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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, USAb Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, USA
Low-Q region Rietveld analyses were performed on C-S-H synchrotron XRD patterns, using the software MAUD.Two different crystal structures of tobermorite 11 Å were used as a starting model: monoclinic ordered Merlinotobermorite, and orthorhombic disordered Hamid tobermorite. Structural modifications were required to adaptthe structures to the chemical composition and the different interlayer spacing of the C-S-H samples. Refinementof atomic positionswas done by using special constraints called fragments that maintain interatomic distances andorientations within atomic polyhedra. Anisotropic crystallite size refinement showed that C-S-H has a nanocrys-talline disordered structure with a preferred direction of elongation of the nanocrystallites in the plane of the Cainterlayer. The quality of the fit showed that the monoclinic structure gives a more adequate representation ofC-S-H, whereas the disordered orthorhombic structure can be considered a more realistic model if the lack oflong-range order of the silica chain along the c-direction is assumed.
Calcium silicate hydrate (C-S-H), the main binding phase inPortland cement matrix, constitutes up to 70% in weight of hardenedordinary cement pastes. Despite the large number of studies and thevast amount of literature available on cementitious materials, theatomic scale structure of C-S-H is still partly unknown owing to itshigh complexity. The optimization of strength and durability of cementcan be obtained through adjustments of the structure of C-S-H at ananometric level [1,2], but this is subject to a detailed knowledge ofthe C-S-H crystal structure.
Several models have been proposed for structure of C-S-H. It isrecognized that it has a multilayer structure composed of calciumlayers and interrupted tetrahedral chains on both sides. Various studieshave indicated structural relationships to tobermorite 14 Å (Ca5Si6O17
9H2O–C5S6H9, plombierite) [3,4] and jennite (Ca9Si6O21 10H2O–C9S6H10) [5,6]. Both structures contain linear silicate chains of the“dreierkette” form in which the silicate tetrahedra are arranged insuch a way as to repeat a kinked pattern after every three tetrahedra.Two of the three tetrahedra share O–O edges with the central Ca–Opart of the layer; these are linked together and are often referred to as‘paired’ tetrahedra (P) (Fig. 1, I). The third tetrahedron, which sharesan oxygen atom at the pyramidal apex of a Ca polyhedron, connectsthe two paired tetrahedra and is called “bridging” (B) [2]. However,C-S-H formed by hydration of portland cement paste, is significantly
teiro).
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different from these crystal structures, basically for the followingreasons:
• The Ca/Si ratio of about 1.75 is higher than that of jennite and muchhigher than that of 14 Å tobermorite.
• Both tobermorite and jennite have long tetrahedral chains, whereasin C-S-H the chains have lengths of 2, 5, 8, … (3n−1) tetrahedra.This pattern results from the repetition of two paired tetrahedraconnected by the bridging tetrahedron. A particular case occurs forn=1, when only two bridging tetrahedra are present and as will bediscussed later, they are referred to as dimer.
• The average chain length of C-S-H increases with age: 29Si NMRexperiments on C3S paste cured at 25 °C show that the mean chainlength after 1 day is 2.1 tetrahedra, 2.6 after 1 month, 3.3 after1 year, and 4.8 after 26 years [7]. However, when the Ca/Si ratio islow, dimer chains (length=2) occur even in mature paste.
In 1986, Taylor proposed a model [3] for C-S-H that consisted of adisordered layer structure, whereby the majority of the layers werestructurally similar to those of jennite and others were related to14 Å tobermorite. In both types of layer, the structures were modifiedby the omission of silicate tetrahedra. This is an effective solution becauseit is possible to obtain the expected chain length and correct the Ca/Siratio of C-S-H. The tobermorite-type structure was found to be moresuitable to describe the lower Ca/Si ratio C-S-H, whereas jennite-typestructure was more suitable for the high Ca/Si ratio C-S-H.
In 1992, Richardson andGroves proposed a generalizedmodel for thenanostructure of C-S-H [8] that accounted for the chemical differences be-tween C-S-H and the structures of tobermorite and jennite. Themodel in-cluded the chemical neutrality of the structure by the protonation of
Fig. 1. I) Schematic diagram showing dreierkette chains present in tobermorite: centralcalcium layer (light blue polyhedra) and dreierkette silicon chains (dark blue tetrahedra).The chains have a kinked pattern where some silicate tetrahedra share O–O edges with thecentral Ca–O layer (called ‘paired’ tetrahedra (P)), and others that do not (called ‘bridging’tetrahedra (B). II) Diagrams illustrating tobermorite-based dimer that has the maximumdegree of protonation of the silicate chains (Ca/Si=1.0). (a) A highly schematic diagramdemonstrating chemical accounting of the model. (b) and (c) showmore realistic structuralrepresentations derived using crystal structure data for tobermorite with the silicate chainseither aligned along the plane of the page or perpendicular to thepage. III) Diagrams illustrat-ing the C-S-H model for Ca/Si=1.5 (a), (b) and (c) have the same meaning than in II [9].
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O2− anions, resulting from deleting bridging tetrahedra. This led tosilanol groups in place of the oxygen atoms shared by paired andbridging tetrahedra. In addition, different amounts of calcium polyhedrawere located in place of the deleted bridging tetrahedra to respect thereal Ca/Si ratio of C-S-H under consideration. Later on, Richardsonsuggested detailed structural representations of C-S-H, derived fromcrystal structure data of tobermorite, for three different levels of proton-ation for Ca/Si ratios of 1.0, 1.25 and 1.5 [8,9].
Recently, new understanding of the C-S-H structure has beenachieved using a variety of methods: X-ray diffraction [10–13], totalscattering 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 byMerlino et al. [17]. This conclusion was obtained by simulating thediffuse X-ray scattering due to the nanostructured features of thematerial by a Gaussian shape broadening of the structure factor oftobermorite 11 Å, tobermorite 14 Å, and jennite, that were supposedto resemble the nanostructure of C-S-H in real portland cement concrete.Their results show that the structure of tobermorite 11 Å is strikinglysimilar to that of C-S-H (I). The loss of coherent scattering on the C-S-Hsample above about 3.5 nm might reflect the maximum crystallite sizeof the material. In addition, a Monte Carlo refinement of tobermorite11 Å was done to obtain the best fit of the PDF to the synthetic C-S-Hsample. After the refinement it is still possible to see the distinct multi-layer structure made by a stacking of calcium layers with tetrahedralchains on both sides, and calcium ions in addition to water moleculesin the interlayer space.
