Estimation of the 2D mineralogical distribution in hydrated C3S matrix through coupling of chemical solver with experimental data.

doi:10.21203/rs.3.rs-3344183/v1

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Estimation of the 2D mineralogical distribution in hydrated C3S matrix through coupling of chemical solver with experimental data.

https://doi.org/10.21203/rs.3.rs-3344183/v1

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This paper presents a procedure to identify 2D mineralogical distribution in a cement matrix using EDS quantified maps. The results obtained from EDS analysis are combined with the chemical solver GEMS and the thermodynamic database CEMDATA18 to estimate the mineralogical distribution in pixels, while respecting thermodynamic equilibrium. Anhydrous phases are identified through a clustering process applied to the EDS quantified maps. Homogenization schemes are then employed to calculate in on hand the homogenized mechanical distribution in the studied zone and in one other hand the macroscopic homogenized properties. The proposed method is applied to C₃S hardened cement paste (HCP) at 28 days of hydration. An accompanying experimental campaign confirms the accuracy of the estimated mineralogical and mechanical distributions using this approach.

C3S binder

chemical solver

mineralogical distribution

homogenization scheme

Concrete is a centimeter heterogeneous material made with aggregates embedded in a cement matrix which is also heterogeneous at micrometer scale [1]. The mineralogical distribution within this cement matrix significantly affects most of the chemo-mechanical properties of concrete [2]. For example, Portland cement with a high aluminum content is more susceptible to sulfated attacks [3]. The addition of silica fume to a cement matrix reduces average porosity but also decreases the content of portlandite, which acts as a chemical buffer against degradation [4]. Therefore, understanding the mineralogical changes between different cementitious materials or different degradation states is crucial for comprehending the degradation mechanisms and their impact on critical engineering properties of cement structures.

Various techniques are available to obtain this information. One technique involves the use of a scanning electron microscope (SEM), which has been commonly employed for several decades to study the distribution of hydration products in hardened cement paste (HCP) [5]. In 1970, Marcinkowski et al. [6] demonstrated the potential of this technique by studying fracture surfaces of concrete. Seven years later, Stucke et al. [7] proposed combining SEM with Energy Dispersive Spectroscopy (EDS) to study the evolution of the calcium-to-silicon atomic ratio in C3S (alite) HCP. They performed a series of EDS spot analyses from the core of an alite grain through the matrix, composed of C-S-H and portlandite. A quantification correction scheme was applied to calculate elemental concentrations from X-ray intensities. The following year, Barnes et al. [8] conducted elemental mapping on the four major phases of Portland clinker: alite, belite, celite, and ferrite. Their work highlighted the ability of SEM-EDS to efficiently provide the distribution of chemical elements in cement materials, even minor elements.

Presently, depending on the objectives of the study, SEM-EDS techniques are used by the cement science community in two distinct ways. As proposed by Stucke et al. [7], EDS spot analyses with a quantification correction scheme are performed when the study aims to precisely identify the chemical composition of a specific zone [9, 10]. This approach yields interesting results but can be challenging to apply over a large area. Conversely, when the focus is on the distribution of chemical elements in cementitious materials, elemental mapping is employed [11], as proposed by Barnes et al. [8]. This method is powerful for understanding phenomena related to mineralogical changes. In many recent studies, these two approaches are combined to faithfully estimate the distribution of chemical species in cementitious materials [12, 13]. Coupling with other techniques such as X-ray diffraction, micro-tomography, or nano-indentation is often performed to enhance detection capabilities [14, 15].

Recently, Teng et al. [16] conducted multivariate statistical analysis on SEM-EDS X-ray spectral images of rare earth minerals to efficiently extract phase recognition. Ford et al. [17] proposed an approach based on EDS, backscattered electron (BSE) imaging, and nano-indentation measurements, which combines clustering and machine learning to estimate the 2D mineralogical distribution in a cement matrix. Georget et al. [18] presented an approach that enables phase separation and chemical analysis using EDS and BSE analysis. This method involves denoising of quantified maps and clustering.

In all the approaches presented above, only pure phases are sampled. In the present study, we propose a method that combines a chemical solver with EDS quantified maps to estimate the mixture of phases in each pixel. The mineral composition of pixels, along with a homogenization scheme, allows for the estimation of the mechanical elastic properties of the representative elementary volume (REV). The method is applied to C3S HCP, a cement matrix that offers simplicity. All modeling results are validated by a dedicated experimental campaign.

3.1. Materials

The sample examined in this paper is a hardened cement paste (HCP) consisting of pure silicate tricalcium (C₃S) mixed with pure water at a water-to-binder ratio of 0.5. The sample was cured for 28 days at 20°C and under high relative humidity (approximately 100%) in a sealed bag. Prior to conducting solid analyses, the sample was lyophilized and freeze-dried to stop the hydration process.

The XRD diagram in Fig. 1 highlights the mineralogical composition of the C₃S HCP, which consists solely of Portlandite, C-S-H, and anhydrous C₃S. A volume weight of 2200 kg/m3 was estimated using mercury intrusion porosity (MIP) analysis.

