Rietveld Refinement of the Structures of 1.0 C-S-H and 1.5 C-S-H

Cement and Concrete Research 42 (2012) 1534–1548

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Cement and Concrete Research

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Rietveld reﬁnement of the structures of 1.0 C-S-H and 1.5 C-S-H

Francesco Battocchio a, Paulo J.M. Monteiro b,⁎, Hans-Rudolf Wenk a a Department of Earth and Planetary Sciences, University of California, Berkeley, CA 94720, USA b Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, USA article info abstract

Article history: Low-Q region Rietveld analyses were performed on C-S-H synchrotron XRD patterns, using the software MAUD. Received 1 November 2010 Two different crystal structures of tobermorite 11 Å were used as a starting model: monoclinic ordered Merlino Accepted 26 July 2012 tobermorite, and orthorhombic disordered Hamid tobermorite. Structural modifications were required to adapt the structures to the chemical composition and the different interlayer spacing of the C-S-H samples. Refinement Keywords: of atomic positions was done by using special constraints called fragments that maintain interatomic distances and Calcium-silicate-hydrate (C-S-H) (B) orientations within atomic polyhedra. Anisotropic crystallite size refinement showed that C-S-H has a nanocrys- X-ray diffraction (B) Crystal size (B) talline disordered structure with a preferred direction of elongation of the nanocrystallites in the plane of the Ca Crystal structure (B) interlayer. The quality of the fit showed that the monoclinic structure gives a more adequate representation of C-S-H, whereas the disordered orthorhombic structure can be considered a more realistic model if the lack of long-range order of the silica chain along the c-direction is assumed. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction different from these crystal structures, basically for the following reasons: Calcium silicate hydrate (C-S-H), the main binding phase in Portland cement matrix, constitutes up to 70% in weight of hardened • The Ca/Si ratio of about 1.75 is higher than that of jennite and much ordinary cement pastes. Despite the large number of studies and the higher than that of 14 Å tobermorite. vast amount of literature available on cementitious materials, the • Both tobermorite and jennite have long tetrahedral chains, whereas atomic scale structure of C-S-H is still partly unknown owing to its in C-S-H the chains have lengths of 2, 5, 8, … (3n−1) tetrahedra. high complexity. The optimization of strength and durability of cement This pattern results from the repetition of two paired tetrahedra can be obtained through adjustments of the structure of C-S-H at a connected by the bridging tetrahedron. A particular case occurs for nanometric level [1,2], but this is subject to a detailed knowledge of n=1, when only two bridging tetrahedra are present and as will be the C-S-H crystal structure. discussed later, they are referred to as dimer. Several models have been proposed for structure of C-S-H. It is • The average chain length of C-S-H increases with age: 29Si NMR recognized that it has a multilayer structure composed of calcium experiments on C3S paste cured at 25 °C show that the mean chain layers and interrupted tetrahedral chains on both sides. Various studies length after 1 day is 2.1 tetrahedra, 2.6 after 1 month, 3.3 after have indicated structural relationships to tobermorite 14 Å (Ca5Si6O17 1 year, and 4.8 after 26 years [7]. However, when the Ca/Si ratio is 9H2O–C5S6H9, plombierite) [3,4] and jennite (Ca9Si6O21 10H2O– low, dimer chains (length=2) occur even in mature paste. C9S6H10) [5,6]. Both structures contain linear silicate chains of the “dreierkette” form in which the silicate tetrahedra are arranged in In 1986, Taylor proposed a model [3] for C-S-H that consisted of a such a way as to repeat a kinked pattern after every three tetrahedra. disordered layer structure, whereby the majority of the layers were Two of the three tetrahedra share O–O edges with the central Ca–O structurally similar to those of jennite and others were related to part of the layer; these are linked together and are often referred to as 14 Å tobermorite. In both types of layer, the structures were modiﬁed ‘paired’ tetrahedra (P) (Fig. 1, I). The third tetrahedron, which shares by the omission of silicate tetrahedra. This is an effective solution because an oxygen atom at the pyramidal apex of a Ca polyhedron, connects it is possible to obtain the expected chain length and correct the Ca/Si the two paired tetrahedra and is called “bridging” (B) [2].However, ratio of C-S-H. The tobermorite-type structure was found to be more C-S-H formed by hydration of portland cement paste, is signiﬁcantly suitable to describe the lower Ca/Si ratio C-S-H, whereas jennite-type structure was more suitable for the high Ca/Si ratio C-S-H. In 1992, Richardson and Groves proposed a generalized model for the nanostructure of C-S-H [8] that accounted for the chemical differences be- ⁎ Corresponding author. tween C-S-H and the structures of tobermorite and jennite. The model in- E-mail address: [email protected] (P.J.M. Monteiro). cluded the chemical neutrality of the structure by the protonation of

O2− anions, resulting from deleting bridging tetrahedra. This led to silanol groups in place of the oxygen atoms shared by paired and bridging tetrahedra. In addition, different amounts of calcium polyhedra were located in place of the deleted bridging tetrahedra to respect the real Ca/Si ratio of C-S-H under consideration. Later on, Richardson suggested detailed structural representations of C-S-H, derived from crystal structure data of tobermorite, for three different levels of proton- ationforCa/Siratiosof1.0,1.25and1.5[8,9]. Recently, new understanding of the C-S-H structure has been I) achieved using a variety of methods: X-ray diffraction [10–13], total scattering methods using the pair distribution function (PDF) [14], and molecular dynamics (MD) [15,16]. The analysis of Skinner's work [14] had been performed on a sample of C-S-H 1.0 aged 4 months, which revealed that the nanostructure of synthetic calcium hydrate re- sembled the crystal structure of natural tobermorite 11 Å, as refined by Merlino et al. [17]. This conclusion was obtained by simulating the diffuse X-ray scattering due to the nanostructured features of the material by a Gaussian shape broadening of the structure factor of tobermorite 11 Å, tobermorite 14 Å, and jennite, that were supposed to resemble the nanostructure of C-S-H in real portland cement concrete. Their results show that the structure of tobermorite 11 Å is strikingly similar to that of C-S-H (I). The loss of coherent scattering on the C-S-H sample above about 3.5 nm might reflect the maximum crystallite size of the material. In addition, a Monte Carlo refinement of tobermorite 11 Å was done to obtain the best fit of the PDF to the synthetic C-S-H sample. After the refinement it is still possible to see the distinct multi- layer structure made by a stacking of calcium layers with tetrahedral chains on both sides, and calcium ions in addition to water molecules II) in the interlayer space. X-ray patterns can be analyzed in two ways: One is to determine the distribution of atom pairs. This method is mainly applied to amorphous materials with no long-range order or highly disordered structures [18]. The second method is based on the structure factor, which relates the diffraction pattern to lattice planes in the long-range ordered crystal [19]. Interestingly C-S-H is in between. In this study we will use the standard crystallographic Rietveld method [20] to further quantify the nanostructure of this material. Two different samples of C-S-H aged 4 months are used. The first is the same used by Skinner et al. [14] for C-S-H with a Ca/Si ratio of 1.0, and the second refers to C-S-H with Ca/Si of 1.5. Two crystal structures for C-S-H were used as a starting model in the Rietveld analysis: (a) the monoclinic tobermorite 11 Å refined by Merlino et al. [17] and successfully applied in Skinner's work [14], characterized by high structural order, and (b) the orthorhombic version of the tobermorite structure refined by Hamid [21] that, due to the smaller unit cell, presents a lower degree of order and therefore is more likely to capture the real structure of C-S-H. The detailed description of the two tobermorite crystal structures is out of the scope of this presentation and we refer to the papers by Merlino et al. [17] and Hamid [21] and references within them for further details. However, for clarification, we note that tobermorite is a mineral that occurs with a substantial degree of disorder in its X-ray diffraction pattern that is expressed by diffuse reflections along the c-axis. Hamid attributed this feature to the different positions and orientations that the silicate chains can take. In particular, it was pro- III) posed that a chain can be statistically displaced by b/2 and that each tetrahedron of the chain can be statistically tilted in two alternative Fig. 1. I) Schematic diagram showing dreierkette chains present in tobermorite: central positions. The structure can be represented in an orthorhombic calcium layer (light blue polyhedra) and dreierkette silicon chains (dark blue tetrahedra). subcell with space group Imm2. Merlino refined two polytypes for Thechainshaveakinkedpatternwhere some silicate tetrahedra share O–O edges with the the ordered tobermorite structure: one orthorhombic with space central Ca–O layer (called ‘paired’ tetrahedra (P)), and others that do not (called ‘bridging’ group F2dd, and the other monoclinic symmetry with space group tetrahedra (B). II) Diagrams illustrating tobermorite-based dimer that has the maximum degree of protonation of the silicate chains (Ca/Si=1.0). (a) A highly schematic diagram B11m. demonstrating chemical accounting of the model. (b) and (c) show more realistic structural This investigation applies the Rietveld method to the refinement representations derived using crystal structure data for tobermorite with the silicate chains of the atomic structures and nanostructures of C-S-H by using the either aligned along the plane of the page or perpendicular to the page. III) Diagrams illustrat- software “MAUD” [22], which is particularly suitable to implement a ing the C-S-H model for Ca/Si=1.5 (a), (b) and (c) have the same meaning than in II [9]. new approach for the analysis of nanostructured materials. 1536 F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548