X-ray patterns can be analyzed in twoways: One is to determine thedistribution of atom pairs. This method is mainly applied to amorphousmaterials with no long-range order or highly disordered structures [18].The second method is based on the structure factor, which relates thediffraction pattern to lattice planes in the long-range ordered crystal[19]. Interestingly C-S-H is in between. In this study we will use thestandard crystallographic Rietveld method [20] to further quantifythe nanostructure of this material. Two different samples of C-S-Haged 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 toC-S-H with Ca/Si of 1.5. Two crystal structures for C-S-H were usedas a starting model in the Rietveld analysis: (a) the monoclinictobermorite 11 Å refined byMerlino et al. [17] and successfully appliedin Skinner's work [14], characterized by high structural order, and(b) the orthorhombic version of the tobermorite structure refinedby Hamid [21] that, due to the smaller unit cell, presents a lower degreeof order and therefore is more likely to capture the real structure ofC-S-H.
The detailed description of the two tobermorite crystal structuresis out of the scope of this presentation and we refer to the papers byMerlino et al. [17] and Hamid [21] and references within them forfurther details. However, for clarification, we note that tobermorite isa mineral that occurs with a substantial degree of disorder in its X-raydiffraction pattern that is expressed by diffuse reflections along thec-axis. Hamid attributed this feature to the different positions andorientations that the silicate chains can take. In particular, it was pro-posed that a chain can be statistically displaced by b/2 and that eachtetrahedron of the chain can be statistically tilted in two alternativepositions. The structure can be represented in an orthorhombicsubcell with space group Imm2. Merlino refined two polytypes forthe ordered tobermorite structure: one orthorhombic with spacegroup F2dd, and the other monoclinic symmetry with space groupB11m.
This investigation applies the Rietveld method to the refinementof the atomic structures and nanostructures of C-S-H by using thesoftware “MAUD” [22], which is particularly suitable to implement anew approach for the analysis of nanostructured materials.
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2. Experimental methods
2.1. Materials
C-S-H samples were synthesized using CaO and amorphous SiO2
(Carbosil) mixed at a water-solid ratio of about 12.9. The CaO reagentwas prepared by heating pure CaCO3 (Fisher) at 900 °C for 24 h.
Two samples of C-S-H with different stoichiometric calcium/siliconratios (1.0 and 1.5) were prepared. The samples were cured for fourmonths.
2.2. Experiments at APS and Rietveld refinement
X-ray diffractionmeasurements weremade at the Advanced PhotonSource, beamline BESSRC 11‐ID‐C at Argonne National Laboratory. Theenergy of the monochromatic incident beam was 114.95 keV, corre-sponding to a wavelength of 0.1079 Å. Details of the experiments aredescribed elsewhere [14].
Before collecting diffraction images of the C-S-H samples, a calibra-tion was performed at the same conditions in order to refine detectordistance and orientation. Using CeO2 by NIST as standard, this calibrationwas performed with the MAUD software. Two Caglioti coefficients andone Gaussianity coefficient were refined to determine a Pseudo‐Voigtinstrumental function. These parameters were then used and kept con-stant for the analyses of the C-S-H samples.
Subsequently the diffraction spectrum of the sample, obtained byazimuthally averaging diffraction rings in Fit2D [23] was entered intoMAUD as well as the starting structure of tobermorite 11 Å [17]. Thecrystal structure has a monoclinic unit cell with space group B11m(monoclinic, first setting was used). Since the quality of the data wasnot sufficient to refine temperature factors, a single isotropic B valuewas used for silicon (0.6), for calcium (0.8), for oxygen (1.2) and forwater (5), in accordance with values observed in hydrous silicates. Bvalues were kept intentionally lower compared to values that usuallyoccur in the refinement of calcium silicate hydrate structure, in order toseparate the peakbroadening related to the dimensionof nanocrystallites
Fig. 2. (a) Plots of the experimental spectra of C-S-H 1.0 (crosses) and calculated spectra (contibackground and instrumental intensity (scale factor), in addition to setting of isotropic grain stogether with refinement of background, instrumental intensity (scale factor), isotropic g
from the broadening attributable to the thermal vibrations. In the atomicstructures water molecules are represented by the ion O2−and labeledO(W), similarly groups \OH are represented by the ion O− and labeledO(H).
3. Monoclinic tobermorite refinement
3.1. C-S-H 1.0
3.1.1. Refinement procedureThe experimental spectrum of C-S-H and the calculated spectrum
of tobermorite 11 Å at the beginning of the analysis are shown inFig. 2a. Only a manual correction of the first background parameterand the scale factor was done. As a starting point, the crystallite sizewas set to 100 nm. Note that after this rough correction, the calculatedspectrum of tobermorite 11 Å fits reasonably well the experimentalspectrum of C-S-H, providing us with a sound basis upon which to im-prove the refinement. This result is in good agreement with the workof Skinner et al. [14], even though it was obtained in a different way.
From this point, the best least squaresfit of the experimental spectrumwith the calculated spectrumwas obtained by adding parameters refinedin this order:
1. Background coefficients, image center, tilting angles and scale factor2. Isotropic grain size3. Cell parameters.
Comparison of calculated and experimental spectrum after refine-ment (Fig. 2b) shows a definite improvement of the analysis; however,the quality of the fit of the main peaks is still insufficient to get reliableinformation about the nanostructure and atomic structure of C-S-H. Amuch better fit of the peak shape is necessary to determine the dimen-sion of the grains, and a goodfit of the peak positions and their intensityare indispensable to obtain information about the cell parameters andthe structure factor. The classic Rietveld method refines a model func-tion to fit the observed diffraction spectrum. The model function is de-fined by parameters describing the instrument, background, crystal
nuous line) of tobermorite 11 Å at the beginning of the analysis after a rough correction ofize to 100 nm; and (b) after calibration of image center coordinates and tilting anglesrain size, and cell parameters. The diffraction image is inserted in (a).
Fig. 3. Schematic drawing of the atomic structure of Tobermorite 11 Å using crystalsystem, space group, and atomic coordinates proposed by Merlino [2] (square spots),and thedefinition ofmeaningful polyhedra. Polyhedra verticeswithout spot are generatedafter application of the symmetry operation of the space group B11m (Table 1). Here, thesilicon tetrahedra are dark gray, the irregular calcium polyhedra are light gray, and themonoclinic cell edges a, b, and c are the continuous lines. The orthogonal referenceframe (x,y,z) has the x axis aligned with a and the z axis aligned with c.