3.2. Characterization Methods

SEM-EDS analyses were carried out on carbon metallized polished sections. The acquisition was performed on SEM Hitachi S3500N with two EDS Brüker 5030 XFlash at 15 keV with x100 magnification. The pixel size is 2.4 µm, at the limit of the interaction pair for this type of material [19]. The average number of photons per pixel is greater than 4000, allowing rework of signals from hyperspectral images. Each of the three images carried out contains 736 000 pixels. A Phi-Rho-Z quantification methods [20] was applied on hyperspectral images to extract in each pixel the atomic ratio between calcium, silicon and oxygen. The hydrogen content was voluntary excluded due to the limit of analysing of EDS for light elements.

The XRD analyses presented in this paper were carried out with an Aeris Research Edition diffractometer from Malvern-Panalytical in θ − 2θ configuration, under CuKα radiation (λ = 1.54 Å) with a divergence slot of 0.5° and a rotation platform.

The distribution of young modulus in the HCP was estimated by nano-indentation measurements with an Anton & Paar NHT3 nano-indentor equipped with a Berkovitch tip. The grid nano-indentation technique was used to map the mechanical property distribution of the C₃S HCP. Tests were displacement-controlled with nanoindentation depth equal to 0.30 µm to focus on the mechanical properties of the HCP constituents without macroscale effect [21]. Poisson’s ratio was assumed to be constant and equal to 0.24. A loading and unloading speed of 12mN/s was applied for each indent. Quasi-static measurement with Olivier & Phar technique [22, 23] was performed four times on a 35 x 35 grid with grid spacing of 10 µm.

TGA analyses were performed with a Netzsch STA 409 PC LUXX thermobalance. Samples were crushed, 120 mg were introduced into alumina crucibles. Analyses were started with a 10mn initial step at 30°C, followed by a 10°/min increment between 30 and 1145°C, for a total analysis time of 2h. The mass variations are then deduced, using the Proteus Thermal Analysis software from the same manufacturer, via the Marsh tangential method [24].

Microstructure was assessed by mercury intrusion porosimetry (MIP) using a Micromeritics Autopore IV on a sample section of 0.5g. A contact angle of 130° for a surface tension of 0.48 N/m were put as Washburn’s equation input parameter. Data were gradually collected to a maximum pressure of 414 MPa.

The chemical modelling was performed with the python module xgems version 1.0.7 based on the Gibbs Energy Minimization Software GEMS [25] coupled with CEMDATA18 [26]. All the calculations were performed with Google Colab platform.

The Fig. 2 provides a detailed overview of the sequence of operations performed on quantitative EDS images to determine the 2D mineralogical and mechanical distribution in the C₃S HCP.

One of the aims of this paper is to combine a chemical solver with EDS to determine the mineralogical distribution. However, a major drawback of this approach is the inability to preserve a portion of the anhydrous species, as observed experimentally [27, 28]. To partially address this problem, a clustering scheme called the "Gaussian mixture model" [29] is applied to the calcium, silicon, and oxygen maps, based on atomic ratios, to identify pixels containing only anhydrous C₃S. In practice, seven clusters are defined, representing different possible mineralogical arrangements (CH, C-S-H, C₃S, CH + C-S-H, C₃S + C-S-H, CH + C₃S, CH + C-S-H + C₃S). The cluster with a mean of its Gaussian distribution for calcium-to-silicon atomic ratio (C/S atomic ratio) and oxygen atomic ratio close to those of C₃S is considered the pure C₃S cluster, and all pixels belonging to this cluster are identified as anhydrous C₃S. To confirm this choice, the hydration model by Parrot & Killoh [27, 28] is applied to estimate the proportion of remaining anhydrous C₃S after 28 days of hydration.

The calculation of oxide mass ratios is performed on the remaining pixels, based on the atomic ratio maps of calcium for calcium oxide (CaO) and silicon for silicon dioxide (SiO₂). For each pixel, the water content is calculated by multiplying the total mass by the water-to-binder ratio. This approach assumes that all HCP samples had the same amount of water available for hydration.

A chemical calculation is then conducted using the chemical solver GEMS [25] to determine the atomic mineralogical distribution for each pixel. The residual porosity is estimated by calculating the difference between the total mass of species in the pixel and its initial total mass. Two modeling cases are considered in this study: one considering the chemical properties of dry C-S-H and another considering the properties of saturated C-S-H.

The volume ratio calculation allows for an estimation of the distribution of elastic mechanical properties in the C₃S HCP. Depending on the difference in mechanical properties between each phase, either the Mori-Tanaka (MT) or self-consistent (SC) scheme [30] is applied.

Finally, the self-consistent scheme is applied to the dataset of homogenized mechanical properties to estimate the macroscopic mechanical properties of the material.

The Fig. 3 illustrates the atomic proportions of calcium, silicon, and oxygen for the three zones of the C₃S HCP analyzed using EDS. Notably, to validate the accuracy of EDS quantification, it is interesting to observe that the average C/S atomic ratio aligns with the composition of the raw materials, thereby confirming the relevance of EDS quantification.

In appendix A, a calculation based on the mean value of oxygen (64%) is proposed to estimate the total porosity in the analyzed zone.

Except for C₃S phases deducted from clustering method, the CEMDATA18 thermodynamic database [26] is used in the chemical solver. Table 1 provides a list of all the phases considered during the GEMS calculation, along with their respective thermodynamic properties.

