Structure modeling and quantitative X-ray diffraction of C-(A)-S-H

Atomistic structure models of nanocrystalline calcium silicate hydrate [C-(A)-S-H] that account for nanostructural features such as isolated layers, turbostratic disorder and fibrils are obtained via a supercell approach using TOPAS. Quantitative X-ray diffraction of C-(A)-S-H is demonstrated for autoclaved aerated concrete and intermediates.


Introduction
Nanocrystalline calcium silicate hydrate (C-S-H) and its aluminium-substituted variants (C-A-S-H) have been intensively studied owing to their importance as the main binding phase in Portland cement concrete. They display a wide range of chemical compositions, structures and crystallinities, usually categorized in terms of their Ca/Si ratio (Richardson, 2014). C-(A)-S-H as found in Portland cement concrete marks the upper limit at Ca/Si ' 1.8 and shows moderate aluminium substitution at Al/Si ratios of 0.05 (Richardson et al., 2010). Water contents are often described in terms of H 2 O/Si ratios, which show considerable variation between analysis methods since they also account, for example, for hydroxyl ions or adsorbed water (Lothenbach & Nonat, 2015;Roosz et al., 2016). The morphologies as observed by transmission electron microscopy (TEM) range from lamellar to fibrillar and are influenced by Ca/Si ratio and chemical environment (Richardson, 2004;Richardson et al., 2010;Tajuelo Rodriguez et al., 2015). C-(A)-S-H is termed nanocrystalline in respect to its diffuse X-ray diffraction (XRD) patterns, which share some characteristics with those of crystalline calcium silicate hydrates, especially the layered mineral tobermorite (Nonat, 2004). Asymmetric hk bands can be attributed to isolated layers or, in the presence of a 001 reflection, to turbostratic disorder, which is the random translation and rotation of layers parallel to each other (Grangeon, Claret, Lerouge et al., 2013;Grangeon, Claret, Linard & Chiaberge, 2013;Grangeon et al., 2016Grangeon et al., , 2017. The most diffuse XRD patterns can be attributed to minute crystallite sizes along the a and c dimensions, which would be equivalent to fibrils (Richardson, 2014). Despite common terminology, C-(A)-S-H phases are not completely X-ray amorphous, nor do they contain amorphous and crystalline domains as suggested by the term semicrystalline.
Tobermorite [Ca 4.5 Si 6 O 14 (OH) 3 Á2H 2 O] is the main binding phase in autoclaved aerated concrete (AAC) and other hydrothermally cured building materials (Gundlach, 1973;Hamid, 1981). C-(A)-S-H, as the main intermediate phase in the production process, either originates from the hydration of Portland cement or forms under hydrothermal conditions from portlandite and silicate, e.g. quartz (Schober, 2020). The hydrothermal formation of tobermorite has been studied by both ex situ and in situ methods at temperatures of 453 K (Mitsuda, 1982), 463 K (Bernstein, 2011;Kikuma et al., 2011;Matsui et al., 2011;Schober, 2005) and 463-583 K (Shaw et al., 2000). For example, the ex situ study of Schober (2005) provided phase quantifications including C-(A)-S-H for 30 or 60 min intervals. Quantifications used an internal standard and assumed a constant Ca/Si ratio for the C-(A)-S-H phase (Schober, 2020). The more recent in situ XRD studies of Matsui et al. (2011) and Kikuma et al. (2011) presented improved data with measurement intervals of 4.15 min. However, their evaluation of phases relied entirely on normalized integrated intensities.
Direct quantification of C-(A)-S-H via Rietveld refinement would improve the significance of in situ XRD studies and advance the quality control of AAC products, where larger amounts of C-(A)-S-H are known to affect key parameters such as drying shrinkage (Gundlach, 1973). Owing to the lack of a defined crystal structure, quantitative XRD has, however, proven difficult and relies on the successful application of internal or external standard methods. For instance, Garbev et al. (2008) simulated the C-S-H XRD pattern by whole powder pattern decomposition assuming lattice constants similar to those of tobermorite, but the actual quantification relied on the addition of corundum as an internal standard. In contrast, the external standard methods of Snellings et al. (2014) and Maddalena et al. (2019) used pseudo-Voigt profiles for quantitative comparison of peak areas relative to a reference sample. Furthermore, Jansen et al. (2011) examined alite hydration by the G-factor method, which was combined with an hkl phase model for C-S-H by Bergold et al. (2013) and used for the study of AAC paste hydration by Schreiner et al. (2018).
The importance of Portland cement concrete has led to the proposal of various structure models for C-(A)-S-H, many of them derived from tobermorite or clinotobermorite. The Ca/Si ratios were usually adjusted by the omission of bridging Si tetrahedra and increased occupancy of interlayer Ca. For example, Geng et al. (2017) applied anisotropic crystallite size and strain broadening to 14 Å tobermorite, whereas Renaudin et al. (2009) and Battocchio et al. (2012) used 11 Å tobermorite; the latter also constrained certain geometries (e.g. tetrahedral geometry for silicate) and refined atom positions without loss of coordination or inappropriate modification of bond lengths. Attempts towards a more realistic description of C-(A)-S-H including the nanostructure led to large-scale atomistic structure models composed of building blocks that cover different chemical compositions or variable configurations, or even represent surfaces (Kunhi Mohamed et al., 2018). Furthermore, Richardson (2014) derived multiple hypothetical C-S-H structure models by chemical and geometrical reasoning.
The most elegant simulation of asymmetric hk bands, however, did not require any anisotropic crystallite size or strain broadening and was obtained from 11 Å tobermorite solely by considering turbostratic disorder (Grangeon, Claret, Lerouge et al., 2013;Grangeon, Claret, Linard & Chiaberge, 2013;Grangeon et al., 2016Grangeon et al., , 2017Drits & Tchoubar, 1990). The utilized mathematical formalism may not be applicable for phase quantification, but other turbostratically disordered phases such as smectite and nontronite have been described by Rietveld-compatible supercell approaches (Ufer et al., 2004;Wang et al., 2012). Therefore, the objective of this study is to develop and test a similar supercell approach for C-(A)-S-H phases using TOPAS (Coelho, 2018) and to assess its applicability in refining ex situ and in situ XRD measurements related to the hydrothermal curing of AAC.