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structure and microstructure. C-S-H has a large unit cell with manyatoms. Because of the nanocrystalline microstructure, diffraction peaksare broad and each peak is composed of many individual reflectionsfrom lattice planes hkl (lines at bottom of spectrum). It is not possibleto refine atomic coordinates of all atoms as would be done in a conven-tional analysis. We will have to introduce rigid components or “frag-ments” to combine atoms into larger units.
The analysis demands that any constraints that are introduced willnot only lead to a realistic structure, but also avoid the loss ofinteratomic distances as described by the PDF. For instance, it is wellknown that due to the covalent character of the atomic bond, silicon
Fig. 4. (a) An example of the construction of a fragment using MAUD. 1. Definition of the atomcrystalline cell reference frame, and definition of the rotational Eulerian starting angles (roll, pitused to define the atomic positions in the fragment linked to Silicon 1.
and oxygen atoms form tetrahedra, with silicon in the center and fouroxygen atoms in the vertices. Therefore, it is necessary to preservethese geometric characteristics during the refinement process.
Based on the atomic coordinates presented by Merlino et al. fortobermorite 11 Å [17], three significant tetrahedra were identified,Si1, Si2 and Si3 (Fig. 3). The squares symbolize the atomic positionsreported byMerlino that could be refined, while the remaining verticesrepresent the atomic positions generated from the previous iterationsby the symmetry operations of the space group B11m. Constraintswere applied only to the Si atoms in the tetrahedra. Coordinates of thecalcium and water atoms Ca1, Ca3 and O(W)6 were refined. For thecalcium 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 orderto keep these atoms inside the interlayer, even after the structure hadbeen refined, it was decided to refine the fractional atomic coordinatesx and y, while z was fixed equal to zero, i.e. these atoms were confinedto the x–y plane. The polyhedra used to define the constraints in MAUDare called fragments. The software permits to fix their shape and allowsrefining as follows:
• Coordinates X, Y, Z of the center of the local Cartesian frame placedin the fragment, expressed on the absolute reference frame of themonoclinic cell;
• Orientation of the fragment through three Euler angles roll, pitch,and yaw; and
• Expansion or the shrinkage through the definition of a bond lengthparameter.
The process of defining a fragment in MAUD (Fig. 4a) requires firstintroducing the fractional coordinates for every atom within thepolyhedron, which is defined in the local frame of the fragment. Thesecoordinates give the shape of the fragment as well as the value of thebond length. Second, the position of the center of the fragments, previ-ously defined in the local coordinate system, has to be defined in the ab-solute frame of the unit cell, whose origin is located at the intersection ofthe axes b and c of the monoclinic cell. Note that this absolute frame isorthonormal, and that its axes are not aligned with the axes a, b and cof the monoclinic cell. Thus, in order to locate the fragments and theiratoms in correct positions, it is mandatory to convert the coordinatesfrom the reference system of the monoclinic cell to the orthonormalone. Finally, the orientation of the fragment is defined by assigningthree Euler angles.
coordinates in the local frame. 2. Definition of the position of the local frame center in thech and yaw, in degrees). 3. Definition of the normalized length parameter. (b) Local frame
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This method is amenable to allocating the fragments in rough posi-tions and then refining them later. It assumes that the shape of fragmentis known before the refinement. In practice, this is accomplished using aperfect polyhedron where the vertex coordinates are known and thedistances between vertices and center are constant (which is the bondlength parameter in MAUD). But this process was not appropriate inthis study for the following reasons:
• The starting point was not an empty structure, but rather a well-defined structure, at an advanced stage of refinement (Fig. 2)
• The introduction of the fragments implies substitution of the atomicpositions already refined with the atomic position defined by thefragments
• In order to replace the atoms in their perfect positions it is necessary toknow the exact values of the center of the fragments, their orientations,and their shape at the last stage of refinement.
This problem was solved by drawing the monoclinic cell with andthe atomic positions connected to the polyhedra using Rhinoceros, a3-D design software [24]. At this point, the atomic positions of allthe silicon and oxygen atoms inside the tetrahedra could be extrapolated,defined in the local frame of the fragments, and the center of the localframes of every tetrahedron expressed on the absolute reference frameof the monoclinic cell. For this method no knowledge of the startingEulerian angles is needed, as the fragments are already built in theirexact orientation. It was decided to set them to a default value veryclose but different from zero to avoid singularities when the code isrunning (Fig. 4a).
Tetrahedra defined as fragmentsmay not be perfectly regular (i.e. theedges are slightly different). Thus the method applied here allows fordefining arbitrary polyhedral shapes. Using fragments reduces the com-putational effort because the number of refinable parameters is lowered.Without using fragments it would be necessary to refine three atomiccoordinates (x, y, z) for every atom of the asymmetric unit. Thus,for example, for four oxygen atoms coordinated in a tetrahedron,therewould be 4×3=12 parameters to refine. Using fragments, the re-quired parameters are three for the absolute coordinates of the center ofthe fragment, three for the Euler angles, and one for the bond distance,resulting in a total of 7 parameters for every fragment used.
Fig. 4b shows one of the tetrahedra drawn in Rhinoceros and definedby a fragment inside the unit cell. The center of the local frame is placedin the atom, labeled silicon 1 (Si1), with fragment coordinates (0, 0, 0);the value of the bond length is a value between the maximum of all thecoordinates x, y, z of the vertices of the tetrahedron in the local frame.Dividing the coordinates x, y, z by the bond length provides the fractionalcoordinates of the vertices of the fragment (to be fixed later by the soft-ware). In this sense, the bond length parameter does not represent somuch a length as a parameter used for normalization. From now on, itwill be called the normalization length. Again, the variation of its valueduring the refinement process represents the expanding/shrinkingdynamics of the polyhedron.
Although this way to define fragments using MAUD was found tobe a very powerful tool in the refinement of the atomic structure, itwas not sufficient in regard to the structure of C-S-H. The tetrahedrain tobermorite are bonded in chains with a peculiar spatial configura-tion called dreierkette, where two kinds of tetrahedra with differentspatial orientation are distinguished: two paired tetrahedra and onebridging tetrahedron [9] (Fig. 1-I). Fig. 5a shows the portion of chaininside themonoclinic cell of tobermorite 11 Å,made up of three tetrahe-dra: every tetrahedron shares two oxygen atomswith the adjacent ones;and even after refinement these atoms must be shared. In this figure, aswell as in the figures that follow, the dimension of the spheres, whichrepresent the different atoms, is based on the atomic radius of thedifferent species.