Table 1

Cemdata18 database: Standard thermodynamic properties at 25°C and 1 bar of solid solution models C-S-H named C-S-HQ and Portlandite.
Phase		ΔfG° (kJ/mol)	ΔfH° (kJ/mol)	S° (J/K/mol)	a0 (J/K/mol)	a1 (J/mol/K²)	a3 (J/K^0.5/mol)	V° (cm³/mol)	Ref
C-S-HQ	TobH C_2/3⁻S⁻H_1.5	-1668.56	-1841.5	89.9	141.6			55	[31]
	TobD C_5/6⁻S⁻H_1.83	-1570.89	-1742.4	121.8	166.9			48	[31]
	JenH C_1.33⁻S⁻H_2.17	-2273.99	-2506.3	142.5	207.9			76	[31]
	JenD C_1.5⁻S⁻H_2.5	-2169.56	-2400.7	173.4	232.8			81	[31]
	NaSH: N_0.5-S_0.2-H_0.45	-431.20	-478.0	41.2	37.9			10.5	[32]
	KSH: K_0.5-S_0.2-H_0.45	-440.80	-489.6	48.4	40.6			12.4	[32]
Portlandite CH		-897.01	-985	83	187	-0.022	-1600	33	[26]

Next to the GEMS calculation, the atomic distribution is converted to mass, volume, volume ratio, and Young's modulus distribution based on the properties provided in Table 2.

Table 2

Chemical, mechanical and transport properties of individual cement paste phases, taken from several sources in the literature or calculated in this study (calc).
Name	Chemical properties					Mechanical properties			Transport properties
Name	Composition	C/S	H/S	ρ (kg/m³)	M (g/mol)	E (Gpa)	υ	Ref	D_i (m²/s) in HTO
C_1.62-S₁-H₁ (dry)	1.24 CaO, 0.76 SiO2, 0.76 H2O	1.62 [calc]	1 [calc]	2650 [33]	128.72	57	0.31	[34]	-
C_1.62-S₁-H_1.5 (dry)
C_1.62-S₁-H₄ (saturated)	1.24 CaO + 0.76 SiO₂ + 3.04 H₂O	1.62 [calc]	4 [35, 36]	1950 [35, 36]	169.76	28	0.24	[calc]	2.24e-12 [37]
C_1.62-S₁-H₆ (C-S-H LD)	1.24 CaO, 0.76 SiO2, 4.64 H2O	1.62 [calc]	6 [36]	1190 [36]	198.56	18.1	0.24	[38]	5.2e-12 [39]
C_1.62-S₁-H_3.2 (C-S-H HD)	1.24 CaO + 0.76 SiO₂ + 2.48 H₂O	1.62 [calc]	3.2 [36]	1790 [36]	159.68	31	0.24	[38]	1.1e-13 [39]
C₃S	3 CaO + SiO₂	3	-	3123 [40]	228	117	0.31	[41]	0
Portlandite	1 CaO + 1 H2O	-	-	2241 [40]	74.09	42.3	0.324	[41]	0
Aqueous	H2O	-	-	1000	18	0	0	-	2e-09 [39]

The atomic ratio between Jennites and Tobermorites of C-S-HQ model [26] is homogenized in C-S-H dry (C_1.62-S₁-H₁) or saturated (C_1.62-S₁-H₄). The C/S atomic ratio presented in Table 2 is calculated using the chemical solver with the CSHQ model for C₃S HCP. The H/S atomic ratio of dry C-S-H is adjusted to ensure the oxygen atomic content in dry C-S-H aligns with the main peak oxygen atomic distribution observed in the EDS analysis (Fig. 5b).

The mechanical properties of dry C-S-H are obtained from the atomistic structural models of C-S-H gel developed by Pellenq et al. [34]. The mechanical properties of saturated C-S-H are estimated using the Mori-Tanaka homogenization scheme, considering the properties of dry C-S-H and a porosity calculation based on the H/S atomic ratio. With this approach, the porosity of saturated C-S-H with an H/S atomic ratio of 4 is estimated to be 35%.

The chemical properties of C-S-H LD (low-density) and HD (high-density) are provided in Table 2 for the overall calculation, based on the method by Tennis and Jennings [42], using properties derived from the work of Kurihara & al. [36]

This section describes the main results obtained with this approach; first part is focus on global properties.

4.1. Global Chemo-mechanical properties

This section focuses on the mineralogical distribution of a representative elementary volume (REV) of C₃S HCP after 28 days of hydration in a sealed bag at 20°C. The Table 3 presents three methods used to estimate the mineralogical distribution:

An approach based on a hydration calculation using the Parrot & Killoh model, assuming a specific surface area of 300 m²/g for the anhydrous C₃S.
A combination of experimental results obtained from XRD, TGA, and EDS techniques. The porosity estimated from EDS results is calculated using the mathematical approach detailed in appendix A. However, capillary porosity measurement was not performed in the experimental campaign.
The ‘‘EDS-based calculation’’ described in the previous section. The values provided in Table 3 represent the average values within the REV. In this paper, no solution was found to accurately estimate the 2D distribution of C-S-H LD and HD, which is why the corresponding cases are blank in the table.