2. Experimental methods from the broadening attributable to the thermal vibrations. In the atomic structures water molecules are represented by the ion O2−and labeled 2.1. Materials O(W), similarly groups \OH are represented by the ion O− and labeled O(H).

C-S-H samples were synthesized using CaO and amorphous SiO2 (Carbosil) mixed at a water-solid ratio of about 12.9. The CaO reagent 3. Monoclinic tobermorite refinement was prepared by heating pure CaCO3 (Fisher) at 900 °C for 24 h. Two samples of C-S-H with different stoichiometric calcium/silicon 3.1. C-S-H 1.0 ratios (1.0 and 1.5) were prepared. The samples were cured for four months. 3.1.1. Refinement procedure The experimental spectrum of C-S-H and the calculated spectrum 2.2. Experiments at APS and Rietveld refinement of tobermorite 11 Å at the beginning of the analysis are shown in Fig. 2a. Only a manual correction of the first background parameter X-ray diffraction measurements were made at the Advanced Photon and the scale factor was done. As a starting point, the crystallite size Source, beamline BESSRC 11‐ID‐C at Argonne National Laboratory. The was set to 100 nm. Note that after this rough correction, the calculated energy of the monochromatic incident beam was 114.95 keV, corre- spectrum of tobermorite 11 Å fits reasonably well the experimental sponding to a wavelength of 0.1079 Å. Details of the experiments are spectrum of C-S-H, providing us with a sound basis upon which to im- described elsewhere [14]. prove the refinement. This result is in good agreement with the work Before collecting diffraction images of the C-S-H samples, a calibra- of Skinner et al. [14],eventhoughitwasobtainedinadifferentway. tion was performed at the same conditions in order to refine detector From this point, the best least squares fit of the experimental spectrum distance and orientation. Using CeO2 by NIST as standard, this calibration with the calculated spectrum was obtainedbyaddingparametersrefined was performed with the MAUD software. Two Caglioti coefficients and in this order: one Gaussianity coefficient were refined to determine a Pseudo‐Voigt 1. Background coefficients, image center, tilting angles and scale factor instrumental function. These parameters were then used and kept con- 2. Isotropic grain size stant for the analyses of the C-S-H samples. 3. Cell parameters. Subsequently the diffraction spectrum of the sample, obtained by azimuthally averaging diffraction rings in Fit2D [23] was entered into Comparison of calculated and experimental spectrum after refine- MAUD as well as the starting structure of tobermorite 11 Å [17]. The ment (Fig. 2b) shows a definite improvement of the analysis; however, crystal structure has a monoclinic unit cell with space group B11m the quality of the fit of the main peaks is still insufficient to get reliable (monoclinic, first setting was used). Since the quality of the data was information about the nanostructure and atomic structure of C-S-H. A not sufficient to refine temperature factors, a single isotropic B value much better fit of the peak shape is necessary to determine the dimen- was used for silicon (0.6), for calcium (0.8), for oxygen (1.2) and for sion of the grains, and a good fit of the peak positions and their intensity water (5), in accordance with values observed in hydrous silicates. B are indispensable to obtain information about the cell parameters and values were kept intentionally lower compared to values that usually the structure factor. The classic Rietveld method refines a model func- occur in the refinement of calcium silicate hydrate structure, in order to tion to fit the observed diffraction spectrum. The model function is de- separate the peak broadening related to the dimension of nanocrystallites fined by parameters describing the instrument, background, crystal

Fig. 2. (a) Plots of the experimental spectra of C-S-H 1.0 (crosses) and calculated spectra (continuous line) of tobermorite 11 Å at the beginning of the analysis after a rough correction of background and instrumental intensity (scale factor), in addition to setting of isotropic grain size to 100 nm; and (b) after calibration of image center coordinates and tilting angles together with reﬁnement of background, instrumental intensity (scale factor), isotropic grain size, and cell parameters. The diffraction image is inserted in (a). F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548 1537

and oxygen atoms form tetrahedra, with silicon in the center and four oxygen atoms in the vertices. Therefore, it is necessary to preserve these geometric characteristics during the refinement process. Based on the atomic coordinates presented by Merlino et al. for tobermorite 11 Å [17], three significant tetrahedra were identified, Si1, Si2 and Si3 (Fig. 3). The squares symbolize the atomic positions reported by Merlino that could be refined, while the remaining vertices represent the atomic positions generated from the previous iterations by the symmetry operations of the space group B11m.Constraints were applied only to the Si atoms in the tetrahedra. Coordinates of the calcium and water atoms Ca1, Ca3 and O(W)6 were refined. For the calcium atoms and water molecules in the interlayer Ca2, O(W)1, O(W)2 and O(W)3, however, a different approach was taken. In order to keep these atoms inside the interlayer, even after the structure had been refined, it was decided to refine the fractional atomic coordinates x and y,whilez was fixed equal to zero, i.e. these atoms were confined to the x–y plane. The polyhedra used to define the constraints in MAUD are called fragments. The software permits to fix their shape and allows refining as follows:

• Coordinates X, Y, Z of the center of the local Cartesian frame placed in the fragment, expressed on the absolute reference frame of the monoclinic cell; Fig. 3. Schematic drawing of the atomic structure of Tobermorite 11 Å using crystal • Orientation of the fragment through three Euler angles roll, pitch, system, space group, and atomic coordinates proposed by Merlino [2] (square spots), and the definition of meaningful polyhedra. Polyhedra vertices without spot are generated and yaw; and after application of the symmetry operation of the space group B11m (Table 1). Here, the • Expansion or the shrinkage through the definition of a bond length silicon tetrahedra are dark gray, the irregular calcium polyhedra are light gray, and the parameter. monoclinic cell edges a, b,andc are the continuous lines. The orthogonal reference frame (x,y,z)hasthex axis aligned with a and the z axis aligned with c. The process of defining a fragment in MAUD (Fig. 4a) requires first introducing the fractional coordinates for every atom within the polyhedron, which is defined in the local frame of the fragment. These structure and microstructure. C-S-H has a large unit cell with many coordinates give the shape of the fragment as well as the value of the atoms. Because of the nanocrystalline microstructure, diffraction peaks bond length. Second, the position of the center of the fragments, previ- are broad and each peak is composed of many individual reflections ously defined in the local coordinate system, has to be defined in the ab- from lattice planes hkl (linesatbottomofspectrum).Itisnotpossible solute frame of the unit cell, whose origin is located at the intersection of to refine atomic coordinates of all atoms as would be done in a conven- the axes b and c of the monoclinic cell. Note that this absolute frame is tional analysis. We will have to introduce rigid components or “frag- orthonormal, and that its axes are not aligned with the axes a, b and c ments” to combine atoms into larger units. of the monoclinic cell. Thus, in order to locate the fragments and their The analysis demands that any constraints that are introduced will atoms in correct positions, it is mandatory to convert the coordinates not only lead to a realistic structure, but also avoid the loss of from the reference system of the monoclinic cell to the orthonormal interatomic distances as described by the PDF. For instance, it is well one. Finally, the orientation of the fragment is defined by assigning known that due to the covalent character of the atomic bond, silicon three Euler angles.