Sample preparation
An AAC mixture was prepared similarly to industrial recipes from 55.6 wt% quartz sand (99.1 wt% SiO 2 ; D50 23 mm), 22.2 wt% ordinary Portland cement (CEM I 52.5 R), 22.2 wt% lime (93 wt% CaO), 0.1 wt% aluminium powder and water with a water-to-solid ratio of 0.72. A specimen Ca/Si ratio of 0.63 was calculated from the chemical composition of the raw materials [evaluated by X-ray fluorescence (XRF)]. Compared with industrial recipes, a lower SO 3 content of 0.56 wt% was obtained by partially replacing cement with lime and refraining from calcium sulfate addition. The mixture was homogenized using a dissolver blade stirrer for 3 min at 3000 min À1 and cast into a polytetrafluoroethylene (PTFE) mold. Foaming, subsequent setting and accelerated cement hydration took place at 340 K for 6 h. In order to produce C-(A)-S-H without much tobermorite, the mixture was briefly cured in saturated steam; heating from 373 K to ca 458 K (1.1 MPa) took 200 min, and the temperature of 458 K was maintained for 90 min. After cooling below 340 K and before opening, the autoclave was flushed with nitrogen to minimize carbonatization. Further drying took place at <310 K for 4 h.
Experimental mixtures for in situ XRD were prepared likewise from 52.8 wt% quartz sand (99.1 wt% SiO 2 ; D50 23 mm), 21.0 wt% ordinary Portland cement (CEM I 52.5 R), research papers 21.0 wt% lime (93 wt% CaO), 5.1 wt% calcium fluoride (>99 wt%; sintered at 970 K for 3 h), 0.1 wt% aluminium powder and water with a water-to-solid ratio of 0.7. In contrast to industrial recipes, calcium fluoride was added to serve as an internal standard as it was assumed to be inert. A specimen Ca/Si ratio of 0.69 was obtained, whereas the calculation without calcium fluoride would obtain Ca/Si of 0.63. The mixtures were homogenized and cast into sample trays, usually three at a time, and covered with a PTFE thread-seal tape, polyethylene (PE) foil and a planar weight. The foaming yielded an even surface of ca 40 Â 20 mm. The samples were kept at 340 K for the first 6 h and then at room temperature for at least 36 h prior to in situ measurement.