Table 3

Mineralogical distribution of alite matrix based on different approaches (Calc: calculated in this study).
Chemical element		Hydration calculation	Experimental	EDS based calculation
C-S-H	C_1.62-S₁-H₁ (dry)	0.39	0.44 (XRD)	0.35
	C_1.62-S₁-H₄ (saturated)	0.58	0.64 (Calc)	0.66
	C_1.62-S₁-H_3.2 (C-S-H HD)	0.17	0.18 (Calc)
	C_1.62-S₁-H₆ (C-S-H LD)	0.41	0.46 (Calc)
Portlandite		0.23	0.24 (XRD/TGA)	0.21
C₃S		0.07	0.07 (XRD/TGA)	0.04
Porosity	Total	0.30	0.4 (105°C [43]) / 0.38 (EDS)	0.40
Porosity	Capillary	0.11		0.09

All the modeling approaches presented in the paper are consistent with the experimental results, indicating approximately 40% dry C-S-H and 20% Portlandite in the mineralogical distribution. However, there are some discrepancies between the modeling and experimental data regarding the total porosity. The hydration calculation underestimates the total porosity, while the EDS-based calculation cannot account for pixels containing a portion of C₃S, resulting in a lower C₃S content compared to the experimental observations. The estimated capillary porosity values align with around 10% in both modeling cases, but there were no specific experimental measurements conducted to confirm this value. As anticipated for a water-to-binder ratio of 0.5, the estimation of the proportions of HD and LD C-S-H based on the Tennis and Jennings model reveals a higher proportion of LD C-S-H, which is consistent with expectations [42].

The Fig. 4 provides a detailed illustration of the various mineralogical configurations in terms of volume ratio depending on the C/S atomic ratio within a pixel. It also shows the consequences on the homogenized properties, including mechanical and diffusion properties of the pixel. The figure presents different options for C-S-H modeling, taking into account the H/S atomic ratio. The porosity calculation follows the methodology described in Fig. 2. Depending on the specific C-S-H chemical composition considered, the porosity can be either total porosity or capillary porosity.

In all cases, an increase in the C/S atomic ratio from 1.62 resulted in a decrease in the C-S-H volume ratio in favor of portlandite. Simultaneously, a decrease in total porosity was observed in both modeling approaches using the dry C-S-H chemical composition. This reduction in porosity can be attributed to the difference in density between dried C-S-H (2.650) and portlandite (2.204). However, when saturated C-S-H with an H/S atomic ratio of up to 3.26 was considered, the opposite trend occurred. The density of saturated C-S-H was lower than that of portlandite, leading to an increase in capillary porosity depending on the C/S atomic ratio within the pixel.

It is worth noting that a pixel with a C/S atomic ratio of 1.62 and a mass water-to-cement ratio of 0.5 has just enough space to precipitate saturated C-S-H with an H/S atomic ratio of 4. The precipitation of C-S-H LD with an H/S atomic ratio of 6.1 in a pixel with a C/S atomic ratio of 1.62 would require a water content higher than the average E/C of 0.5 used during casting. This observation suggests that the distribution of water within the sample is not homogeneous.

Regarding the homogenized Young's modulus, the gap observed in the MT homogenization results from the transition of the matrix from C-S-H to portlandite. The MT scheme requires a minimum volume ratio of the matrix, which is typically set at 30%. As the C-S-H content decreases in the pixel with varying C/S atomic ratios, there is a tipping point where portlandite becomes the dominant species and should be considered as the matrix. It is interesting to note that the magnitude of the gap is consistently lower than 1.5 GPa, which is comparable to the uncertainty in the measurement of mechanical properties for each species [41, 44].

In both modeling cases using the dry C-S-H chemical composition, the Young's modulus of C-S-H is equal to 57 GPa. In this scenario, similar to the findings of Hou et al. [45], the self-consistent (SC) scheme underestimates the homogenized Young's modulus when the porosity exceeds 25%. Therefore, the homogenized modulus calculated using the SC scheme in these cases ranges between 5 and 10 GPa for a pixel with a C/S atomic ratio of 1.62. In all other modeling cases, the SC and MT schemes yield similar results, with homogenized Young's modulus values ranging from 24 to 27 GPa. In the EDS-based calculation, the MT scheme is favored for estimating the homogenized Young's modulus of pixels containing dry C-S-H with an H/S ratio of 1 to maintain consistency regardless of the C/S atomic ratio. Conversely, the SC scheme is applied when the ‘‘EDS-based calculation’’ involves saturated C-S-H with an H/S ratio of 4.

It is interesting to observe that regardless of the C-S-H chemical composition considered, the homogenized Young's modulus remains between 21 and 27 GPa for C/S atomic ratios ranging from 1.62 to 10. For a C/S atomic ratio of 10, more than half of the pixels contain portlandite with a relatively high Young's modulus. The presence of portlandite is compensated by the porosity resulting from the density difference between C-S-H and portlandite.

4.2. Clustering based on gaussian mixture method

The Fig. 5 shows the 2D distribution, the oxygen atomic ratio, the surface ratio and the C/S atomic ratio of the seven clusters estimated from clustering process.

The first observation highlighted by Fig. 5a is the zonation of each cluster. Clusters 6, 3, and 2 are grouped together, forming particles that resemble anhydrous C₃S grains. The larger size of these grains may be attributed to the hand-mixing preparation method. Cluster 5 forms an intergranular matrix, within which cluster 4 appears as inclusions. As shown in Fig. 5c, the surface ratio of clusters 1 and 0 is not significant enough to be observed in Fig. 5a.

