and 19.86%, respectively. Pore size distribution is between 0.003 and 0.05 μm.

The general idea of model construction can be summarized as follows: by setting a threshold, a continuous Gaussian random field will be transformed into a two-phase field to represent the pores and the matrix of the porous medium.

As illustrated in Fig. 5, the two-phase transformation on random fields RF₁ and RF₂ is to select a threshold D and then two sets P₁ and P₂ are obtained where the value of field is within [D, +∞) and (−∞, D). This operation can be expressed as: (3)P1≜{x|RF≥D};P2≜{x|RF<D}

According to the study of Adler et al. [20,21], For a Gaussian random field that satisfies the standard normal distribution (expectation = 0, standard deviation = 1), the probability density function fRF can be expressed using Eq. (4). When the Gaussian random field is generated in a cubic domain with a side length of a, the volume of P₁ and P₂ can be expressed using V_P1 and V_P2 after the two-phase transformation, as presented in Eq. (5). ψ(D) and Φ(D) are the volumetric fractions of phases P₁ and P₂, and they can be estimated using the Gaussian cumulative density function and its tail function in accordance with the chosen threshold D, as shown in Eqs. (6) and (7). Moreover, the Gaussian cumulative density function can be expressed using the Gaussian error function or the error complementary function [22], as shown in Eq. (8). After calculation, an expression for the relationship between the selected threshold D and the volumetric fraction of phase P₁ is given in Eq. (9). (4)fRF=12πe−x22 (5)VP1=a3ψ(D);VP2=a3Φ(D) (6)ψ(D)=12π∫D∞e−x22dx;Φ(D)=12π∫−∞De−x22dx (7)ψ(D)+Φ(D)=1 (8)ψ(D)=12erfc(D2)=12[1−erf(D2)] (9)D=2erfinv[1−2ψ(D)]

Suppose that phase P₁ represents the pores and phase P₂ stands for the matrix, then the two-phase field is capable of representing porous media. The function ψ(D) means the probability of the event “Value of RF not smaller than D”, thus its value equals to the volumetric fraction of phase P₁, and can also be interpreted as the porosity of material.

Notably, the fluctuation of a Gaussian random field is determined by correlation length (L_C) [20,21]. That is, for any couple of points a and b in a Gaussian random field, the correlation degree of RF(a) and RF(b) is measured by the covariance function: (10)Cov=exp⁡(−‖a−b‖2Lc2)

If a and b denote different points (a≠b), LC→0 leads to Cov→0, and then the random field exhibits a weak correlation. LC→∞ leads to Cov→1, and then the random field exhibits a powerful correlation. The larger the L_C, the more correlated the random field and the smoother the fluctuation, and then the pore phase after two-phase transformation is composed of larger pores, as RF₁ shown in Fig. 5. Conversely, the smaller the L_C, the less correlated the random field and the wavier the fluctuation, and then the pore phase is made up of smaller pores, as RF₂ in Fig. 5. In the resulting two-phase field, the size of the generated pores is distributed within a certain range around Lc. These pores do not have a uniform geometry, but are spatially curved objects with random shapes. As indicated by the results of the MIP test (Fig. 4), the pore size span of cement paste and mortar is large (30, 67, 17 times difference between the maximum and minimum pores in P-0.3, P-0.4/0.5, M-0.3/0.4/0.5, respectively). Therefore, multiple Gaussian random fields with different Lc values are created, thresholds are given and then the created fields are changed into two-phase fields to represent pores of different size ranges. Subsequently, the superposition of these fields will generate a pore structure model that satisfies the real pore size distribution of the material. The selected Lc values of random fields should be distributed within the studied pore size range and the threshold D of each field is determined by porosity, which in turn is derived from the variation of the experimental pore size distribution. Moreover, it should be noted that during the superposition of fields, large pores may cover parts of small pores, then leads to a total porosity lower than the sum. To avoid this effect, the porosities of the fields with small pores should be slightly greater than the initial values. For example, there are three fields (F₁, F₂ and F₃) with gradually decreasing Lc values. Following the two-phase transformation (porosities are φ1, φ2 and φ3) and superposition, some of the pores in F₂ may be covered by the pores in F₁, and some of pores in F₃ may be covered by the pores in F₁ and F₁. Hence, φ2 needs to be modified to φ2/(1−φ1) and φ3 needs to be modified to φ3/(1−φ1−φ2) to offset the pore overlap effect. The modeling parameters used in this work are as follows:

The cement paste models (P-0.3, P-0.4, P-0.5) were built in a cube with a side length of 1 μm and a mesh division of 400 × 400 × 400, leading to a cell size of 0.0025 μm. Therefore, the model can cover the pore size ranges of 0.003–0.09 μm and 0.003–0.2 μm. The cement paste models are the results of the superposition of eight independent fields. The mortar models (M-0.3, M-0.4, M-0.5) were generated in cube of 0.3 μm with a grid division of 300 × 300 × 300. The cell size is 0.001 μm, and the investigated pore size range is 0.003–0.05 μm. The pore size range of mortar is relatively smaller compared with that of cement paste; thus, the model is created by superimposing six independent fields with different Lc values. The related parameters are provided in Table 4, and the generated models are shown in Fig. 6. It should be noted that the total porosity of a model includes open and close pores. By contrast, only open porosity can be measured by the MIP test, and thus, the open porosity of the generated model should be further verified (refer to Subsection 3.2).

Model validation is accomplished by verifying pore size distribution and open porosity by using the image analysis method of mathematical morphology. Mathematical morphology is a theory and technique for analyzing and processing geometric structures; it is commonly used in binary images [23–25]. It contains basic operations such as erosion, dilation, morphological opening, morphological closing and geodesic reconstruction. As shown in Fig. 7a, the morphological opening operation can compare an image with a certain shape (structure element) and eliminate parts of the initial image that are smaller than or equal to it. This operation can be expressed as: (11)PMO=(I⊖E)⊕E

PMO is the processed image after morphological opening. I is the initial image. E is the structural element. ⊖ and ⊕ are the basic morphological operations of erosion and dilation by the operator E. Erosion and dilation resemble Minkowski addition and subtraction introduced in the theory of integral geometry. More details can be found in the work of Morales [26]. The structure element shown in the illustration is selected as a cell. After the morphological opening, isolated blue blocks and blue blocks with only one side connected to others are found and deleted. By applying this idea to the 3D pore space model, the structure element can be set as a cube or other 3D symmetrical shapes. Then, the morphological opening can eliminate pores that are smaller than or equal to the size of the defined structure element. By setting a series of gradually increasing structure elements, the pore network will be sieved to obtain the pore size distribution curve. Fig. 7b presents two examples of processing: the P-0.3 model with a structure element 5 times the cell size (5 times the cell size = 0.0125 μm, pore smaller than or equal to this value are erased) and a structure element 9 times the cell size (9 times the cell size = 0.0225 μm, pore smaller than or equal to this value are erased).

Geodesic reconstruction is used to extract components that are connected to the boundary of an image. In the current work, it is numerically implemented and applied to extract open pores in a pore space model. As illustrated in Fig. 8a, the mechanism of this operation is a recursive algorithm of boundary element dilation for the size of the selected structure element, followed by an intersection with the initial image. The calculation ends when no additional part is identified after the intersection; thus, the components are connected to the boundary will be extracted. This operation can be expressed as: (12)PGR=∪Y∩Xi≠0Xi

PGR is the processed image after geodesic reconstruction. Y stands for the pores on the boundary of model and Xi is the connected pores in the pore network. More details can be found in the work of Matheron [23]. In accordance with this principle, open pores can be investigated, and open porosity can be verified. Fig. 8b presents the results of the application of geodesic reconstruction to the P-0.3 and M-0.3 models. Through calculation, the open porosities of P-0.3, P-0.4, P-0.5, M-0.3, M-0.4, M-0.5 are 19.76%, 23.71%, 31.46%, 11.19%, 17.27%, 19.82%, respectively, which are close to the experimental values. Based on the obtained open pore network, a series of morphological opening operations with gradually expanding structure elements were numerically applied. The open pore network was sieved, and the pore volume variation as the pore size changes was estimated. The obtained pore size distribution curves are compared with the MIP data in Fig. 9. The generated models nearly fit the real pore space characteristics thus are supposed to be qualified for the following simulation of the adsorption-desorption process.

In summary, the model generation and validation are finished following the five steps: 1. investigate the pore space characteristics of the material (open porosity and open pore size distribution obtained from the MIP test) and preset Lcs of random field distributed within the studied pore size range; 2. Investigate the experimental pore size distribution curve, preset a porosity for each field and calculate the threshold; 3. Perform the two-phase transformation; 4. Combine two-phase fields; 5. Numerically investigate the open porosity (using the geodesic reconstruction operation) and the pore size distribution (using the morphological opening operation) of the generated model, and then adjust parameters. The methods of adjustment include: increase or decrease the Lc of each field, increase or decrease the porosity of each field, add or delete field to guarantee all the pores in the studied size range can be simulated with correct proportions.

d denotes the minimum size of the drained pore during the desorption process and the maximum size of the water-filled pore during the absorption process; it can be estimated via the Kelvin-Laplace equation [27,28]: (13)d=−4γMRTρln(RH)

Condition: Standard atmospheric pressure and 20°C.