X-ray fluorescence
Dried powder samples were mixed with lithium tetraborate in a ratio of 1:8 and fused into glass beads. The loss on ignition was determined in parallel at 1243 K for 1 h. The chemical composition was determined using a Bruker S8 Tiger wavelength dispersive X-ray fluorescence spectrometer calibrated against 27 glass bead standards.

Scanning electron microscopy
For scanning electron microscopy (SEM) analysis, brokenoff pieces (<10 mm) were fixed to sample stubs using a conductive carbon tab; no surface coating or polishing was applied. Secondary electron images at a magnification of 10 000 were obtained with an acceleration voltage of 5 kV using an FEI QUANTA 250 field emission gun in high-vacuum mode. Quantitative energy-dispersive X-ray spectroscopy (EDX) was carried out on a Zeiss EVO MA 10 with an acceleration voltage of 15 kV and a beam current of 1.5 nA. Spectra were recorded at imprints of larger quartz grains at a magnification of 5000, with an acquisition time of 100 s. Those quartz grains are usually surrounded by homogeneous C-(A)-S-H or tobermorite which are exposed if grains are broken off. Uncertainties remain about the sample topography, which may cause inaccuracies due to microabsorption. The X-ray absorption of lighter elements increases disproportionately at longer absorption path lengths, e.g. underestimating oxygen and exaggerating calcium (Newbury & Ritchie, 2013). Therefore the theoretical 66.7 at.% oxygen present in mixtures of Ca(OH) 2 and SiO 2 was used as a selection criterion (Fig. S1).

X-ray diffraction
X-ray diffraction was carried out on a Bruker D8 operated in Bragg-Brentano geometry. An X-ray source (Cu 40 kV, 40 mA) with a long fine focus and a linear position-sensitive detector (LynxEye) with an opening of 2.947 2 were mounted at radii of 280 mm. The anti-scattering slit was set to 2.56 and an Ni K filter was employed. Conventional powder samples with a diameter of 25 mm were front loaded and measured under rotation by continuous scans from 5 to 70 2 with a step size of 0.01 , a measurement time per step of 0.5 or 0.95 s, and Soller slits of 2.5 on the primary and secondary paths. The variable divergence slit was set to illuminate either 13 or 22 mm on the sample. Atomic displacement parameters were investigated by measurements up to 135 2 with a reduced sample illumination of 5 mm.
In situ experiments were carried out on the same instrument equipped with an autoclave chamber (Mesecke et al., 2020). Each measurement was recorded within 14 min (or initially 11 min) by continuous scans from 14.2 to 50.8 2 (or 8.6 to 37.1 2) with a step size of 0.01 , a measurement time per step of 0.2 s, a fixed divergence slit of 0.9 and a Soller slit of 2.5 on the secondary path. Sample trays were unwrapped and inserted into the chamber. Unirradiated parts were covered with a specifically shaped PTFE foil in order to prevent condensate leaching later on. After the front face screw-on lid was sealed, the still unpressurized system was pre-heated using a circulation thermostat. At 363-373 K the chamber was flushed with steam from an external boiler to ensure saturated water vapor, after which the whole system was sealed (0 min). The temperature increased steadily at a rate of 0.51-0.54 K min À1 until constant conditions were approached (ca 200 min). The studied temperatures (457, 466 and 473 K) were maintained at AE1 K for at least 6 h. Afterwards, the chamber cooled at a rate of ca 1 K min À1 and at <330 K the chamber was opened.
All measurements were refined sequentially. In situ measurements severely affected by condensate droplets on the beryllium window were excluded. Fifth-order Chebyshev polynomials were used to refine the background. The sample displacement parameter had an overall margin of À0.5 to 0.0 mm and changed in relation to temperature by ca 0.1 mm during particular in situ experiments.