The Fig. 5b and Fig. 5d provide information about the species present in each cluster. The C/S atomic ratio of cluster 6 has a mean value of 1.7, which is close to that of C-S-H. As explained in the previous section, the H/S value of dry C-S-H is fixed at 1 to match its oxygen atomic ratio with cluster 6.

Cluster 0 exhibits similarities with anhydrous C₃S in terms of C/S and oxygen atomic ratios. However, some parts of the anhydrous C₃S are smaller than the pixel size and are thus mixed with other species. Considering that not all of the C₃S is observable at this pixel size, the surface ratio estimated for cluster 0 in Fig. 5c is consistent with the Parrot & Killoh calculation presented in Table 3. As explained in Fig. 2, pixels belonging to cluster 0 are not considered in the solver calculation step and are treated as entirely composed of anhydrous C₃S.

4.3. Mineralogical distribution

The Fig. 6 shows the output results of the solver calculation steps considering dry C-S-H with C/S and H/S atomic ratios of 1.62 and 1, respectively. As explained in the previous section, Fig. 6d corresponds to pixels belonging to cluster 0 without any chemical calculation.

In a more general way, Fig. 6 highlights that at a pixel size of 2.4 µm, very few pixels are composed of 100% of a single mineral, while most of them are a combination of different species. Similar to what was observed for clusters 6, 3, and 2 in Fig. 5a, Fig. 6a shows that C-S-H is mainly located in structures resembling anhydrous C₃S grains. This grouping of C-S-H is called C-S-H rich zone in the following sections. On the other hand, portlandite (Fig. 6c) tends to occupy the intergranular spaces. The total porosity (Fig. 6b) is slightly lower in the intergranular spaces due to the lower content of dry C-S-H compared to other zones. As observed in Fig. 4, in this modeling case, the porosity increases when the C/S atomic ratio of the pixel decreases, mainly due to the density difference between dry C-S-H and portlandite. It is interesting to note that at 28 days of hydration, despite hand mixing, no large particles of anhydrous C₃S are visible in Fig. 6d.

The Fig. 7 proposes considering saturated C-S-H with an H/S atomic ratio of 4 in the solver calculation steps instead of dry C-S-H. The calculated porosity corresponds to the capillary pores. In this approach, the distribution of portlandite and C₃S is not affected by the H/S atomic ratio of C-S-H.

Pixels with low content of portlandite in Fig. 6c are filled with saturated C-S-H in Fig. 7a. This is because, as shown in Fig. 4, a pixel with a C/S atomic ratio of 1.62 and an initial water-to-binder mass ratio of 0.5 has a close to zero porosity. As a result, the capillary porosity is not uniform in the C₃S HCP matrix, as shown in Fig. 7b. It is interesting to note that the highest capillary porosity in Fig. 7b is found in pixels with high portlandite content. This observation is consistent with Fig. 4, where the porosity increases with the C/S atomic ratio of the pixel.

In the following sections, the C₃S, matrix is divided in two main zones: the intergranular space and the C-S-H rich zone, as observed on Fig. 6, Fig. 7 and Fig. 9.

4.4. Estimation of the distribution of homogenized properties

The homogenization scheme is applied to the results obtained from Fig. 6 and Fig. 7, using the mechanical properties provided in Table 2, in order to estimate the distribution of the Young's modulus in the C₃S HCP. The Fig. 8 displays the 2D distribution and histogram of the Young's modulus, depending on the chosen modeling approach for the C-S-H chemical composition. These histograms are then compared to nano-indentation measurements.

First observation: The distribution of the Young's modulus is similar in both modeling cases and in good agreement with experimental results, with a mean between 20 and 26 GPa. No pixel with a Young's modulus higher than 26 GPa is observed.

In the first modeling case considering dry C-S-H, the high Young's modulus of portlandite (42 GPa) located in the intergranular space, and the dry C-S-H (57 GPa) from the C-S-H rich zone, are compensated by the total porosity. The average Young's modulus of the C-S-H rich zone is 20 GPa, which is close to that of LD C-S-H (Table 2).

In the second modeling case, the Young's modulus of saturated C-S-H is 28 GPa, combined with the low porosity in the C-S-H rich zone, resulting in an average Young's modulus of 25 GPa. This value falls within the range of LD C-S-H and HD C-S-H (Table 2).

The self-consistent scheme performed on the datasets allows an estimation of the macroscopic mechanical properties of the REV. A Young's modulus of 22.8 GPa and a Poisson's ratio of 0.27 are calculated for the modeling considering dry C-S-H. In the second modeling case with saturated C-S-H, the Young's modulus is 26.2 GPa and the Poisson's ratio is 0.25. In both cases, the macroscopic properties are in good agreement with the scientific literature [46].

In this section, we will address several questions of interest. Clear answers or initial thoughts will be provided based on the results detailed in the previous section.

Q1 What is the relevance of the approach ‘‘EDS based calculation’’ proposed in this study?

The C₃S HCP was selected in the study due to its simplicity in order to confirm the relevance of this approach. The Fig. 9 summarizes the 2D distribution of the solid species. A qualitative colormap with seven grades is applied to differentiate the ratio between dry C-S-H and portlandite in each pixel (For example, a yellow pixel is composed of 20 to 40% dry C-S-H and 80 to 60% portlandite). Pixels containing only anhydrous C₃S are plotted in orchid.