Unfortunately, the basic MAUD software does not have an optionfor bonding the different fragments in their particular configuration.A solution was found by considering first the chemical composition
of C-S-H. The Ca/Si ratio of the sample was 1.0, while the Ca/Si ratiosof tobermorite 11 Å range from 0.67 to 0.83. Clearly, even if a verygood fit was obtained, the crystalline structure of tobermorite 11 Åwould not be suitable to represent the sample because of this chemicalinconsistency. Richardson [9] suggested the modified tobermoritestructure shown in Fig. 1-II, in which the bridging tetrahedra were de-leted as well as the calcium atom in the interlayer, Ca2, and where thetwo bridging oxygen atoms O1 and O7 were replaced by two O–Hgroups, O(H)1 and O(H)7, in order to maintain the neutrality of thecharge (Fig. 5b). The y fractional atomic coordinates of the atomsbelonging to the silicon 1 tetrahedronwere shifted all by the cell param-eter b. Although this did not lead to an improvement of the analysis—because the crystal lattice remained the same—it did provide a bettervisualization of the new features of the atomic structure. Now it isclear that there are only two tetrahedra inside the monoclinic cell,around the atoms Si1 and Si3, and they are joined by the O2 oxygenatom (Fig. 5c).
At this point it was decided to perform a fundamental modificationof the definition of the fragments representing the two tetrahedra.The atomic positions of all atoms belonging to each fragment wereredefined in a local Cartesian framewhose origin was located in the po-sition occupied by the shared oxygen atom O2, rather than in the centerof the tetrahedra, occupied by the silicon atom. This modificationmadeit possible to constrain the O2 atom to be bonded to both fragments.With this new setting, the fragments are forced to move togetherthrough the refinement of the X, Y, Z coordinates of the new commonorigin, namely the O2 atom, defined again on the absolute frame ofthemonoclinic cell. No restriction is made about the rotation, which re-mains independent between the two tetrahedra. In accordancewith thenomenclature established by Richardson [9], the new organization ofthe tetrahedra and the new crystal structure, fromnowonwill be calledrespectively dimer tetrahedra and tobermorite-based dimer.
Finally, the atomic structure was modified to increase the interlayerspacing to 12.5 Å, based on Taylor's [5] results for C-S-H I. This wasaccomplishedfirst by increasing the c-cell parameter of the monocliniccell to 12.5 Å. Then, the z fractional atomic coordinates of all the atomsoutside the interlayerweremodified between atoms inside the tetrahe-dra and between the tetrahedra and the calcium layer, to preserve theirinter-atomic distances. This last modification was not applied to atomsin the interlayer because in space group B11m they are in special posi-tions and not subject to repetitions of the space group listed inTable 1. The new modified structure was called tobermorite 12.5 Åbased dimer.
A comparison of the structure before and after the modifications isshown in Fig. 6. Both structures are drawn using the same symmetryoperations of the space group B11m (Table 1). Note that the structureof tobermorite 11 Å (on the left) reflects all the features described byMerlino et al. [17].
Themodificationsmade to obtain the tobermorite 12.5 Å based dimerstarting from tobermorite 11 Å Merlino are summarized as follows:
• the monoclinic cell is expanded along the c axis from 11 Å to 12.5 Åwhile the other cell parameters are unchanged.
• the z atomic coordinates of the atoms outside the interlayer aremodified in order to keep the interatomic distances unchanged.
• the bridging tetrahedron and the Ca2 atom are deleted in order toobtain Ca/Si=1.0 .
• the silicon 1 tetrahedron is rigidly shifted by b in the y direction inorder to highlight the feature of the dimer.
The new structure was entered into MAUD, and a basic refinementof crystallite size and cell parameters was done. The results demon-strate an impressive improvement in the fit (Fig. 7a) compared to thesame stage of refinement with tobermorite 11 Å (Fig. 2b), confirmingthat the new structure is closer to the real structure of C-S-H, evenbefore refining the atomic structure. These results also validateRichardson's [9] and Taylor's [5] results on C-S-H.
Table 1General site positions of the atoms in the monoclinic cell for the space group B11m. x, yand z are the fractional atomic coordinates of the structure.
Operation number Translation along a Translation along b Translation along c
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). Transformationof 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 Si1tetrahedra of an amount equal to b for a better understanding of the paired appearance of the atomic structure (c).
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The refinement of the atomic structure (Fig. 7b) was performedincorporating into MAUD a simple repulsion force field between theatoms to prevent them from overlapping. The overall refinementproceeded as follows:
1. The atomic coordinates x, y, z of the atoms Ca1, Ca3 and O(W)6, aswell as the coordinates X, Y, Z of the center of the fragments andtheir Eulerian angles were refined. For atoms O(W)1, O(W)2 andO(W)3 in the interlayer, only the coordinates x and y were refinedand z was set to 0.
2. The list of previous parameterswas enlarged to include the normaliza-tion length.
Fig. 6. Comparison between tobermorite 11 Å Merlino (left) and tobermorite 12.5 Å baseddimer (right) unit cells as they appear after the application of the symmetry operation ofspace group B11m. In black calcium atoms, in gray silicon atoms, and in light-grayoxygen atoms (the bigger spheres represent water molecules as the ion O2− ).
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3. The structure was checked with the visual output produced byMAUD to verify that the interatomic distances of atoms not includedinside the fragments were respected. The refinement was led almostto convergence. At this point, the average isotropic crystallite sizerecorded was about 61.5 Å.
4. All the parameters were fixed, and the refinement of the anisotropicgrain size was performed using the Popa model which uses a spheri-cal harmonic expansion [25]. The last refined value of isotropic grainsize was used as a starting value for the first coefficient of the spheri-cal harmonic, whereas the other coefficients were refined incremen-tally. It was necessary to fix all other parameters with the exceptionof the anisotropic size parameters, because refining the structure at
Fig. 7. Plotting of the experimental spectra of C-S-H 1.0 (dots) and calculated spectra (continuoutobermorite 11 Å plotted in Fig. 2b (a) and at the final stage of the refinement; (b), Ca/Si ratio=with Ca2 on the interlayer at the final stage of the refinement.
the same time resulted in a divergence to a wrong structure, wherefragments separated from the calcium interlayer andO(W)6 separatedfrom Ca1.