Reaction kinetics
Because there were occasional disturbances from steam and condensate, at least three in situ experiments were combined for each investigated temperature. In order to improve the accessibility of time-resolved phase quantities, the reaction kinetics were modeled by regression analysis using the leastsquares method. The equations are empirical and usually assume a first-order reaction behavior. The parameters include the initial quantity w 0 , the rate constant k, the time of the onset of reaction t 0 and the theoretical quantity after prolonged reaction w 1 .

Chemical composition
On the basis of, for example, 29 Si magic-angle spinning NMR studies, the chemistry of C-(A)-S-H can be summarized as follows. The tobermorite-like main layer comprises opposing sevenfold-coordinated Ca-O polyhedra encompassed by paired silicate tetrahedra with the formula [Ca 2 Si 2 O 7 ] 2À (Gartner et al., 2017). At low Ca/Si ratios, bridging silicate tetrahedra connect those pairs and form long chains where the negative charge is primarily balanced by protons (Table 1). With increasing Ca/Si, fewer bridging silicate tetrahedra are present and larger proportions of the charge are balanced by Ca 2+ ions (Gartner et al., 2017). The positions of the charge-balancing Ca 2+ ions and their ligands remain uncertain. For example, in tobermorite (Hamid, 1981) there is no apparent geometry, but Richardson (2014) proposed an octahedral geometry similar to that of portlandite. In that structure, both sides of the sixfold-coordinated Ca-O octahedra share one O atom with a silicate tetrahedron and two O atoms with the main-layer Ca-O polyhedra. In isolated layers, one side is instead coordinated by water or hydroxyl groups (Gartner et al., 2017). At Ca/Si ratios of 1.0, half of the bridging tetrahedra are occupied and the negative charge of the main layer is balanced by identical proportions of Ca 2+ ions and protons. At Ca/Si = 1.3, the bridging silicate occupancy decreases further and protons are replaced by Ca 2+ ions completely (Richardson, 2014). Assuming an occupancy of 20%, a neutral charge without protons or hydroxyl groups is maintained at Ca/Si = 1.364 (Table 1). Even higher Ca/Si ratios of up to $1.8 occur in saturated Ca(OH) 2 solution, such as freshly hydrated Portland cement concrete. Because of the high alkalinity, an increasing fraction of divalent Ca 2+ ions are replaced by two monovalent CaOH + ions (Table 1) (Gartner et al., 2017). For example, at Ca/Si = 1.7 the expected ratio of CaOH + to the total amount of Ca is 0.4, whereas a ratio of 0.46 was determined experimentally (Thomas et al., 2003). For hydrothermal solutions, the Ca(OH) 2 saturation concentration at 453 K is one order of magnitude lower than that at room temperature (Peppler & Wells, 1954) and its alkalinity is slightly reduced (Bates et al., 1956).
Aluminium replaces silicon predominantly at bridging sites. At moderate alkalinity its tetrahedral geometry is maintained, whereas at high alkalinity the availability of hydroxyl ligands induces an octahedral coordination (Kunhi Mohamed et al., 2020). The additional negative charge due to Al 3+ instead of Si 4+ requires additional charge-balancing ions, which are Ca 2+ or potentially alkali ions (Richardson, 2014). Since aluminium and silicon are similar in terms of their X-ray scattering ability, the addition of structural sites for Al 3+ seems unnecessary at Al/Si 0.05.
Since structural sites for coordinating water and hydroxyl groups are incorporated into the structure models, H 2 O/Si ratios at high Ca/Si account for mild drying conditions (Lothenbach & Nonat, 2015;Richardson, 2014;Roosz et al., 2016). On the other hand, the positions of adsorbed water predominant at low Ca/Si are more difficult to assume and respective H 2 O/Si ratios account for more severe drying conditions (Richardson, 2014).