At the REV scale, as shown in Table 3, the "EDS-based calculation" is in good agreement with experimental results and hydration calculations. The C₃S HCP composition, after 28 days of hydration, consists of 35% dry C-S-H, 21% portlandite, 4% anhydrous C₃S, 40% total porosity, and 9% capillary porosity. Additionally, the overall total porosity is estimated to 38% by considering the atomic distribution in the REV and the water-to-binder ratio (see appendix A).

At the pixel size of 2.4 µm, Fig. 9 highlights two main precipitation zones. The first zone, referred to as the "C-S-H rich zone," primarily consists of particles of saturated C-S-H that resemble the structure of old anhydrous C₃S grains. The second zone, known as the "intergranular space," is composed of portlandite, capillary porosity, and some saturated C-S-H. This zonation can already be observed in the clustering results shown in Fig. 5. However, clusters often consist of multiple minerals, and specific names cannot be assigned to them, as indicated in Table 4.

Table 4

Average volume ratio of solid species in each cluster estimated by GMM method in C3S HCP
Cluster ID	Total solid	Portlandite	C_1.62-S₁-H₁ (dry)	Anhydrous C₃S
0	1	0	0	1
1	0.65	0.64	0.36	0
2	0.56	0.21	0.79	0
3	0.52	0.03	0.97	0
4	0.72	0.87	0.13	0
5	0.61	0.47	0.53	0
6	0.53	0.06	0.94	0

Only clusters 3 and 6 can be considered as a single species, as suggested by their C/S atomic ratio (Fig. 5d), which indicates the presence of dry C-S-H. Clusters 4 and 2 mainly consist of portlandite but still contain a significant amount of dry C-S-H. Pixels belonging to clusters 5 and 1, which form an intergranular matrix, are composed of a mixture of dry C-S-H and portlandite. The validity of these findings is confirmed by the comparison with the results of the "EDS based calculation" (Table 4). Clustering methods based on BSE, EDS, or nano-indentation measurements are common techniques used to analyze the 2D distribution in cement-based materials [17, 45, 47, 48]. For example, Georget et al. [18] applied clustering processes to grayscale combined maps generated from color elemental maps obtained through EDS analysis. Once the clusters were determined, the C/S atomic ratio and the aluminum-to-calcium atomic ratio from EDS spectra were calculated for a few pixels in each cluster to identify the main representative mineral. Like the work of Georget et al. [18], all existing studies consider pure phases within a pixel and not a mixture of phases.

The strength of the "EDS based calculation" approach lies in its ability to treat each pixel as a combination of mineral species, thanks to the introduction of a chemical solver. The Fig. 6 and Fig. 9 demonstrate that pixels are almost always composed of a mixture of species. The simplifications made for cluster 3 and 6 in Table 4 are based on global averaging and may not hold true locally.

However, the "EDS based calculation" encounters difficulties in properly estimating anhydrous C₃S particles smaller than the pixel size. Thermodynamic rules tend to favor the dissolution of clinker species in favor of hydrates. The clustering process estimates only 4% of anhydrous C₃S, compared to the experimental value of 7% (Table 3). There is still 3% of anhydrous C₃S grains smaller than the pixel size present in the pixels, which cannot be accounted for in the solver calculation steps. Therefore, a portion of the low content of portlandite in the C-S-H rich zone and C-S-H in the portlandite-rich zone, observed in the Fig. 9, could be attributed to these C₃S grains. As suggested by Stutzman et al. [49], a possible solution would be to perform segmentation on BSE maps at a nanometric scale in the same areas to estimate the proportion of anhydrous C₃S below the EDS pixel size. A dissolving limit could be applied in the chemical solver for each pixel to account for the remaining anhydrous C₃S.

The assumption of homogeneity of the initial water mass in the "EDS based calculation" approach limits the relevance of the homogenization process. The Fig. 8c shows a wide range of nano-indentation measurements, ranging from 0 to 80 GPa. In contrast, the numerical results have a much narrower data range of only a few GPa. In reality, access to the initial water is not the same depending on which part of the matrix is hydrated. For example, water must diffuse through a C-S-H layer to reach the anhydrous core of C₃S, while the intergranular space has no limitations in terms of water consumption. Consequently, some pixels may exhibit high total porosity, as observed in nano-indentation measurements ranging from 0 to 20 GPa, while others may have low total porosity, as observed in measurements ranging from 26 to 80 GPa. In the latter case, the percentage of anhydrous C₃S in the pixel can also significantly increase the young modulus of the pixel. Despite this criticism, the distribution of young modulus is in good agreement with nano-indentation measurements. Furthermore, the self-consistent homogenization scheme applied to all homogenized pixels provides a consistent macroscopic young modulus at the REV scale. Thus, the "EDS based calculation" approach can estimate the macroscopic mechanical properties of a material without the need for mechanical measurements.

The relevance of this approach is demonstrated by its good agreement at the REV scale with experimental and hydration calculation results, as well as at the pixel scale with clustering methods.

Q2 What is the influence of C-S-H chemical properties on the mineralogical distribution.