Fig. 4. (a) An example of the construction of a fragment using MAUD. 1. Definition of the atom coordinates in the local frame. 2. Definition of the position of the local frame center in the crystalline cell reference frame, and definition of the rotational Eulerian starting angles (roll, pitch and yaw, in degrees). 3. Definition of the normalized length parameter. (b) Local frame used to define the atomic positions in the fragment linked to Silicon 1. 1538 F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548

This method is amenable to allocating the fragments in rough posi- of C-S-H. The Ca/Si ratio of the sample was 1.0, while the Ca/Si ratios tions and then refining them later. It assumes that the shape of fragment of tobermorite 11 Å range from 0.67 to 0.83. Clearly, even if a very is known before the refinement. In practice, this is accomplished using a good fit was obtained, the crystalline structure of tobermorite 11 Å perfect polyhedron where the vertex coordinates are known and the would not be suitable to represent the sample because of this chemical distances between vertices and center are constant (which is the bond inconsistency. Richardson [9] suggested the modified tobermorite length parameter in MAUD). But this process was not appropriate in structure shown in Fig. 1-II, in which the bridging tetrahedra were de- this study for the following reasons: leted as well as the calcium atom in the interlayer, Ca2,andwherethe two bridging oxygen atoms O1 and O7 were replaced by two O–H • The starting point was not an empty structure, but rather a well- groups, O(H)1 and O(H)7, in order to maintain the neutrality of the defined structure, at an advanced stage of refinement (Fig. 2) charge (Fig. 5b). The y fractional atomic coordinates of the atoms • The introduction of the fragments implies substitution of the atomic belonging to the silicon 1 tetrahedron were shifted all by the cell param- positions already refined with the atomic position defined by the eter b. Although this did not lead to an improvement of the analysis— fragments because the crystal lattice remained the same—it did provide a better • In order to replace the atoms in their perfect positions it is necessary to visualization of the new features of the atomic structure. Now it is know the exact values of the center of the fragments, their orientations, clear that there are only two tetrahedra inside the monoclinic cell, and their shape at the last stage of refinement. around the atoms Si1 and Si3, and they are joined by the O2 oxygen This problem was solved by drawing the monoclinic cell with and atom (Fig. 5c). the atomic positions connected to the polyhedra using Rhinoceros,a At this point it was decided to perform a fundamental modification 3-D design software [24]. At this point, the atomic positions of all of the definition of the fragments representing the two tetrahedra. the silicon and oxygen atoms inside the tetrahedra could be extrapolated, The atomic positions of all atoms belonging to each fragment were defined in the local frame of the fragments, and the center of the local redefined in a local Cartesian frame whose origin was located in the po- frames of every tetrahedron expressed on the absolute reference frame sition occupied by the shared oxygen atom O2, rather than in the center of the monoclinic cell. For this method no knowledge of the starting of the tetrahedra, occupied by the silicon atom. This modification made Eulerian angles is needed, as the fragments are already built in their it possible to constrain the O2 atom to be bonded to both fragments. exact orientation. It was decided to set them to a default value very With this new setting, the fragments are forced to move together close but different from zero to avoid singularities when the code is through the refinement of the X, Y, Z coordinates of the new common running (Fig. 4a). origin, namely the O2 atom, defined again on the absolute frame of Tetrahedra defined as fragments may not be perfectly regular (i.e. the the monoclinic cell. No restriction is made about the rotation, which re- edges are slightly different). Thus the method applied here allows for mains independent between the two tetrahedra. In accordance with the defining arbitrary polyhedral shapes. Using fragments reduces the com- nomenclature established by Richardson [9], the new organization of putational effort because the number of refinable parameters is lowered. the tetrahedra and the new crystal structure, from now on will be called Without using fragments it would be necessary to refine three atomic respectively dimer tetrahedra and tobermorite-based dimer. coordinates (x, y, z) for every atom of the asymmetric unit. Thus, Finally, the atomic structure was modified to increase the interlayer for example, for four oxygen atoms coordinated in a tetrahedron, spacing to 12.5 Å, based on Taylor's [5] results for C-S-H I. This was there would be 4×3=12 parameters to refine. Using fragments, the re- accomplishedfirst by increasing the c-cell parameter of the monoclinic quired parameters are three for the absolute coordinates of the center of cell to 12.5 Å. Then, the z fractional atomic coordinates of all the atoms the fragment, three for the Euler angles, and one for the bond distance, outside the interlayer were modified between atoms inside the tetrahe- resulting in a total of 7 parameters for every fragment used. dra and between the tetrahedra and the calcium layer, to preserve their Fig. 4b shows one of the tetrahedra drawn in Rhinoceros and defined inter-atomic distances. This last modification was not applied to atoms by a fragment inside the unit cell. The center of the local frame is placed in the interlayer because in space group B11m they are in special posi- in the atom, labeled silicon 1 (Si1), with fragment coordinates (0, 0, 0); tions and not subject to repetitions of the space group listed in the value of the bond length is a value between the maximum of all the Table 1. The new modified structure was called tobermorite 12.5 Å coordinates x, y, z of the vertices of the tetrahedron in the local frame. based dimer. Dividing the coordinates x, y, z by the bond length provides the fractional A comparison of the structure before and after the modifications is coordinates of the vertices of the fragment (to be fixed later by the soft- shown in Fig. 6. Both structures are drawn using the same symmetry ware). In this sense, the bond length parameter does not represent so operations of the space group B11m (Table 1). Note that the structure much a length as a parameter used for normalization. From now on, it of tobermorite 11 Å (on the left) reflects all the features described by will be called the normalization length. Again, the variation of its value Merlino et al. [17]. during the refinement process represents the expanding/shrinking The modifications made to obtain the tobermorite 12.5 Å based dimer dynamics of the polyhedron. starting from tobermorite 11 Å Merlino are summarized as follows: Although this way to define fragments using MAUD was found to • the monoclinic cell is expanded along the c axis from 11 Å to 12.5 Å be a very powerful tool in the refinement of the atomic structure, it while the other cell parameters are unchanged. was not sufficient in regard to the structure of C-S-H. The tetrahedra • the z atomic coordinates of the atoms outside the interlayer are in tobermorite are bonded in chains with a peculiar spatial configura- modified in order to keep the interatomic distances unchanged. tion called dreierkette, where two kinds of tetrahedra with different • the bridging tetrahedron and the Ca2 atom are deleted in order to spatial orientation are distinguished: two paired tetrahedra and one obtain Ca/Si=1.0 . bridging tetrahedron [9] (Fig. 1-I). Fig. 5a shows the portion of chain • the silicon 1 tetrahedron is rigidly shifted by b in the y direction in inside the monoclinic cell of tobermorite 11 Å, made up of three tetrahe- order to highlight the feature of the dimer. dra: every tetrahedron shares two oxygen atoms with the adjacent ones; and even after refinement these atoms must be shared. In this figure, as The new structure was entered into MAUD, and a basic refinement well as in the figures that follow, the dimension of the spheres, which of crystallite size and cell parameters was done. The results demon- represent the different atoms, is based on the atomic radius of the strate an impressive improvement in the fit(Fig. 7a) compared to the different species. same stage of refinement with tobermorite 11 Å (Fig. 2b), confirming Unfortunately, the basic MAUD software does not have an option that the new structure is closer to the real structure of C-S-H, even for bonding the different fragments in their particular configuration. before refining the atomic structure. These results also validate A solution was found by considering first the chemical composition Richardson's [9] and Taylor's [5] results on C-S-H. F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548 1539