The supercell approach
Nanostructures like isolated layers, turbostratically disordered layers and fibrils can be simulated by a supercell approach using TOPAS. Structural sites in TOPAScompatible syntax are most conveniently edited in spreadsheets (see supplementary material). Initially, all structural sites of tobermorite (Hamid, 1981; COD 8103550) were described individually within space group P1. Orthotobermorites were deemed unsuitable because of the implausible coordination of charge-balancing Ca ions located between layers (Richardson, 2014), but when isolated or turbostratically disordered layers are considered, this issue becomes irrelevant. In order to obtain an isolated layer, the tobermorite structure was reduced to one of its two identical  Table 1 Idealized compositional parameters for isolated C-(A)-S-H layers summarized from previous studies (Gartner et al., 2017;Richardson, 2014 main layers. All developed models are based on this further modified isolated layer. The bridging silicate tetrahedra on one side of the C-(A)-S-H layer were slightly adjusted such that they are symmetric to the other side. Structural sites for charge-balancing Ca ions and respective octahedrally coordinating sites for water or hydroxyl groups were split along a and statistically distributed on both sides of the C-(A)-S-H layer (Fig. 1). Occupancies and vectors for certain atom groups were defined in dummy sites for later adjustment of Ca/Si ratios or atom positions. The maximum bridging silicate tetrahedron occupancy is 0.25 as in tobermorite (Hamid, 1981), whereas for charge-balancing Ca ions it is 0.5 (Table 1). Atomic displacement parameters were adopted from the original structure.
Within the supercell, atom positions and interatomic distances are preserved by defining fractional coordinates independent of the lattice constants (Ufer et al., 2004). For this purpose, variables have to be assigned to the lattice constants and to their original values; the latter are replicated to a dummy site. Multiplication of fractional coordinates with the original lattice constant and division by the current lattice constant cancels out any changes to the latter, as during refinement each fractional coordinate is automatically multiplied by the current lattice constant. Elongation of lattice constants to at least twice their original size creates a partially empty supercell with new hkls simulating even severely asymmetric bands (Ufer et al., 2004). Consequently, the structure itself accounts for the anisotropic size and macros to account for the anisotropy are no longer necessary. Larger structures within the supercell are created by duplication of structural sites and increment of fractional coordinates; however this may exceed the computational limits of TOPAS 5.0. The automatic approximation of hkls using significantly fewer peaks was introduced by TOPAS 6.0 (Coelho et al., 2016).

Modification and testing
3.3.1. Single-layer case. Isolated C-(A)-S-H layers are simulated by elongation of the supercell along c. This can be understood as a two-dimensional crystal structure since the translational symmetry along one dimension is broken. A sufficient number of hkls is produced for c = 200 Å [ Fig. 2(a)], ca ten times the original lattice constant. Complete elimination of oscillations is achieved at c > 400 Å . This simplification of turbostratic disorder successfully simulates most asymmetric hk bands. A distinct 001 reflection is absent; instead a non-physical 00l band appears at <8 2 Cu K due to the diffraction contribution from adjacent supercells (Ufer et al., 2004). In real systems there would be no empty space and nearby atoms cause absorption or diffuse scattering. The redundant 00l band can be removed by scaling respective reflections to zero (see Supplementary material A).
3.3.2. Multi-layer case. C-(A)-S-H layer connectivity can occur at low Ca/Si via bridging silicate tetrahedra or at higher Ca/Si via Ca-O octahedra . Respective turbostratic disorder is implemented by duplication of the C-(A)-S-H layer subjected to stacking vectors. The 001 reflection appears [ Fig. 2(b)] and shifts as expected towards 7.4 2 Cu K with increasing stack size (Grangeon, Claret, Linard & Chiaberge, 2013;Grangeon et al., 2016). The nonphysical diffraction contribution at low 2 is suppressed, but removal of redundant reflections was still applied. Since turbostratic disorder involves random translations and rotations (Ufer et al., 2004), the disorder obtained from a single set of stacking vectors is insufficient and causes aberrations at >20 2 Cu K. Layer rotation may not be implementable, but a combination of ca ten multi-layer models with random stacking vectors can average out most aberrations.
3.3.3. Fibrils. Fibrils extending in the b direction are simulated by elongation of the lattice constants a and c. This can be understood as a one-dimensional crystal structure since the translational symmetry along two dimensions is broken. The fibril cross section comprises several original unit cells along a, accomplished by duplication of structural sites with an increment of x, which relates to the fibril width. The crystallite size parameter can then be interpreted as the fibril length. Small cross sections only produce a broad main band at ca 29 2 and a 020 reflection at 49.5 2 Cu K [ Fig. 2(c)]. Increased fibril width increases the translational symmetry along a until the XRD patterns resemble the single-layer case. The nonphysical diffraction contribution at low 2 is removed by scaling all h0l reflections to zero, which would be equivalent to h < 1 and l < 4 in the original structure (see Supplementary material A).  than paired silicate tetrahedra. In such a scenario the occupancy of central bridging silicate sites and the mean chain length of silicate tetrahedra could vary without impact on Ca/ Si ratios.