γ is the surface tension of water (0.072 N/m).

M is the molar mass of water (18 g/mol).

R is the universal gas constant [8.314 J/(mol⋅K)].

T is the temperature (20°C = 293.15 K).

ρ is the density of water (1 g/cm³).

RH is the relative humidity.

For example, RH=90% corresponds to d=0.02 μm, RH=70% corresponds to d=0.006 μm and RH=50% corresponds to d=0.003 μm.

In accordance with the experimental result, a hysteresis phenomenon occurs between the isothermal adsorption and desorption processes. This phenomenon is attributed to the fact that the sequence of pore drainage depends not only on size but also on arrangement. Large pores that are connected outward through small pores will not drain at the corresponding RH because the path is blocked by small pores. “Large pores” are called “ink-bottle” pores, and this phenomenon is known as the “ink-bottle” effect. The drainage of ink-bottle pores occurs when RH decreases to the value that corresponds to the smallest size in the water migration path. The “ink bottle” effect should be considered in numerical simulation.

A schematic diagram of the desorption process with decreasing RH is shown in Fig. 10. Pores A1 and pore A2 have the same size; however, at the same RH value (third image in the figure), pore A1 drains while pore A2 does not, because pore A1 is connected outward through the larger pore C1, while pore A2 (the belly) is connected outward through the smaller pore B2 (the neck), forming an “ink-bottle” pore that will drain with pore B2 at a lower RH.

The scenario of the numerical simulation of desorption is designed on the basic of the image analysis operation of mathematical morphology. In the 1st step, the characteristic diameter d at different RH values is calculated using the Kelvin-Laplace equation and set as the size of structure element, as shown in Fig. 11. The pore network is then processed via the morphological opening operation to filter out pores larger than the structure element. In the 2nd step, the processed image is subjected to geodesic reconstruction to extract pores connected to the boundary, from which the drainage mass is obtained at each RH during the desorption process. In the case of a large pore connected to the outside through a small pore, the small pore will be deleted after morphological opening. Following the geodesic reconstruction, the large pore will be judged as a close pore and thus is without drainage at the current RH. It will wait for a lower RH to be able to drain. m1, m2 and m3 denote the cumulative desorption mass of water as RH decreases.

A schematic diagram of adsorption characteristics with increasing RH is shown in Fig. 12. The order of filling during adsorption differs from the order of draining during desorption due to the presence of ink-bottle pores. When RH rises, the small pore B2 is filled first, followed by the large pore A1, and then C1 and C2. The belly of “ink-bottle” A2 will eventually be filled following the dissolution and diffusion of gas molecules.

The idea of the numerical simulation of adsorption is illustrated in Fig. 13. To calculate filling mass of water from RH A to RH B (B > A) during the adsorption process, the 1st step is to obtain the desorption image at RH A (e.g., m1, m2 and m3 in Fig. 11). The 2nd step is to apply the morphological opening operation to the open pore network (the result of geodesic reconstruction on the initial pore space model). The size of the structural element is the size of the characteristic diameter d at RH B. The 3rd step is to perform a subtraction between the results of the 1st and 2nd steps, and then the new filling parts from RH A to RH B are obtained (blue parts m4, m5, and m6). The combination of Figs. 12 and 13 indicate that m4 = mass of water filling B2, m5 = mass of water filling A1, and m6 = mass of water filling C1 + C2. To provide a specific example of calculating the filling mass of water from 50% RH to 60% RH, the 1st image of desorption at 50% RH is indispensable, and then, the morphological opening is performed on the open pore space by using the size determined by 60% RH (0.004 μm) to obtain the 2nd image. Finally, the 2nd image is subtracted from the 1st image to calculate the filling mass of water from 50% RH to 60% RH.