Fig. 5. Drawing of dreierkette chain within the monoclinic cell of tobermorite 11 Å. Tetrahedra that share a O–O edge with the central Ca–O layer octahedra are called “paired” tetrahedra (P), and the others that do not are called “bridging” tetrahedra (B) [3]. Oxygen atoms that are shared between two adjoining tetrahedra are shown in black (a). Transformation of the atomic structure in agreement with Richardson [3]: deleting of the bridging tetrahedra Si2 and protonation of the not shared oxygen atoms O1 and O7 (b), translation of Si1 tetrahedra of an amount equal to b for a better understanding of the paired appearance of the atomic structure (c).

The refinement of the atomic structure (Fig. 7b) was performed incorporating into MAUD a simple repulsion force field between the atoms to prevent them from overlapping. The overall refinement proceeded as follows: Table 1 General site positions of the atoms in the monoclinic cell for the space group B11m. x, y 1. The atomic coordinates x, y, z of the atoms Ca1, Ca3 and O(W)6,as and z are the fractional atomic coordinates of the structure. well as the coordinates X, Y, Z of the center of the fragments and Operation number Translation along a Translation along b Translation along c their Eulerian angles were refined. For atoms O(W)1, O(W)2 and 1+x +y +z O(W)3 in the interlayer, only the coordinates x and y were refined 2+x +y −z and z was set to 0. 3+x+0.5 +y +z+0.5 2. The list of previous parameters was enlarged to include the normaliza- − 4+x+0.5 +y z+0.5 tion length. 1540 F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548

thesametimeresultedinadivergencetoawrongstructure,where fragments separated from the calcium interlayer and O(W)6 separated from Ca1. 5. After convergence of the grain shape was reached, the crystal structure and the atomic positions were refined again. 6. Finally, the occupancy of all the atoms was refined in this sequence: first, a short refinement of occupancies and previous parameters was done. Then, the occupancies of atoms whose refined value was bigger than one were fixed to one, while the others were refined. The occupancy of atoms belonging to the same fragment was bound to a unique value to preserve the atomic coordination. This last step was performed with two goals in mind: (1) to improve the fit through the addition of another refinable parameter; and (2) to simulate the disorder inside the structure, a relevant issue for nanostructured or amorphous materials. Even nanostructured materials contain a wide amorphous domain along grain boundaries. Furthermore, as pointed out by Egami and Billinge [18], every X-ray diffraction method, including the Rietveld method, is only capable of describing the structure of crystalline long-range ordered materials. Because X-ray diffrac- Fig. 6. Comparison between tobermorite 11 Å Merlino (left) and tobermorite 12.5 Å based dimer (right) unit cells as they appear after the application of the symmetry operation of tion is based on Bragg's law, it can be applied rigorously only under space group B11m. In black calcium atoms, in gray silicon atoms, and in light-gray the assumption that the crystal lattice is a perfect periodic repetition oxygen atoms (the bigger spheres represent water molecules as the ion O2− ). of the unit cell in three-dimensional space. Therefore the solution to refine the occupancy in the last step of the analysis can be viewed as an attempt to extend the use of the Rietveld method to the study of 3. The structure was checked with the visual output produced by nanocrystals, until now a field explored predominantly through the MAUD to verify that the interatomic distances of atoms not included use of PDF. Note that the temperature factor was not refined. inside the fragments were respected. The refinement was led almost The resulting refinement is shown in Fig. 7b. The calculated spectrum to convergence. At this point, the average isotropic crystallite size fits almost perfectly the experimental spectrum of C-S-H and reliable in- recorded was about 61.5 Å. formation about structure and microstructure of C-S-H can be summa- 4. All the parameters were fixed, and the refinement of the anisotropic rized [5]: grain size was performed using the Popa model which uses a spheri- cal harmonic expansion [25]. The last refined value of isotropic grain • The good fit of the shape of the peak allows to obtain the dimension size was used as a starting value for the first coefficient of the spheri- and the shape of crystallites. cal harmonic, whereas the other coefficients were refined incremen- • The fit of peak positions gives the new cell parameters. tally. It was necessary to fix all other parameters with the exception • The fit of the intensity of the peaks results in determining the structure of the anisotropic size parameters, because refining the structure at factor.

Fig. 7. Plotting of the experimental spectra of C-S-H 1.0 (dots) and calculated spectra (continuous line) of tobermorite 12.5 Å b–d (based dimer) at the same degree of refinement than the tobermorite 11 Å plotted in Fig. 2b (a) and at the final stage of the refinement; (b), Ca/Si ratio=1.08. Plot of calculated and experimental spectra of tobermorite 12.5 Å b–d (based dimer) with Ca2 on the interlayer at the final stage of the refinement. F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548 1541

Table 2 Results of the reﬁnement of crystal structure and nanostructure of tobermorite 12.5 Å based dimer.

Sample C-S-H 1.0

Empirical formula Ca8Si7.4O18.5(OH)7.4·7.8H2O Crystal system Monoclinic Space group B11m Unit cell dimension a=6.69(1)Å b=7.34(1)Å c=24.77(4)Å γ=122.93(4)° Isotropic grain size D=61.5(4)Å

Anisotropic grain size (Popa model) R0 =55.7(6)Å

R1 =−42.2(7)Å

R2 =−9.8(0)nm

R3 =−9.7(7)Å Rw 6.58% Rwnb 6.70% R 4.87% Rnb 4.90%

• Although each broad diffraction peak is the sum of many individual Fig. 8. Schematic drawing of the atomic structure at the end of the refinement process. Fragments and atoms are respectively transparent and solid before and after the refinement Bragg reflections (indicated Fig. 7b), nevertheless many structural de- of the atomic structure. tails can be resolved.

3.1.2. Results of refinement of the fit demonstrates an improvement of the structure, the application of constraints maintained the multilayered feature of the overall The results of the refinement of the structure are listed in Tables 2 structure. In particular, the distance between fragments is maintained and 3. The cell parameters are not very different from the starting through the bond in correspondence with the oxygen atom O2 (see values. This means that the structure of tobermorite 11 Å was a good Fig. 9). Note that after refinement, the atoms are still grouped together starting model and, increasing the interlayer distance in accordance within the three groups of water interlayer, calcium layer, and fragments; with Taylor, was appropriate. Table 3 gives the atomic structure at the no constraints were applied to maintain the distances between Ca1–Ca3, end of the refinement process. The atoms can be divided into three and Ca1–O(W)6. The fragments–calcium layer, fragments–water inter- groups: the calcium layer, water interlayer and tetrahedral fragments. layer and calcium layer–water interlayer remained unchanged. For the atoms in the calcium layer, all the atomic coordinates were set The position of water molecules O(W)2 and O(W)3 changed much free during the refinement; for the water molecules in the interlayer more than any other atomic position. In this class of multilayered only the coordinates x and y were refined. The refinement containing material, the water molecules in the interlayer are bonded through the fragments led to different results: the coordinates of the atoms van der Waals interactions that are weaker than the covalent-ionic inside the polyhedra were fixed and the atomic positions could be bonding existing in the rest of the structure. Correspondingly the calculated only indirectly using the refined value of the origin of water molecules moved by a larger amount, which has a physical the local frame tied to the fragment as well as the Euler angles and meaning, further confirming the reliability of the structure and the the normalized length. consistency of the method. After refinement the position of the molecules The occupancies of the atoms Ca1, Ca3 and O(W)6 were set equal O(W)2 and O(W)3 converged almost to the same position. Considering to one after observing that during the refinement the value diverged that the occupancy for both the molecules is 0.5 and the pronounced beyond one which has no physical meaning. With regard to the molecules thermal vibration of water, they are supposed to be a single water mole- of water O(W)2 and O(W)3, although their occupancy was set free, cule, as confirmed when the occupancy of the two molecules did not however, it did not shift from the initial value of 0.5. change from their initial value of 0.5 during the refinement process. Fig. 8 shows the structure of tobermorite 12.5 Å based-dimer at the beginning and at the end of the refinement process. While the quality

Table 3 Reﬁned tobermorite 12.5 Å based dimer structure. *Fixed temperature factor (not reﬁned).