Implementation
The developed structure models facilitate the evaluation of experimental results. Layered and fibrillar nanostructures are distinguishable by the shape of the main band at 29.5 , by distinct reflections at 32 and 55 2 Cu K, or by the additional presence of a 001 reflection. Considering reported XRD patterns, many laboratory preparations with Ca/Si < 1.5 relate to layers (Black et al., 2006;Garbev et al., 2008;Maddalena et al., 2019;Roosz et al., 2016), whereas the XRD patterns of specimens with Ca/Si > 1.5 relate to fibrillar structures (Bergold et al., 2013;Snellings et al., 2014). Further improvements and specimen-specific modifications will be necessary to account for the whole range of C-(A)-S-H phases. Since even diffuse bands at 42-45 2 are precisely simulated, knowledge about the Ca/Si ratio or related occupancies would allow indirect modification of atom groups via refinement of associated vectors. For instance, the positions of charge-balancing Ca 2+ ions may shift according to Ca/Si with influence on the main band. Although, Grangeon et al. (2016) linked the intensity at 16.1 2 to the occupancy of bridging silicate tetrahedra, Ca/Si ratios should be distinguished by complementary methods as their determination via XRD would be ambiguous [ Fig. 2(d)].  Calculated XRD patterns for C-(A)-S-H featuring isolated layers (a), turbostratically disordered layers (11.5 Å layer-to-layer distance) (b) and fibrils (c), (d). The simulations account for a variable divergence slit, an isotropic crystallite size of 30 nm and b = 3.675 Å . The Ca/Si ratio was kept at 1.0 (a)-(c). A non-physical diffraction contribution at low 2 was removed (b)-(d).