5. After convergence of the grain shape was reached, the crystalstructure 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 parameterswas done. Then, the occupancies of atoms whose refined value wasbigger than one were fixed to one, while the others were refined.The occupancy of atoms belonging to the same fragment was boundto a unique value to preserve the atomic coordination.
This last stepwas performedwith two goals inmind: (1) to improvethe fit through the addition of another refinable parameter; and (2) tosimulate the disorder inside the structure, a relevant issue for nano-structured or amorphousmaterials. Even nanostructuredmaterials con-tain awide amorphous domain along grain boundaries. Furthermore, aspointed out by Egami and Billinge [18], every X-ray diffraction method,including the Rietveld method, is only capable of describing the struc-ture of crystalline long-range ordered materials. Because X-ray diffrac-tion is based on Bragg's law, it can be applied rigorously only underthe assumption that the crystal lattice is a perfect periodic repetitionof the unit cell in three-dimensional space. Therefore the solution to re-fine the occupancy in the last step of the analysis can be viewed as anattempt to extend the use of the Rietveld method to the study ofnanocrystals, until now a field explored predominantly through theuse of PDF. Note that the temperature factor was not refined.
The resulting refinement is shown in Fig. 7b. The calculated spectrumfits almost perfectly the experimental spectrum of C-S-H and reliable in-formation about structure and microstructure of C-S-H can be summa-rized [5]:
• The good fit of the shape of the peak allows to obtain the dimensionand the shape of crystallites.
• The fit of peak positions gives the new cell parameters.• The fit of the intensity of the peaks results in determining the structurefactor.
s line) of tobermorite 12.5 Å b–d (based dimer) at the same degree of refinement than the1.08. Plot of calculated and experimental spectra of tobermorite 12.5 Å b–d (based dimer)
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 refinementof the atomic structure.
Table 2Results of the refinement of crystal structure and nanostructure of tobermorite 12.5 Åbased dimer.
Sample C-S-H 1.0
Empirical formula Ca8Si7.4O18.5(OH)7.4·7.8H2OCrystal system MonoclinicSpace group B11mUnit cell dimension a=6.69(1)Å
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• Although each broad diffraction peak is the sum of many individualBragg reflections (indicated Fig. 7b), nevertheless many structural de-tails can be resolved.
3.1.2. Results of refinement
The results of the refinement of the structure are listed in Tables 2and 3. The cell parameters are not very different from the startingvalues. This means that the structure of tobermorite 11 Å was a goodstarting model and, increasing the interlayer distance in accordancewith Taylor, was appropriate. Table 3 gives the atomic structure at theend of the refinement process. The atoms can be divided into threegroups: the calcium layer, water interlayer and tetrahedral fragments.For the atoms in the calcium layer, all the atomic coordinates were setfree during the refinement; for the water molecules in the interlayeronly the coordinates x and y were refined. The refinement containingthe fragments led to different results: the coordinates of the atomsinside the polyhedra were fixed and the atomic positions could becalculated only indirectly using the refined value of the origin ofthe local frame tied to the fragment as well as the Euler angles andthe normalized length.
The occupancies of the atoms Ca1, Ca3 and O(W)6 were set equalto one after observing that during the refinement the value divergedbeyondonewhich has nophysicalmeaning.With regard to themoleculesof water O(W)2 and O(W)3, although their occupancy was set free,however, it did not shift from the initial value of 0.5.
Fig. 8 shows the structure of tobermorite 12.5 Å based-dimer at thebeginning and at the end of the refinement process. While the quality
Table 3Refined tobermorite 12.5 Å based dimer structure. *Fixed temperature factor (not refined).
of the fit demonstrates an improvement of the structure, the applica-tion of constraints maintained the multilayered feature of the overallstructure. In particular, the distance between fragments is maintainedthrough the bond in correspondence with the oxygen atom O2 (seeFig. 9). Note that after refinement, the atoms are still grouped togetherwithin the three groups ofwater interlayer, calcium layer, and fragments;no constraints were applied to maintain the distances between Ca1–Ca3,and Ca1–O(W)6. The fragments–calcium layer, fragments–water inter-layer and calcium layer–water interlayer remained unchanged.
The position of water molecules O(W)2 and O(W)3 changed muchmore than any other atomic position. In this class of multilayeredmaterial, the water molecules in the interlayer are bonded throughvan der Waals interactions that are weaker than the covalent-ionicbonding existing in the rest of the structure. Correspondingly thewater molecules moved by a larger amount, which has a physicalmeaning, further confirming the reliability of the structure and theconsistency of themethod. After refinement the position of themoleculesO(W)2 and O(W)3 converged almost to the same position. Consideringthat the occupancy for both the molecules is 0.5 and the pronouncedthermal vibration of water, they are supposed to be a single water mole-cule, as confirmed when the occupancy of the two molecules did notchange from their initial value of 0.5 during the refinement process.
Fig. 9. Detail of rotation of tetrahedra Si1 and Si3 before the refinement of the atomicstructure (transparent) and after the refinement (solid).
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 andtilting angles, together with a refinement of background, instrumental intensity (scale factor), isotropic grain size, and cell parameters. Inset: Two-dimensional diffraction imageof C-S-H 1.5 aged 4 months.
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The effectiveness of the improvedmethod to define the fragments isdemonstrated yet again by the fact that they remained bonded togetherthrough the oxygen bridging atom O2, even though they changed theirposition. Likewise, it is interesting to observe that the overall orienta-tion of the fragment changed only slightly (Fig. 9), even if the value ofthe new Eulerian angles of the tetrahedron Si1 (not shown here)changed by a large amount. This is due to how the three Eulerian anglesare defined:
• First a rotation around the z axis (roll)• Second a rotation around the new x′ axis (pitch)• Third a rotation around the new z″ axis (yaw).
Finally, the dimension of the tetrahedra also changed slightly,allowing for the maintenance of the interatomic distance betweenthe vertices of the fragments and the other atoms in the unit cell.