Atom site label xy zOccupancy B* factor

Ca1 0.2294(8) 0.4089(7) 0.2337(2) 0.94(6) 0.8 Ca2 0.7417(4) 0.8751(8) 0.3119(7) 1.00(0) 0.8 Si1 0.7308(8) 0.3556(2) 0.1606(1) 0.85(8) 0.6 Si3 0.7552(2) 0.9318(9) 0.1746(9) 0.84(0) 0.6 O1 0.7710(0) 0.4176(9) 0.0964(8) 0.85(8) 1.2 O2 0.7225(9) 0.1501(6) 0.1542(0) 0.84(9) 1.2 O3 0.9349(2) 0.5205(0) 0.1981(3) 0.85(8) 1.2 04 0.4982(0) 0.3111(1) 0.1860(2) 0.85(8) 1.2 O(W)6 0.2096(4) 0.4129(6) 0.1370(1) 0.94(6) 5 O7 0.7265(3) 0.8182(1) 0.1105(8) 0.84(0) 1.2 O8 0.5257(2) 0.7864(1) 0.2163(5) 0.84(0) 1.2 O9 0.9468(0) 0.9973(6) 0.2112(9) 0.84(0) 1.2 O(W)1 0.0810(6) 0.2140(1) 0.0000(0) 0.47(1) 5 O(W)2 0.6880(7) −0.2044(6) 0.0000(0) 1.00(0) 5 Fig. 9. Detail of rotation of tetrahedra Si1 and Si3 before the reﬁnement of the atomic O(W)3 0.4232(8) 0.1453(8) 0.0000(0) 0.61(8) 5 structure (transparent) and after the reﬁnement (solid). 1542 F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548

Fig. 10. Plots of the experimental spectrum of C-S-H 1.5 (dots) and calculated spectrum (continuous line) of tobermorite 11 Å after calibration of image center coordinates and tilting angles, together with a reﬁnement of background, instrumental intensity (scale factor), isotropic grain size, and cell parameters. Inset: Two-dimensional diffraction image of C-S-H 1.5 aged 4 months.

The effectiveness of the improved method to define the fragments is the dimer is changed from the initial condition in which the dimers demonstrated yet again by the fact that they remained bonded together are aligned with b. However, the dimensions of the crystallite must be through the oxygen bridging atom O2, even though they changed their interpreted with some caution for two reasons: first, they represent position. Likewise, it is interesting to observe that the overall orienta- only average values; and second, the large amount of disordered grain tion of the fragment changed only slightly (Fig. 9), even if the value of boundary material, typical of all nanostructured materials, does not the new Eulerian angles of the tetrahedron Si1 (not shown here) allow for defining an accurate crystallite shape. changed by a large amount. This is due to how the three Eulerian angles are defined: 3.2. C-S-H 1.5 • First a rotation around the z axis (roll) 3.2.1. Structural modifications • Second a rotation around the new x′ axis (pitch) Because modeling C-S-H with a calcium to silica ratio of 1.0 was so • Third a rotation around the new z″ axis (yaw). successful, the same method was applied to a sample C-S-H with a Finally, the dimension of the tetrahedra also changed slightly, calcium to silica ratio of 1.5, aged for 4 months. allowing for the maintenance of the interatomic distance between The two-dimensional diffraction image of the sample, after sub- the vertices of the fragments and the other atoms in the unit cell. traction of the scattering from the sample holder (Fig. 10 inset) is In the Popa model, the average crystallite is approximated by an similar to the CSH 1.0 sample, with the same nanostructural features ellipsoid. For each reflection hkl, the radius of the crystallite along the (Fig. 2a inset). The one-dimensional spectrum (Fig. 10) is also similar direction [hkl]Dhkl is calculated. The multilayered feature of nanostruc- to the one from C-S-H 1.0, although some differences in the intensity ture is confirmed by the fact that the dimension along the direction of of the peaks between 2.5° and 4.0° indicate a change in the atomic the stacking of the layers, i.e. the c axis of the unit cell, is shortest structure.

(D001 =29.2 Å) compared with the other dimensions. In addition, the The analysis performed on the CSH 1.5 sample was similar to that maximum dimension of the grain (D−460 =74.4 Å) is parallel to the performed on the CSH 1.0 sample. The ﬁrst step was to begin with plane of the interlayer, i.e. the plane z=0. Tobermorite crystallites usually tobermorite 11 Å and a reﬁnement of the image parameters, instru- occur as platelets elongated along the b axis (the direction [010]). It is mental intensity, background, isotropic grain size, and cell parameters interesting to observe that this is not the case here since we have until convergence was almost reached (Fig. 10). Again, it can be seen

D010 =69.4 Å. There could be two explanations for this discrepancy. that, although the structure of tobermorite 11 Å was a reasonable The first is that, since the tetrahedral chains are now substituted by di- starting point, additional refinement of the atomic positions was mers, the crystallite does not develop anymore along the direction of necessary. Other modifications of the atomic structure were also the chains, while in tobermorite they are aligned with the b axis. The necessary in order to satisfy the Ca/Si ratio of this sample, which was second is that, at the end of the fragment refinement, the direction of double compared to tobermorite 11 Å. Richardson [9] proposed a possible