Application
this stage (Fig. 4). These observations are in agreement with previous studies (Bernstein, 2011;Mitsuda et al., 1992) and could be explained by initial restriction of silicate ions to the vicinity of slowly dissolving quartz grains in an otherwise Ca(OH) 2 -saturated hydrothermal solution (Qu et al., 2018). In contrast, comparable spots in thoroughly cured AAC [Figs. 3(b) and (d)] contain typical lath-like tobermorite (Schreiner et al., 2019).
4.1.2. Elemental composition. Two samples were analyzed by SEM-EDX: one was prepared with calcium fluoride and did not yield any tobermorite (Table S1), whereas the other did yield small amounts of tobermorite ( Table 2). The obtained Ca/Si and Al/Si ratios are similar to results reported by Mitsuda et al. (1992). The hydrogen and water contents were calculated from oxygen that was not attributable to other quantified oxides. Since this calculation neglects unquantified carbonate, the H 2 O/Si ratios are slightly higher than for other mildly dried samples (Richardson, 2014). As the hydrothermal reaction proceeds and the quantity of tobermorite increases, the study of C-(A)-S-H by SEM-EDX is hindered and the elemental composition would have to be determined at higher magnifications by TEM-EDX (Mitsuda et al., 1992;Richardson, 2014).
4.1.3. Refinement. The observed XRD patterns were best fitted using the 4.5 nm-width fibrillar model (Fig. 4). The small reflection at 7.9 2 was attributable to traces of tobermorite, and the refinement range was reduced to 14-62 2. Furthermore, reduction of the supercell lattice constants to a = 200 Å and c = 100 Å was necessary to comply with computational limitations of TOPAS 5.0. Prior to refinement, the Ca/Si ratio was set to 1.3 according to derived parameters (Table 1). The crystallite size was reduced to 20 nm in order to match the shape of the 020 reflection at 49.5 2. Stepwise refinement, no matter in which order, led to a main-layer compression of 0.5 Å , an x adjustment for charge-balancing Ca 2+ ions and a z adjustment for its coordinating water. In tobermorite (Hamid, 1981) just half of the main-layer Ca-O polyhedra contain a planar square of O atoms; the other half exhibit a slightly pyramidal geometry. By reducing the fractional coordinate z of the latter, the Ca-O polyhedra can be made to share a uniform geometry.
4.1.4. Phase quantification. The direct quantification of C-(A)-S-H was assessed by the internal standard method (Fig. 5). Besides quartz and C-(A)-S-H the original sample contains traces of katoite, calcite, larnite, portlandite and tobermorite (Table S2). Here, fifth-order Chebyshev polynomials were chosen; lower orders would slightly exaggerate the C-(A)-S-H quantities and specimen Ca/Si ratios. For quartz, no bias in relation to corundum addition is observed and variations seem to be random, but for C-(A)-S-H there is a small bias. At low corundum additions the trend suggests less then quantified, whereas at high additions the trend suggests more. In conclusion, the direct quantification of C-(A)-S-H via Rietveld refinement yields plausible results, but compared   Table 2 Elemental composition of C-(A)-S-H analyzed by SEM-EDX for a sample after 1.5 h at 458 K (290 min).
Hydrogen quantification (*) assumes excess oxygen to be present as water. The results of 17 spectra were averaged.

Figure 4
Rietveld refinement of a sample after 1.5 h at 458 K (290 min) with 20 wt% subsequently added corundum.
with highly crystalline phases a higher detection limit is expected and interference with background polynomials may occur.

Analysis of hydrothermal reactions
4.2.1. Formation of C-(A)-S-H and tobermorite. As indicated by the results of in situ XRD (Fig. 6), the hydrothermal reaction of quartz, portlandite and Portland cement can be roughly divided into two stages. The first stage yields C-(A)-S-H and here starts around 425 K (100 min) (Table 3). This C-(A)-S-H and that from Portland cement hydration were fitted using the refined fibrillar structure model. The Ca/ Si ratios of the structure model have a negligible effect on quantitative results and will be discussed later on. For portlandite dissolution, an exponential equation was chosen (Shaw et al., 2000), which adequately accounts for the temporarily constant reaction rate (100-200 min) and the slowed dissolution of larger remnant crystallites. The dissolution of moderately reactive quartz can be considered as the rate-limiting partial reaction . The second stage is marked by tobermorite formation and conversion of C-(A)-S-H; in the corresponding kinetic models, the increase in tobermorite is balanced by a decrease in C-(A)-S-H content. At 466 and 473 K (ca 260 min), tobermorite formation started soon after complete portlandite consumption, whereas at 457 K it was significantly delayed. Since the determined reaction rates do not reflect this temperature correlation, the onset of tobermorite formation is seen as the most useful parameter for the comparison of experiments. The surface sensitivity of in situ XRD measurements using Bragg-Brentano geometry can be problematic (Mesecke et al., 2020). The phase quantities before hydrothermal reaction were calculated from the raw materials, assuming cement hydration yielded 25 wt% portlandite (Taylor, 1997), and afterwards control measurements were performed on homogenized samples (Table 4). From the resulting data it can be concluded that the surface composition differed from the bulk, and in situ measurements observe ca 10 wt% more C-(A)-S-H and 10 wt% less quartz. Hence the specimen Ca/Si ratios (Fig. 7) are higher than expected.
4.2.2. Evolution of Ca/Si ratios. The calculated specimen Ca/Si ratios reflect the varying Ca/Si ratios of C-(A)-S-H (Fig. 7). Assumption of constant Ca/Si ratios would imply an implausible increase of specimen Ca/Si ratios (300-400 min). Initially, Ca/Si = 1.7 can be expected for C-(A)-S-H from Portland cement hydration (Taylor, 1997).  Quantitative XRD of a sample after 1.5 h at ca 458 K (290 min) with various amounts of subsequently added corundum. The solid line accounts for 4 wt% amorphous material (e.g. adsorbed water). Specimen Ca/Si ratios were calculated on the basis of a Ca/Si ratio of 1.3 for C-(A)-S-H.