Thermodynamic, chemical, and mechanical properties of C-S-H are critical inputs in the 'EDS based calculation.' C-S-H models available in the thermodynamic database CEMDATA18 [26] influence the C/S ratio of the C-S-H matrix and, consequently, the atomic ratio between C-S-H and portlandite. In this study, considering the nature of HCP, the chosen C-S-HQ model estimates the C/S atomic ratio at 1.62. Another C-S-H model, such as the C-S-H II model, would provide a different C/S atomic ratio. The ‘‘EDS based calculation’’ assumes that water is in excess in each pixel. Therefore, the H/S atomic ratio of C-S-H proposed by the thermodynamic database has no effect on the modeling results. The selected H/S atomic ratio is used after the solver calculation to estimate the molar mass, density, and elastic mechanical properties.

The H/S atomic ratio has an impact on the value of the molar mass, which in turn affects the transition from atomic to mass. In this study, the H/S atomic ratio is equal to 1 for dry C-S-H and 4 for saturated C-S-H. As summarized in Fig. 10, Allen & al [50] propose a description of levels of porosity in C-S-H.

In this description, solid C-S-H corresponds to the particles inside the black line containing the interlayer physically bound water. Saturated C-S-H is composed of solid C-S-H and gel porosity. The H/S atomic ratio of solid C-S-H with a C/S atomic ratio of 1.7 is globally estimated to be equal to 1.8 [33, 50, 51]. Saturated C-S-H reaches an H/S atomic ratio between 4 and 5. As shown in Table 5, this value is highly influenced by the drying protocol applied before the analysis.

Table 5

H/S atomic ratio and density of C-S-H with C/S atomic ratio of 1.7 depending on its drying state
Drying state	H/S atomic ratio	Density g/cm3	Ref
Vacuum drying at room temperature	1.2	2.86	[51]
Oven 105°C/110°C	1.4	-	[52]
Oven 105°C/110°C	1.5	2.6	[36]
solid C-S-H (not dry)	1.8	2.65 / 2.73	[33, 50, 51]
	2.1	-	[53]
	2.3	-	[52]
	1.83	-	[35]
11% HR	2.5	2.4	[36]
Saturated	4	1.95	[36, 53]
	4.34	1.85	[35]
	5	1.8–2.1	[33]

Process at 11% of HR tends to eliminate gel water while conserving the interlayer water [35]. With this technique, the H/S atomic ratio is close to that of the solid C-S-H. Drying the sample in an oven at 110°C degrades not only the gel porosity but also some of the interlayer water. As a result, the H/S atomic ratio is lower than that of the solid C-S-H. Vacuum drying at ambient temperature is even more detrimental to the internal porosity of C-S-H.

The choice of drying protocol is not intended to preserve C-S-H porosity but to be compatible with the analysis technique. In this study, SEM-EDS analysis at 28 days of hydration requires freeze-drying. Due to the limited knowledge about the impact of freeze-drying on the H/S atomic ratio of C-S-H, the value is fitted based on the main peak oxygen atomic distribution from the EDS analysis. Considering this technique as one of the most damaging to the C-S-H structure [54], the estimation is consistent with the H/S ratio estimated with other drying methods.

The drying state of C-S-H also affects its density, which influences the transition from mass to volume in ‘‘EDS-based calculation’’. According to the work of Muller [33], the density of solid C-S-H ranges from 2650 to 2730 kg/m3 depending on the degree of hydration. Saturated C-S-H has a density between 1800 and 2100 kg/m3. In this study, the density of dry and saturated C-S-H is respectively 2650 kg/m3 and 1950 kg/m3 (Table 2).

C-S-H is the main component in the C₃S HCP, and fluctuations in its mechanical properties have a significant influence on the distribution of the Young's modulus in the sample. Pellenq et al. [34] calculated the Young's modulus of solid C-S-H to be 57.1 GPa for a C/S atomic ratio of 1.7 and an H/S atomic ratio of 1.6 using atomistic simulation. This result serves as the basis for many studies to estimate the mechanical properties of saturated, HD, or LD C-S-H [45, 55–57]. Self-consistent or Mori-Tanaka homogenization schemes are used to calculate the effective Young's modulus of the saturated mineral by considering the solid C-S-H and porosity as a coupled system. An alternative option is to perform nano-indentation measurements to statistically estimate the Young's modulus of HD and LD C-S-H [38, 41, 44, 47, 58]. Most of the results agree on a Young's modulus between 18 and 25 GPa for LD C-S-H and between 29 and 36 GPa for HD C-S-H. These results are related to the packaging density of HD and LD C-S-H, which are respectively equal to 64% and 74% in the scientific papers [47, 55, 59]. Even in solid C-S-H, Dongshuai et al. [56] highlighted the influence of the H/S atomic ratio on its mechanical properties.

Except for the paper by Liu et al. [60], all these authors considered HD and LD C-S-H in 1D modeling. The LD/HD C-S-H volume ratio is estimated by the Tennis and Jennings model [42] or by nano-indentation measurement. Then, a homogenization scheme is applied to estimate the mechanical properties of the HCP. Stora et al. [1] used a generalized self-consistent scheme in which the HCP is constructed according to the following pattern: an anhydrous grain embedded in a layer of HD C-S-H, which is also embedded in a layer of LD C-S-H. On the other hand, Gao et al. [58] proposed constructing homogenized C-S-H as a matrix of LD C-S-H containing inclusions of HD C-S-H. This difference in modeling choices is due to the lack of knowledge regarding the 2D distribution of HD and LD C-S-H. Garrault et al. [53] considered that LD C-S-H precipitates first as a layer around the anhydrous C₃S, and HD C-S-H appears through the diffusion of water molecules through the LD C-S-H layer. Liu et al. [60] calculated the 2D distribution of LD and HD C-S-H using high-resolution, high-speed nanoindentation mapping. This approach suggests that the pixels in the maps contain only pure phases, which is not always true (Table 4). Due to the difficulties in properly modeling the 2D distribution of HD and LD C-S-H with the "EDS based calculation," this study only considered dry and saturated C-S-H for the homogenization scheme. The Young's modulus of dry C-S-H is considered equal to that of solid C-S-H calculated by Pellenq et al. [34]. For the saturated C-S-H, the gel porosity is calculated using the H/S atomic ratio, and the Young's modulus is estimated through homogenization.