Fig. 11. (a) Bridging calcium atom coordinated with oxygen atoms and water molecules as extrapolated from Richardson [9]; (b) atomic structures of tobermorite 13 Å silicon tetrahedra dimer with calcium bridging atom. F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548 1543 qualitative structure of C-S-H 1.5 where the bridging tetrahedra in the the presence of the calcium atom Ca2 in the interlayer (with z=0), chain is replaced by calcium atoms, coordinated with oxygen atoms and which for the space group B11b represents a special position. water molecules. The sketch of this structure is shown in Fig. 1-III. The new structure was introduced into MAUD, and the same basic The new atomic structure was devised as follows: refinement used for tobermorite 11 Å was executed (Fig. 12a). Com- pared with the previous result, there is a marked improvement in the • The polyhedron relative to the new calcium atom, marked as Ca2, fit of all the peaks, confirming again the validity of the method. was first drawn qualitatively (Fig. 11a), using the sketch proposed by Richardson [9]. Then, the interatomic distances between the atom Ca2 in the center of the polyhedron and the oxygen atoms, 3.2.2. Results of refinement as well as the water molecules in the vertices of the polyhedra, The refinement of the atomic position was performed as in the were arranged qualitatively to values dictated by the atomic radii previous analysis with one exception. Here, the new bridging polyhedron of the different atoms. was not defined as a fragment for two reasons: (1) the initial position of • In place of the deleted bridging tetrahedron, the newly created the atoms of the polyhedron is too approximate to apply constraints that polyhedron was placed inside the structure already constructed as fix their relative distance. Application of constraints at this early stage a starting model for the tobermorite 12.5 Å based dimer. This was of the analysis would probably lead to wrong or unreliable results; done without changing the location of atoms O1 and O7, which rep- and (2) the shape of the calcium polyhedron is less severely defined resent the bridging oxygen atoms between the new polyhedron and than the shape of silicon tetrahedra. the tetrahedra with central atoms Si1 and Si3, respectively. Second, The result of the refinement of the atomic structure is shown in the water molecule O(W)6 was shared between the new atom Ca2 Fig. 12b. After the refinement procedure, the calculated spectrum and the atom Ca3. Finally, the water molecules at the opposite matches to the experimental spectrum and the value of the residual side of the fragment (Fig. 11) were located in the water interlayer is sufficiently low. The quality of the fit at the end of the refinement to maintain the character of multilayered structure, but were replaced procedure is even better when compared with the previous sample by molecules O(W)1 and O(W)3 (which were already present in the (Fig. 7b), demonstrating that the model used was an excellent structure but in different positions). starting point. • The location of the atoms of the new polyhedron was adjusted qual- The results of the refinement are listed in Table 4. Again, the cell itatively to preserve the inter-atomic distance between all the atoms parameters changed slightly. The c cell parameter, whose value was present within the cell. set only qualitatively before the refinement process, increased only • The whole crystal structure was obtained using the general site posi- slightly. The dimension of the crystallites was slightly larger com- tions of space group B11m, which revealed an overlap between the pared to the sample C-S-H 1.0. Considered the nanostructured nature new Ca2 atoms. This problem was solved extending the cell along of the material, in which nanocrystallites are composed by a small its c axis from 25 Å to 26 Å (i.e., raising the interlayer distance to number of cells, a possible explanation for this phenomenon is that 13 Å). A final correction was made to preserve all the atomic positions the larger interlayer spacing leads to larger crystallites. The refine- to maintain the proper inter-atomic distances. ment process of the anisotropic shape demonstrated that in this case the convergence was reached after refinement of the first three Note that adding the atom Ca2 as bridging polyhedron, led to a Ca/Si Popa coefficients. Finally, the R-values for this refinement were ratio of exactly 1.5, which is the same value of Ca/Si ratio of the C-S-H lower than the ones obtained from the sample C-S-H 1.0, demonstrating sample. Using tobermorite 11 Å prevented this possibility because of a better fit.

Fig. 12. Plots of the experimental spectrum of C-S-H 1.5 (crosses) and calculated spectrum (continuous line) of tobermorite 13 Å with calcium bridging atom at the same degree of refinement than tobermorite 11 Å plotted in Fig. 10; and (b) at the final stage of the refinement, Ca/Si ratio=1.5. Compared to Figs. 7–13, the spectrum is cut at 4.6° because the broad peak at 4.4° is the last distinct reflection in the experimental pattern of C-S-H 1.5. 1544 F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548

Table 4 Results of the reﬁnement of crystal structure and nanostructure of tobermorite 13 Å calcium bridging atom.

Sample C-S-H 1.5

Empirical formula Ca12Si8O20·15H2O Crystal system Monoclinic Space group B11m Unit cell dimension a=6.782(5)Å b=7.329(4)Å c=26.36(2)Å γ=124.56(2)° Isotropic grain size D=71.5(4)Å

Anisotropic grain size (Popa model) R0 =67.7(3)Å

R1 =−13.7(7)Å

R2 =−16.7(8)Å Rw 4.04% Rwnb 4.22% R 3.06% Rnb 3.14%

The refined atomic positions are listed in Table 5; a sketch of the structure before and after the refinement process is shown in Fig. 13. This time the refinement of the atomic occupancies in the last step did not change, which is not surprising as the Ca/Si ratio was exactly the same in the model structure as in the sample. Furthermore, the quality of the fit prior to the last step was very good, therefore, the refinement of an additional parameter did not lead to an improvement. After the refinement (Fig. 13), the two protonated oxygen atoms initially called O(H)5 and O(H)6, moved into a different position. Since the contribution of the hydrogen atom to X-ray scattering is almost negligible, compared to that of the oxygen, the scattering of OH group and water molecule is very similar, and they are routinely replaced respectively by O2− anions. Besides, the O(H)5 group moved down to the interlayer, which in tobermorite structure, and in general in multi-layer ceramic materials, is occupied by water molecules and cations. For these reasons we chose to refer to it as a water molecule. In the software it was represented as an O2− anion, and labeled O(W) 5. In contrast, the group O(H)6 and the atom Ca2,movedupwards. The new chemical surroundings led us to consider the group O(H)6 as a water molecule and to label it O(W)6-b. Consequently, the old water molecule O(W)6 was labeled O(W)6-a. Fig. 13. Schematic drawing of the atomic structure of tobermorite 13 Å (a) with calcium fi Some characteristics of the structure after the re nement process bridging atom at the end of the refinement, (b) with detail about tetrahedra; and are similar to what was found in the previous analysis of C-S-H 1.0: (c) calcium bridging atom and water interlayer. The polygons before the refinement process of the atomic structure are transparent, whereas the solid polygons depict the • The atoms Ca1 and Ca3 moved upwards but their interatomic distance new atomic structure at the end of the refinement process. remained almost unchanged.

Table 5 • Refined tobermorite 13 Å calcium bridging atom structure.*Fixed temperature factor Once again the tetrahedra Si1 and Si3 changed their position and (not refined). orientation slightly. • The position of the water atoms in the interlayer changed much Atom site label xyzOccupancy B* factor more than the other atomic positions. Ca1 0.3888(9) 0.4838(2) 0.2452(8) 1.00(0) 0.8 Ca2 0.3629(6) 0.4949(5) 0.0869(5) 1.00(0) 2.7 Ca3 0.8976(0) 0.9627(6) 0.3056(9) 1.00(0) 0.8 The relative positions of the tetrahedra and the new atom Ca2 Si1 0.8119(1) 0.4006(3) 0.1545(2) 1.00(0) 0.6 have changed significantly. Although the Ca2 polyhedron can no longer Si3 0.8389(8) 0.9954(2) 0.1653(4) 1.00(0) 0.6 possibly be considered as a bridging polyhedron, it is important to note O1 0.7846(3) 0.5144(6) 0.1011(7) 1.00(0) 1.2 that, after the refinement process, the position of the atom did not move O2 0.8110(9) 0.1971(5) 0.1284(2) 1.00(0) 1.2 towards the interlayer. This is critical in maintaining the Ca/Si ratio O3 0.0693(3) 0.5781(1) 0.1847(9) 1.00(0) 1.2 O4 0.5800(9) 0.2953(8) 0.1944(7) 1.00(0) 1.2 equal to 1.5, because the interlayer represents a special position in O7 0.9380(7) 0.8997(3) 0.1195(3) 1.00(0) 1.2 space group B11m. O8 0.5507(6) 0.8028(6) 0.1824(4) 1.00(0) 1.2 The refinement of anisotropic crystallite size shows some similarities O9 0.9618(0) 0.0858(8) 0.2122(4) 1.00(0) 1.2 with that of the C-S-H 1.0 sample. Again, the shortest dimension is along O(W)1 0.4553(0) 0.3784(9) 0.0000(0) 0.50(0) 5.0 O(W)2 0.8841(9) 0.5713(8) 0.0000(0) 0.50(0) 5.0 the c axis of the crystalline cell (D001 =52.4 Å), while the largest dimen- O(W)3 0.5003(9) 0.3645(7) 0.0000(0) 0.50(0) 5.0 sion (D−230 =87.7 Å) lies on the interlayer; the crystallite has a plate-like O(W)5 0.5202(1) 0.8407(2) 0.0000(0) 1.00(0) 5.0 shape with a preferential direction of elongation. The direction of the O(W)6-a 0.1733(3) 0.3995(1) 0.1495(8) 1.00(0) 5.0 maximum dimension [−230] is the same as in the previous sample O(W)6-b 0.3509(3) 0.0498(1) 0.1080(1) 1.00(0) 5.0 [−460]. Note that the magnitude of the dimension of the crystallite of F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548 1545

Fig. 14. (a) Atomic position of Hamid tobermorite as taken from [26], and (b) modiﬁed dimer structure: the oxygen atoms O1′ and O2′, which result from the atoms O1 and O2 through Imm2 spatial group transformation, are plotted here to highlight the dimer tetrahedra.