In the Popa model, the average crystallite is approximated by anellipsoid. For each reflection hkl, the radius of the crystallite along thedirection [hkl] Dhkl is calculated. The multilayered feature of nanostruc-ture is confirmed by the fact that the dimension along the direction ofthe stacking of the layers, i.e. the c axis of the unit cell, is shortest(D001=29.2 Å) compared with the other dimensions. In addition, themaximum dimension of the grain (D−460=74.4 Å) is parallel to theplane of the interlayer, i.e. the plane z=0. Tobermorite crystallites usuallyoccur as platelets elongated along the b axis (the direction [010]). It isinteresting to observe that this is not the case here since we haveD010=69.4 Å. There could be two explanations for this discrepancy.The first is that, since the tetrahedral chains are now substituted by di-mers, the crystallite does not develop anymore along the direction ofthe chains, while in tobermorite they are aligned with the b axis. Thesecond is that, at the end of the fragment refinement, the direction of
Fig. 11. (a) Bridging calcium atom coordinatedwith oxygen atoms andwatermolecules as extrdimer with calcium bridging atom.
the dimer is changed from the initial condition in which the dimersare aligned with b. However, the dimensions of the crystallite must beinterpreted with some caution for two reasons: first, they representonly average values; and second, the large amount of disordered grainboundary material, typical of all nanostructured materials, does notallow for defining an accurate crystallite shape.
3.2. C-S-H 1.5
3.2.1. Structural modificationsBecause modeling C-S-H with a calcium to silica ratio of 1.0 was so
successful, the same method was applied to a sample C-S-H with acalcium to silica ratio of 1.5, aged for 4 months.
The two-dimensional diffraction image of the sample, after sub-traction of the scattering from the sample holder (Fig. 10 inset) issimilar to the CSH 1.0 sample, with the same nanostructural features(Fig. 2a inset). The one-dimensional spectrum (Fig. 10) is also similarto the one from C-S-H 1.0, although some differences in the intensityof the peaks between 2.5° and 4.0° indicate a change in the atomicstructure.
The analysis performed on the CSH 1.5 sample was similar to thatperformed on the CSH 1.0 sample. The first step was to begin withtobermorite 11 Å and a refinement of the image parameters, instru-mental intensity, background, isotropic grain size, and cell parametersuntil convergence was almost reached (Fig. 10). Again, it can be seenthat, although the structure of tobermorite 11 Å was a reasonablestarting point, additional refinement of the atomic positions wasnecessary. Other modifications of the atomic structure were alsonecessary in order to satisfy the Ca/Si ratio of this sample, which wasdouble compared to tobermorite 11 Å. Richardson [9] proposed apossible
apolated from Richardson [9]; (b) atomic structures of tobermorite 13 Å silicon tetrahedra
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qualitative structure of C-S-H 1.5 where the bridging tetrahedra in thechain is replaced by calcium atoms, coordinated with oxygen atoms andwater molecules. The sketch of this structure is shown in Fig. 1-III.
The new atomic structure was devised as follows:
• The polyhedron relative to the new calcium atom, marked as Ca2,was first drawn qualitatively (Fig. 11a), using the sketch proposedby Richardson [9]. Then, the interatomic distances between theatom Ca2 in the center of the polyhedron and the oxygen atoms,as well as the water molecules in the vertices of the polyhedra,were arranged qualitatively to values dictated by the atomic radiiof the different atoms.
• In place of the deleted bridging tetrahedron, the newly createdpolyhedron was placed inside the structure already constructed asa starting model for the tobermorite 12.5 Å based dimer. This wasdone without changing the location of atoms O1 and O7, which rep-resent the bridging oxygen atoms between the new polyhedron andthe tetrahedra with central atoms Si1 and Si3, respectively. Second,the water molecule O(W)6 was shared between the new atom Ca2and the atom Ca3. Finally, the water molecules at the oppositeside of the fragment (Fig. 11) were located in the water interlayertomaintain the character ofmultilayered structure, butwere replacedby molecules O(W)1 and O(W)3 (which were already present in thestructure but in different positions).
• The location of the atoms of the new polyhedron was adjusted qual-itatively to preserve the inter-atomic distance between all the atomspresent within the cell.
• The whole crystal structure was obtained using the general site posi-tions of space group B11m, which revealed an overlap between thenew Ca2 atoms. This problem was solved extending the cell alongits c axis from 25 Å to 26 Å (i.e., raising the interlayer distance to13 Å). Afinal correctionwasmade to preserve all the atomic positionsto maintain the proper inter-atomic distances.
Note that adding the atom Ca2 as bridging polyhedron, led to a Ca/Siratio of exactly 1.5, which is the same value of Ca/Si ratio of the C-S-Hsample. Using tobermorite 11 Å prevented this possibility because of
Fig. 12. Plots of the experimental spectrum of C-S-H 1.5 (crosses) and calculated spectrum (refinement than tobermorite 11 Å plotted in Fig. 10; and (b) at the final stage of the refinembroad peak at 4.4° is the last distinct reflection in the experimental pattern of C-S-H 1.5.
the presence of the calcium atom Ca2 in the interlayer (with z=0),which for the space group B11b represents a special position.
The new structure was introduced into MAUD, and the same basicrefinement used for tobermorite 11 Å was executed (Fig. 12a). Com-pared with the previous result, there is a marked improvement in thefit of all the peaks, confirming again the validity of the method.
3.2.2. Results of refinementThe refinement of the atomic position was performed as in the
previous analysiswith one exception. Here, the newbridging polyhedronwas not defined as a fragment for two reasons: (1) the initial position ofthe atoms of the polyhedron is too approximate to apply constraints thatfix their relative distance. Application of constraints at this early stageof the analysis would probably lead to wrong or unreliable results;and (2) the shape of the calcium polyhedron is less severely definedthan the shape of silicon tetrahedra.
The result of the refinement of the atomic structure is shown inFig. 12b. After the refinement procedure, the calculated spectrummatches to the experimental spectrum and the value of the residualis sufficiently low. The quality of the fit at the end of the refinementprocedure is even better when compared with the previous sample(Fig. 7b), demonstrating that the model used was an excellentstarting point.
The results of the refinement are listed in Table 4. Again, the cellparameters changed slightly. The c cell parameter, whose value wasset only qualitatively before the refinement process, increased onlyslightly. The dimension of the crystallites was slightly larger com-pared to the sample C-S-H 1.0. Considered the nanostructured natureof the material, in which nanocrystallites are composed by a smallnumber of cells, a possible explanation for this phenomenon is thatthe larger interlayer spacing leads to larger crystallites. The refine-ment process of the anisotropic shape demonstrated that in thiscase the convergence was reached after refinement of the first threePopa coefficients. Finally, the R-values for this refinement werelower than the ones obtained from the sample C-S-H 1.0, demonstratinga better fit.
continuous line) of tobermorite 13 Å with calcium bridging atom at the same degree ofent, Ca/Si ratio=1.5. Compared to Figs. 7–13, the spectrum is cut at 4.6° because the
Fig. 13. Schematic drawing of the atomic structure of tobermorite 13 Å (a) with calciumbridging atom at the end of the refinement, (b) with detail about tetrahedra; and(c) calcium bridging atom and water interlayer. The polygons before the refinementprocess of the atomic structure are transparent, whereas the solid polygons depict thenew atomic structure at the end of the refinement process.