This study highlights the importance of the H/S atomic ratio in the mechanical properties of C-S-H and emphasizes the impact of the drying protocol on its composition and density. Further efforts are needed to refine the modeling of the 2D distribution of C-S-H and gain a better understanding of its behavior in cementitious materials.

The first aim of this study is to propose a new approach to estimate the 2D mineralogical distribution in the cement matrix based on a combination of EDS quantified maps and a chemical solver. The results, coupled with a homogenization scheme, allow for the calculation of the distribution of mechanical properties at the pixel size and the macroscopic mechanical properties at the REV scale. The method is divided into four steps for each pixel:

First, a mass-based oxide calculation is performed using elemental quantified maps. The water content is calculated by multiplying the water-to-binder ratio by the total mass of oxides.
A chemical calculation is then performed using the GEMS solver coupled with the thermodynamic database CEMDATA18. At the end of this step, the mineralogical composition of each pixel is known. Anhydrous species cannot be deduced from the chemical calculation, so a clustering process is required to determine pixels that are full of anhydrous species. The porosity is estimated by subtracting the mass of minerals from the initial mass.
Two homogenization schemes (Mori-Tanaka or self-consistent) are considered to estimate the homogenized mechanical properties of the pixels, depending on the difference in young modulus between the minerals considered. The self-consistent scheme is applied to the 2D mechanical distribution to estimate the mechanical properties of the REV.

The method is applied to C₃S HCP after 28 days of hydration and compared to a dedicated experimental campaign. The results at the REV scale show good accuracy compared to the hydration calculation and experimental results. The HCP is composed of approximately 40% dry C-S-H, 20% Portlandite, and 40% total porosity. The C₃S content is lower than experimentally observed, which is attributed to the difficulty of the chemical solver in estimating the C₃S content in the pixels. At the pixel scale, the 2D distribution highlights two precipitation zones, as observed by the clustering approach. The average mineralogical distribution from the clustering results confirms the mixture of phases in the clusters. The mechanical distribution is in good agreement with nano-indentation measurements, with a mean value between 20 and 26 GPa. Therefore, the numerical results closely match the experimental distribution, with a data range of just a few GPa. This indicates that the initial water access was not the same for each pixel. The homogenization at the REV scale is consistent with compressive tests, yielding a homogenized young modulus of 22.8 GPa.

This study also discusses the influence of the chemo-mechanical properties of C-S-H in the chemical solver calculation steps. The properties of this mineral vary widely depending on its drying state. In this work, two drying states are considered:

Dry C-S-H with an H/S atomic ratio of 1. According to scientific papers [33, 50, 51], C-S-H without gel porosity has an H/S atomic ratio of 1.8. In this study, the samples were freeze-dried before analysis, and it is consistent to assume that a part of the interlayer water is missing. The value of the H/S ratio for dry C-S-H is fitted to the main peak of the oxygen atomic distribution from EDS analysis. The young modulus is equal to 57 GPa, as proposed by Pellenq [34].
Saturated C-S-H with an H/S atomic ratio of 4. Due to the challenge of accurately modeling the 2D distribution of LD C-S-H and HD C-S-H, an average value of the H/S atomic ratio based on scientific references [36, 53] is chosen. The young modulus is calculated based on that of the dry C-S-H.

This paper highlights the importance of considering the drying state of C-S-H in chemical modeling. The porosity and mechanical properties are clearly influenced by the H/S atomic ratio of C-S-H.

In conclusion, the "EDS-based calculation" allows for the estimation of the macroscopic elastic mechanical properties of the cement sample studied, based on EDS results. This study focuses on C₃S HCP but can be applied to complex cement structures such as CEM I or blended cement. One limitation is related to the ability of the thermodynamic database to accurately model the chemical behavior of the minerals in question. Another limitation is the proportion of anhydrous phases, which is smaller than the pixel size. As explained before, the "EDS-based calculation" does not consider this in the calculation. This limitation could be overcome by using BSE mapping at a very small pixel size, which would allow for the estimation of the proportion of anhydrous phases in the pixels of EDS quantified maps.

The authors declare that no funding was received and they have no conflict of interest.

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Estimation of the 2D mineralogical distribution in hydrated C3S matrix through coupling of chemical solver with experimental data.

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Abstract

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1. Introduction

2. Materials and methods

3.1. Materials

3.2. Characterization Methods

3. Data Processing

4. Results

4.1. Global Chemo-mechanical properties

4.2. Clustering based on gaussian mixture method

4.3. Mineralogical distribution

4.4. Estimation of the distribution of homogenized properties

5. Discussion

6. Conclusion

Declarations

References

Supplementary Files

Status:

Version 1