C-S-H 1.5 is larger than the crystallite of C-S-H 1.0. This is expressed in 4.1. C-S-H 1.0 the more distinct diffraction image. Considering the results of Section 3, we performed a similar analysis 4. Orthorhombic tobermorite refinement for the refinement of the Hamid tobermorite with the C-S-H 1.0 data. The bridging tetrahedra were deleted and, in order to obtain a Ca/Si Fig. 14-a shows the atomic positions of the tobermorite refined by ratio equal to one, the occupancy of the calcium atom Ca3 was set Hamid [21] with space group Imm2. As in Merlino tobermorite, this equal to zero. Fig. 14-b shows the Hamid tobermorite with the deleted structure is still characterized by the dreierketten silicon chains and bridging tetrahedra. A good fit was obtained with an anisotropic refine- the calcium atomic interlayer, however the different space group ment. At the end of the analysis, the average dimension of the crystallite has a higher degree of symmetry compared to that of the monoclinic D100 =43.2 Å,D010 =64.6 Å, and D001 =24.7 Å along c. group. This leads to several atomic superpositions and atomic occupan- The largest dimension is now along b because the direction of the cies lower than 1, especially in the tetrahedra chains, which are both fragments slightly deviated from the direction of the b-axis. The initial tilted by a mirror plane, parallel to the planes b–c, and displaced by an cell dimensions were a=5.586 Å, b=3.696 Å and c=22.779 Å. As amount b/2 along b. This feature was used by Hamid to account for the shown in Table 6, the cell dimensions a and b changed only slightly, diffuse peaks in the X-ray spectrum, that he attributed to a stacking whereas the c dimension increased significantly at the end of the refine- disorder of the chains along the c direction. Because of the atomic super- ment. The refinement of the atomic positions with fragments led to the positions it is difficult to identify the single tetrahedral chain (as done for atomic coordinates presented in Table 7. Note that, although the refine- the Merlino tobermorite in the previous section) Therefore, here we can ment with fragment improved the quality of the fit, the improvement only say that the bridging tetrahedra are in correspondence with the was smaller compared to that observed for the Merlino tobermorite. silicon atoms Si1 and S12, the paired tetrahedra are in correspondence Fig. 15 shows the best fit of the calculated spectrum and compares with the silicon atoms S13 and Si4, and the chains are aligned with it with the experimental values (see the refined atomic structure the direction b. of Table 7). It is worse than that observed for the Merlino model (Fig. 7b).

Table 6 Results of the reﬁnement of crystal structure and nanostructure of Hamid tobermorite applied to CSH 1.0. Table 7 Reﬁned Hamid tobermorite structure applied to CSH 1.0.*Fixed temperature factor. Sample C-S-H 1.0 Atom site label xyzOccupancy B* factor Empirical formula Ca2.25[Si2O5(OH)2]·H2O Crystal system Orthorhombic Ca1 0.0000(0) 0.0000(0) 0.0000(0) 1.00(0) 1.34 Space group Imm2 Ca2 0.0000(0) 0.0000(0) 0.4136(5) 1.00(0) 1.34 Unit cell dimension a=5.59(7)Å Ca3 0.2507(8) 0.5000(0) 0.1968(9) 0.13(0) 4.74 b=3.70(0)Å Si3 0.5116(1) 0.4136(6) 0.3732(7) 0.25(0) 1.18 c=23.49(6)Å Si4 0.5125(3) 0.4247(5) 0.0567(2) 0.50(0) 1.18

Anisotropic grain size (Popa model) R0 =44.3(4)Å O1 0.2804(6) 0.5028(8) 0.1775(4) 0.50(0) 1.34

R1 =−31.2(7)Å O2 0.2783(3) 0.4942(4) 0.4111(7) 0.50(0) 1.34

R2 =−20.3(8)nm O5 0.5005(0) 1.0000(0) 0.0767(0) 0.50(0) 1.89

R3 =0.1(4)Å O6 0.5005(0) 1.0000(0) 0.3482(9) 0.50(0) 1.89 Rw 8.38% O(H)7 0.5098(3) 0.3296(7) 0.3113(0) 0.25(0) 4.18 Rwnb 8.77% O(H)8 0.5057(3) 0.3398(2) 0.1187(2) 0.25(0) 4.18 R 6.37% O(W)1 0.0000(0) 0.0000(0) 0.1108(3) 0.50(0) 3.16 Rnb 6.56% O(W)2 0.0000(0) 0.0000(0) 0.3045(8) 0.50(0) 3.16 1546 F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548

Fig. 15. Plots of the experimental spectrum of C-S-H 1.0 (crosses) and calculated spectrum (continuous line) of Hamid dimer tobermorite at the ﬁnal stage of the reﬁnement.

4.2. C-S-H 1.5 the amorphous fraction in ceramic materials with the Rietveld method. The profile shape function utilized was a convolution of two contribu- Modifications of the structure as done for the Merlino tobermorite tions, taking into account both instrumental and sample aberrations. It here were not successful, which most likely because replacing the is noteworthy to highlight the main differences on the simulation of an bridging tetrahedra with calcium atoms gives an unrealistic structure amorphous material in these two studies. Le Bail simulated the amor- where the two calcium cations superimpose each other. However, the phous feature of the glass using the microstrain, i.e. a very defective crys- calcium atoms Ca3, which is positioned between the two dimer tetra- talline structure. This can be seen as an attempt of adapting to the Rietveld hedra, can already be considered a bridging atom for both the dimer analysis a technique used in molecular dynamics simulations where, tetrahedra Si3 and Si4. Therefore, we used again the Hamid dimer starting from a crystalline model, the amorphous diffraction pattern is structure [21] with the occupancy of the calcium atom Ca3 equal to obtained by distorting the structure by applying a statistical isotropic 0.5 in order to obtain Ca/Si=1.5. Similar to the C-S-H 1.0 analysis, microstrain. Instead, Lutterotti et al. [27] fit the amorphous pattern refin- the larger improvement of the fit was obtained with the anisotropic ing a nanocrystal whose dimension was of the same order of the used refinement of the crystallite dimension as follows: D100 =26.1 Å, crystalline cell without considering microstrain. Although both solutions D010 =70.5 Å, and D001 =26.9 Å. In addition, the cell dimensions revealed themselves valid, they represent an average model, and for this presented in Table 8 show again that the length c increased significantly reason they fail to represent the exact local atomic arrangement. This is to the final value of 25.35 Å. The calculated versus the experimental beyond the purpose of the Rietveld refinement, and subject of studies spectrum at the end of the analysis is shown in Fig. 16. Also in this using PDF analysis [18]. case the refinement is not as good as that executed with the Merlino Unlike these investigations of truly amorphous materials, our study structure (Fig. 12b) : the refined atomic coordinates are listed in confirms a degree of long range order in C-S-H. The results made by Table 9. Here the initial atomic occupancy remained unchanged leading Taylor [2–5] and later by Richardson [7–9] showed a multi-layer nature to a Ca/Si ratio of 1.5. of C-S-H, in which composition and stacking pattern are close to those found in tobermorite. The similarity between C-S-H and the Merlino 5. Discussion tobermorite 11A structure has been demonstrated by Skinner et al. [14], in which the loss of long range order is attributed to small grain It is well known that powder diffraction is suitable to study crystalline size. Here, we can interpret the X-ray diffraction pattern of C-S-H 1.0 materials. However, in 1995 Le Bail [26] refined the microstructure of and 1.5 quantitatively by refining the atomic structure as well as amorphous silica glass with X-ray and neutron diffraction data, based the anisotropic microstructure. The results of the Rietveld refinement, the crystal structure of α-carnegieite. Lutterotti et al. [27] determined using tobermorite as crystal model, lead to the conclusion that C-S-H can be described also as a nanocrystalline ceramic material as well as a semi-amorphous gel. Reliable information about nanostructure and atomic positions Table 8 can only be obtained in steps refining first unit cell, then polyhedral fi Results of the re nement of crystal structure and nanostructure of Hamid tobermorite fi applied to CSH 1.5. positions and atomic positions, and nally particle shapes. It requires high-resolution diffraction patterns that can be obtained only with Sample C-S-H 1.5 synchrotron radiation.