In the simulation work, as illustrated in Figs. 11 and 13, by defining structure elements with different sizes and applying in the morphological opening operation, RH values can be determined according to the Kelvin-Laplace equation, and the cumulative desorption and adsorption mass of water can be obtained via the proposed calculation scenario. The isothermal adsorption and desorption processes were simulated on the basis of the constructed pore structure models of cement paste and mortar. The results are compared with the experimental data in Fig. 14. The simulated desorption curve always locates above the adsorption curve, and the hysteresis behavior exists in the whole RH interval, further verifying the existence of “ink-bottle” pores in the microscopic pore network. Notably, five RH values (20%, 40%, 60%, 80% and 100%) were selected for the adsorption-desorption experiment conducted in this work, and the points of the saturation degree that corresponded to two adjacent humidity levels were connected by a straight line. In fact, the saturation degree does not necessarily satisfy linear variation, and thus, it will cause a certain difference between the experimental and numerical results. The simulated curves can better reflect the characteristic of water transport during the adsorption-desorption process. When 100% RH is reached, the saturation degree of adsorption is about 75%–85%, indicating the existence of 15%–25% ink-bottle pores, which will be filled subsequently following the dissolution and diffusion of gas molecules, as shown in Fig. 12.

In addition, the model was generated on the basis of the pore size distribution obtained from MIP test. The pore size ranges are 0.003–0.09 μm for P-0.3, 0.003–0.2 μm for P-0.4/P-0.5, and 0.003–0.05 μm for M-0.3/M-0.4/M-0.5. No pores smaller than 0.003 μm are found inside. According to the Kelvin-Laplace equation, this size is the characteristic diameter that corresponds to 49% RH. Therefore, the starting point of the simulated adsorption curve and the end point of the desorption curve are chosen to be the experimental values at 49% RH. The investigation is focused on 49%–100% RH interval. The experimental curve shows that a certain adsorption effect still exists in the 0%–49% RH interval. This phenomenon may be attributed to two reasons. 1. The sample may have a part with tiny pores that measure less than 0.003 μm. Limited by the performance of Autopore IV 9500, the smaller the pores, the more difficult they are to detect. The part with tiny pores might not have been traced in the MIP test, and therefore, was omitted from the model creation. However, they will complete water saturation during the growth of 0%–49% RH. 2. Water saturation is a gradual accumulation process. It is not that the corresponding pores are filled instantly when a certain RH value is reached, but water is progressively adsorbed as humidity increases, and the filling is completed in the corresponding pores when the RH value is reached. Consequently, a certain adsorption effect still exists in 0%–49% RH. From this perspective, it also explains why the major part of the simulation curve lies below the experiment curve. In general, the simulation results are close to the experimental ones, verifying the effectiveness of the adsorption-desorption theory and the numerical simulation scenarios.

Figs. 15a–15f show the adsorption and desorption of the P-0.3 model considering 56%, 71% and 78% RH. Meanwhile, (g), (h), (i), (j), (k) and (l) present the results of the M-0.3 model considering 65%, 73% and 79% RH. Since the structure element defined in morphological operations is related to the cell size which is different for the cement paste and mortar models, choosing the same RH values for comparison is impossible, and only values that are closer can be selected. These images illustrate that as RH decreases, an increasing number of small pores and ink-bottle pores are drained. As humidity increases again, large pores are gradually filled, eventually leaving a part of ink-bottle pores to be filled following gas dissolution and diffusion.

The subsequent research can be further improved from the following two aspects: 1. A study on the microstructure of cement-based materials with the use of instruments and means with greater measuring range and accuracy can determine the existence and proportion of tiny pores, and thus, improve the 3D pore structure model of material; 2. With regard to the numerical simulation, the grid division will affect the simulation results to a certain extent. The denser the grid division and the smaller the cell size, the larger the pore size range that the model can contain, on the one hand, and the more accurate the adsorption-desorption simulation will be, on the other hand. As shown in Fig. 16, the principle of searching for open pore space in a model by using the geodesic reconstruction method can be summarized as making the pore cell on the boundary dilate with a certain size (size of the selected structure element) and then intersect with the initial image to determine whether new pore cells are added. The cycle of “dilation + intersection” repeats until the processed image is unchanged and then the open pores are extracted. In Fig. 16, pores a and b have the same size and position but different grid densities. By using the same structure element size (e.g., two times the cell size) for geodesic reconstruction, the calculated open pore in the small grid is only a, while in the large grid, a and b will be defined as connected pores, increasing the proportion of large pores in the model. Therefore, using the smallest element is important in the application of geodesic reconstruction. Meanwhile, a smaller cell size allows the model to contain a larger range of pores and improve the accuracy of the calculation. However, the demands on computer performance will also increase.