Figure 6
Time-resolved phase quantities combining five in situ XRD experiments at 457 K, three at 466 K and three at 473 K. Reaction kinetics are empirically modeled (solid lines) or assumed constant within certain intervals (dashed lines).
formed during the first stage, Ca/Si ratios of 1.41, 1.23 and 1.37 are indicated by the ratio of constant reaction rates for portlandite and quartz dissolution (100-200 min) divided by the respective molar masses. When the Ca(OH) 2 concentration and alkalinity in hydrothermal solution decreases (200-300 min) (Mitsuda, 1982), the Ca/Si ratios of the initial C-(A)-S-H are also expected to decrease (Baston et al., 2012;Lothenbach & Nonat, 2015). Intermediate Ca/Si ratios of 1.31-1.35 were determined by SEM-EDX. An eventual Ca/Si ratio of 1.1, as observed by Mitsuda et al. (1992) using TEM-EDX, would render the specimen Ca/Si ratios of subsequent XRD measurements consistent with expectations (Table 4).

Other phases.
Additionally, in situ experiments observed katoite (Fig. S2), larnite (Fig. S3) and calcite (Fig. S4). Although calcium sulfates and ellestadite-(OH) are common for AAC, none were observed in these sulfatereduced samples. Katoite appeared at 373 K (0 min) alongside the decay of calcium aluminium oxide hydrates with reflections at ca 11 2 Cu K (Jensen et al., 2005). Katoite dissolved slowly during both stages of hydrothermal reaction, and aluminate was presumably incorporated into the formed C-(A)-S-H and tobermorite. Unhydrated larnite from the cement, initially present at ca 2 wt%, dissolved alongside the near-complete portlandite consumption. The calcite quantities increased due to surface carbonatization from ca 2 wt% initially to occasionally 5 wt%. Fluorite (Fig. S5) is not as inert as expected and cannot serve as an internal standard. SEM-EDX confirms calcium fluoride uptake by C-(A)-S-H (Table S1) and the decrease corresponding to near-complete portlandite consumption was also reflected by XRD measurements of subsequently homogenized samples (Table 4).

Analysis of industrially produced AAC
The feasibility of direct C-(A)-S-H quantification is further demonstrated on 11 AAC products (Fig. 8) (Table S3). XRD patterns were best fitted using the 8.9 nm-width fibrillar model, which is attributable to a slightly more lamellar morphology. There is a trend to more C-(A)-S-H at higher specimen Ca/Si ratios, whereas quartz quantities decrease linearly up to Ca/Si = 0.75 (Fig. 8, dashed line). At the time of analysis, the products were 1-6 months old and already contained some vaterite. Since C-(A)-S-H is susceptible to  Table 4 Phase quantities (wt%) before in situ experiments as calculated from the raw materials and afterwards as determined by XRD measurements of homogenized samples.
Specimen Ca/Si ratios were calculated on the basis of a Ca/Si ratio of 1.1 for C-(A)-S-H.