Empirical formula Ca3[Si2O5(OH)2]·H2O The fit can be improved by the refinement of the atomic positions, Crystal system Orthorhombic 4− respecting the coordination and bond angles typical of the [SiO4] tetra- Space group Imm2 hedra and their arrangement in dreierkette chains. This is done using the Unit cell dimension a=5.63(8)Å b=3.67(2)Å fragment option in MAUD that allows to group atomic polyhedra and re- c=25.35(9)Å fining only their position, orientation and expansion/shrinkage. Bonding Anisotropic grain size (Popa model) R0 =32.7(3)Å group of atoms with fragment also permits to reduce the number of − R1 = 49.4(9)Å refinable parameters in the Rietveld analysis. R =−40.6(4)nm 2 However, the results obtained have to be interpreted carefully. R3 =0.1(4)Å Rw 6.53% First, even if the material can be considered nanocrystalline, in mono- Rwnb 6.60% clinic tobermorite a presence of disorder in the distribution of the R 5.33% atoms inside the crystalline cell has been recognized by the improve- Rnb 5.37% ment of the fit with the refinement of the atomic occupancies. The F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548 1547

Fig. 16. Plots of the experimental spectrum of C-S-H 1.5 (crosses) and calculated spectrum (continuous line) of Hamid dimer tobermorite [20] at the final stage of the refinement. same conclusion is valid also for the orthorhombic tobermorite, in 6. Conclusions which the disorder was already present in the structure refined by Hamid [21] through a combination of occupancies lower than one The main conclusions are as follows. and atomic superimpositions. At this point, it is worth mentioning that the refinement of the atomic occupancy is not a trivial step of the 1. The diffraction pattern of C-S-H with Ca/Si ratio of 1.0 and 1.5 aged analysis, because occupancy of anions and cations has to be constrained 4 months can be described by the crystal structure of tobermorite to maintain the coherence of the polyhedra and neutrality of the charge 11 Å when the isotropic crystallite dimension is of the order of the after the refinement. Second, the crystallites in the material obviously crystalline cell dimension. This can be considered an effective do not have a unique value of dimension as can be inferred from this strategy to simulate the nanostructural character of the material. work, but a distribution of sizes. Finally, modeling nanocrystalline However, structural modifications are mandatory in order to obtain material has to take into account that the nano-scale of the crystallites a model which is closer to the real structure of C-S-H and the correct leads to an enormous quantity of boundaries that represent an amor- Ca/Si ratio, that in tobermorite 11 Å ranges from 0.67 to 0.83. phous region that could probably be improved in the analytical method. 2. The best fit of the model calculation to the experimental spectrum The diffuse scattered intensity from the boundary layer cannot be is obtained introducing special constraints, called fragments, that neglected and may influence the refinement of the atomic position. allow the refinement of atomic positions as part of a functional For example the disorder inside the structure can be attributed both blocks which maintain the length of atomic bonds and the relative to the disorder of the atoms inside the nanocrystallites and to the angles between atoms during the refinement. This technique has boundary of the crystallites, leading to two different regions with a dif- already been successfully applied to the Rietveld method in the re- ferent scale of disorder. Since we can only simulate the disorder inside finement of the structure of amorphous materials ab initio. In this the cell by refining the atomic occupancy, this could lead to unreliable work however, fragments are for the first time applied to a com- results. The Rietveld method relies on a least squares fit of a model func- plex structure, in a way such that groups of atoms are replaced in tion with observed data. Especially in complex materials like C-S-H their original positions at an advanced stage of the refinement. great care is necessary to make sure that model parameters are physically 3. The refinement of crystal structures with different levels of disorder meaningful and that the refinement converges into a true minimum [28]. leads to different results in terms of fragment parameters and an- We paid attention to these limitations. isotropic crystallite size. In the refinement of the ordered structure, It has to be pointed out that the hydration reaction that produces i.e. the monoclinic one, the best fit is obtained with a strong refine- the C-S-H can extend for a very long time and C-S-H is a material that ment of fragments position and orientation, followed by the anisotrop- changes with time and further studies have to be done to investigate ic refinement of the crystallite size. On the other hand, the disordered the effect of age on ordering patterns and crystallite size. orthorhombic structure showed that the best fit is obtained when first, the anisotropic crystallite size is refined and second, a weak refinement of fragment parameters is done. This behavior is not surprising if we consider that in the Hamid orthorhombic tobermorite disorder is modeled by allowing a significant overlap of atoms belonging to

Table 9 the tetrahedral chains, as a result of the symmetry constraints of the Refined Hamid tobermorite structure applied to CSH 1.5.*Fixed temperature factor. orthorhombic group and the reduced cell. These results suggest that structural symmetry represents a constraint in the refinement of Atom site label xyzOccupancy B* factor fragments, however more work is needed to investigate the behavior Ca1 0.0000(0) 0.0000(0) 0.0000(0) 1.00(0) 1.34 of fragments in the refinement of other complex atomic structures Ca2 0.0000(0) 0.0000(0) 0.4125(9) 1.00(0) 1.34 with other symmetries. Ca3 0.2750(4) 0.5000(0) 0.1874(0) 0.50(0) 4.74 fi Si3 0.6101(8) 0.4300(4) 0.3529(3) 0.25(0) 1.18 4. The best t of the diffraction pattern of C-S-H 1.0 aged 4 months is Si4 0.4711(0) 0.4203(9) 0.0119(2) 0.25(0) 1.18 obtained by refining the atomic structure of tobermorite 11 Å with O1 0.2444(4) 0.4828(4) 0.9800(7) 0.50(0) 1.34 dimer tetrahedral chains in place of infinite length chain, and a raised O2 0.3809(0) 0.4910(2) 0.3880(7) 0.50(0) 1.34 interlayer spacing. The crystallite average shape is plate-like, with a O5 0.4745(0) 0.0138(1) 0.0298(1) 0.25(0) 1.89 O6 0.6130(0) 0.0193(7) 0.3297(1) 0.25(0) 1.89 preferential direction of elongation and an average (isotropic) di- O7 0.6116(5) 0.3467(9) 0.2953(2) 0.25(0) 4.18 mension of 54 Å. O8 0.4821(6) 0.3390(7) 0.0673(8) 0.25(0) 4.18 5. The best fit of the diffraction pattern of C-S-H 1.5 aged 4 months is O(W)1 0.0000(0) 0.0000(0) 0.1095(0) 0.50(0) 3.16 obtained by including a calcium atom between the dimer tetrahedra, O(W)2 0.0000(0) 0.0000(0) 0.3010(0) 0.50(0) 3.16 and a raised interlayer spacing. The crystallite average shape is again 1548 F. Battocchio et al. / Cement and Concrete Research 42 (2012) 1534–1548

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