Table 4Results of the refinement of crystal structure and nanostructure of tobermorite 13 Åcalcium bridging atom.
Sample C-S-H 1.5
Empirical formula Ca12Si8O20·15H2OCrystal system MonoclinicSpace group B11mUnit cell dimension a=6.782(5)Å
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The refined atomic positions are listed in Table 5; a sketch of thestructure before and after the refinement process is shown in Fig. 13.This time the refinement of the atomic occupancies in the last step didnot change, which is not surprising as the Ca/Si ratio was exactly thesame in the model structure as in the sample. Furthermore, the qualityof the fit prior to the last step was very good, therefore, the refinementof an additional parameter did not lead to an improvement.
After the refinement (Fig. 13), the two protonated oxygen atomsinitially called O(H)5 and O(H)6, moved into a different position. Sincethe contribution of the hydrogen atom to X-ray scattering is almostnegligible, compared to that of the oxygen, the scattering of OHgroup and water molecule is very similar, and they are routinely re-placed respectively by O2− anions. Besides, the O(H)5 group moveddown to the interlayer, which in tobermorite structure, and in generalin multi-layer ceramic materials, is occupied by water molecules andcations. 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, moved upwards.The new chemical surroundings led us to consider the group O(H)6 asa water molecule and to label it O(W)6-b. Consequently, the old watermolecule O(W)6 was labeled O(W)6-a.
Some characteristics of the structure after the refinement processare similar to what was found in the previous analysis of C-S-H 1.0:
• The atoms Ca1 and Ca3movedupwards but their interatomic distanceremained almost unchanged.
Table 5Refined tobermorite 13 Å calcium bridging atom structure.*Fixed temperature factor(not refined).
• Once again the tetrahedra Si1 and Si3 changed their position andorientation slightly.
• The position of the water atoms in the interlayer changed muchmore than the other atomic positions.
The relative positions of the tetrahedra and the new atom Ca2have changed significantly. Although the Ca2 polyhedron can no longerpossibly be considered as a bridging polyhedron, it is important to notethat, after the refinement process, the position of the atomdid notmovetowards the interlayer. This is critical in maintaining the Ca/Si ratioequal to 1.5, because the interlayer represents a special position inspace group B11m.
The refinement of anisotropic crystallite size shows some similaritieswith that of the C-S-H 1.0 sample. Again, the shortest dimension is alongthe c axis of the crystalline cell (D001=52.4 Å), while the largest dimen-sion (D−230=87.7 Å) lies on the interlayer; the crystallite has a plate-likeshape with a preferential direction of elongation. The direction of themaximum dimension [−230] is the same as in the previous sample[−460]. Note that the magnitude of the dimension of the crystallite of
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 O2through Imm2 spatial group transformation, are plotted here to highlight the dimer tetrahedra.
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C-S-H 1.5 is larger than the crystallite of C-S-H 1.0. This is expressed inthe more distinct diffraction image.
4. Orthorhombic tobermorite refinement
Fig. 14-a shows the atomic positions of the tobermorite refined byHamid [21] with space group Imm2. As in Merlino tobermorite, thisstructure is still characterized by the dreierketten silicon chains andthe calcium atomic interlayer, however the different space grouphas a higher degree of symmetry compared to that of the monoclinicgroup. This leads to several atomic superpositions and atomic occupan-cies lower than 1, especially in the tetrahedra chains, which are bothtilted by a mirror plane, parallel to the planes b–c, and displaced by anamount b/2 along b. This feature was used by Hamid to account for thediffuse peaks in the X-ray spectrum, that he attributed to a stackingdisorder of the chains along the c direction. Because of the atomic super-positions it is difficult to identify the single tetrahedral chain (as done fortheMerlino tobermorite in the previous section) Therefore, here we canonly say that the bridging tetrahedra are in correspondence with thesilicon atoms Si1 and S12, the paired tetrahedra are in correspondencewith the silicon atoms S13 and Si4, and the chains are aligned withthe direction b.
Table 6Results of the refinement of crystal structure and nanostructure of Hamid tobermoriteapplied to CSH 1.0.
Sample C-S-H 1.0
Empirical formula Ca2.25[Si2O5(OH)2]·H2OCrystal system OrthorhombicSpace group Imm2Unit cell dimension a=5.59(7)Å
Considering the results of Section 3, we performed a similar analysisfor 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/Siratio equal to one, the occupancy of the calcium atom Ca3 was setequal to zero. Fig. 14-b shows the Hamid tobermorite with the deletedbridging tetrahedra. A good fit was obtained with an anisotropic refine-ment. At the end of the analysis, the average dimension of the crystalliteD100=43.2 Å, D010=64.6 Å, and D001=24.7 Å along c.
The largest dimension is now along b because the direction of thefragments slightly deviated from the direction of the b-axis. The initialcell dimensions were a=5.586 Å, b=3.696 Å and c=22.779 Å. Asshown in Table 6, the cell dimensions a and b changed only slightly,whereas the c dimension increased significantly at the end of the refine-ment. The refinement of the atomic positions with fragments led to theatomic coordinates presented in Table 7. Note that, although the refine-ment with fragment improved the quality of the fit, the improvementwas smaller compared to that observed for the Merlino tobermorite.Fig. 15 shows the best fit of the calculated spectrum and comparesit with the experimental values (see the refined atomic structureof Table 7). It is worse than that observed for the Merlino model(Fig. 7b).
Table 7Refined Hamid tobermorite structure applied to CSH 1.0.*Fixed temperature factor.
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.
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4.2. C-S-H 1.5
Modifications of the structure as done for the Merlino tobermoritehere were not successful, which most likely because replacing thebridging tetrahedra with calcium atoms gives an unrealistic structurewhere the two calcium cations superimpose each other. However, thecalcium atoms Ca3, which is positioned between the two dimer tetra-hedra, can already be considered a bridging atom for both the dimertetrahedra Si3 and Si4. Therefore, we used again the Hamid dimerstructure [21] with the occupancy of the calcium atom Ca3 equal to0.5 in order to obtain Ca/Si=1.5. Similar to the C-S-H 1.0 analysis,the larger improvement of the fit was obtained with the anisotropicrefinement of the crystallite dimension as follows: D100=26.1 Å,D010=70.5 Å, and D001=26.9 Å. In addition, the cell dimensionspresented in Table 8 show again that the length c increased significantlyto the final value of 25.35 Å. The calculated versus the experimentalspectrum at the end of the analysis is shown in Fig. 16. Also in thiscase the refinement is not as good as that executed with the Merlinostructure (Fig. 12b) : the refined atomic coordinates are listed inTable 9. Here the initial atomic occupancy remained unchanged leadingto a Ca/Si ratio of 1.5.
5. Discussion
It iswell known that powder diffraction is suitable to study crystallinematerials. However, in 1995 Le Bail [26] refined the microstructure ofamorphous silica glass with X-ray and neutron diffraction data, basedthe crystal structure of α-carnegieite. Lutterotti et al. [27] determined
Table 8Results of the refinement of crystal structure and nanostructure of Hamid tobermoriteapplied to CSH 1.5.
Sample C-S-H 1.5
Empirical formula Ca3[Si2O5(OH)2]·H2OCrystal system OrthorhombicSpace group Imm2Unit cell dimension a=5.63(8)Å
the amorphous fraction in ceramic materials with the Rietveld method.The profile shape function utilized was a convolution of two contribu-tions, taking into account both instrumental and sample aberrations. Itis noteworthy to highlight the main differences on the simulation of anamorphous material in these two studies. Le Bail simulated the amor-phous feature of the glass using the microstrain, i.e. a very defective crys-talline structure. This canbe seen as an attempt of adapting to theRietveldanalysis a technique used in molecular dynamics simulations where,starting from a crystalline model, the amorphous diffraction pattern isobtained by distorting the structure by applying a statistical isotropicmicrostrain. Instead, Lutterotti et al. [27] fit the amorphous pattern refin-ing a nanocrystal whose dimension was of the same order of the usedcrystalline cell without considering microstrain. Although both solutionsrevealed themselves valid, they represent an average model, and for thisreason they fail to represent the exact local atomic arrangement. This isbeyond the purpose of the Rietveld refinement, and subject of studiesusing PDF analysis [18].
Unlike these investigations of truly amorphous materials, our studyconfirms a degree of long range order in C-S-H. The results made byTaylor [2–5] and later by Richardson [7–9] showed a multi-layer natureof C-S-H, in which composition and stacking pattern are close to thosefound in tobermorite. The similarity between C-S-H and the Merlinotobermorite 11A structure has been demonstrated by Skinner et al.[14], in which the loss of long range order is attributed to small grainsize. Here, we can interpret the X-ray diffraction pattern of C-S-H 1.0and 1.5 quantitatively by refining the atomic structure as well asthe anisotropic microstructure. The results of the Rietveld refinement,using tobermorite as crystal model, lead to the conclusion that C-S-Hcan be described also as a nanocrystalline ceramic material as well asa semi-amorphous gel.
Reliable information about nanostructure and atomic positionscan only be obtained in steps refining first unit cell, then polyhedralpositions and atomic positions, and finally particle shapes. It requireshigh-resolution diffraction patterns that can be obtained only withsynchrotron radiation.
The fit can be improved by the refinement of the atomic positions,respecting the coordination and bond angles typical of the [SiO4]4− tetra-hedra and their arrangement in dreierkette chains. This is done using thefragment option in MAUD that allows to group atomic polyhedra and re-fining only their position, orientation and expansion/shrinkage. Bondinggroup of atoms with fragment also permits to reduce the number ofrefinable parameters in the Rietveld analysis.
However, the results obtained have to be interpreted carefully.First, even if the material can be considered nanocrystalline, in mono-clinic tobermorite a presence of disorder in the distribution of theatoms inside the crystalline cell has been recognized by the improve-ment of the fit with the refinement of the atomic occupancies. The
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.
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same conclusion is valid also for the orthorhombic tobermorite, inwhich the disorder was already present in the structure refined byHamid [21] through a combination of occupancies lower than oneand atomic superimpositions. At this point, it is worth mentioningthat the refinement of the atomic occupancy is not a trivial step of theanalysis, because occupancy of anions and cations has to be constrainedtomaintain the coherence of the polyhedra and neutrality of the chargeafter the refinement. Second, the crystallites in the material obviouslydo not have a unique value of dimension as can be inferred from thiswork, but a distribution of sizes. Finally, modeling nanocrystallinematerial has to take into account that the nano-scale of the crystallitesleads to an enormous quantity of boundaries that represent an amor-phous region that could probably be improved in the analyticalmethod.The diffuse scattered intensity from the boundary layer cannot beneglected and may influence the refinement of the atomic position.For example the disorder inside the structure can be attributed bothto the disorder of the atoms inside the nanocrystallites and to theboundary of the crystallites, leading to two different regions with a dif-ferent scale of disorder. Since we can only simulate the disorder insidethe cell by refining the atomic occupancy, this could lead to unreliableresults. The Rietveldmethod relies on a least squaresfit of amodel func-tion with observed data. Especially in complex materials like C-S-Hgreat care is necessary tomake sure thatmodel parameters are physicallymeaningful and that the refinement converges into a trueminimum [28].We paid attention to these limitations.
It has to be pointed out that the hydration reaction that producesthe C-S-H can extend for a very long time and C-S-H is a material thatchanges with time and further studies have to be done to investigatethe effect of age on ordering patterns and crystallite size.
Table 9Refined Hamid tobermorite structure applied to CSH 1.5.*Fixed temperature factor.
1. The diffraction pattern of C-S-H with Ca/Si ratio of 1.0 and 1.5 aged4 months can be described by the crystal structure of tobermorite11 Å when the isotropic crystallite dimension is of the order of thecrystalline cell dimension. This can be considered an effectivestrategy to simulate the nanostructural character of the material.However, structural modifications are mandatory in order to obtaina model which is closer to the real structure of C-S-H and the correctCa/Si ratio, that in tobermorite 11 Å ranges from 0.67 to 0.83.
2. The best fit of the model calculation to the experimental spectrumis obtained introducing special constraints, called fragments, thatallow the refinement of atomic positions as part of a functionalblocks which maintain the length of atomic bonds and the relativeangles between atoms during the refinement. This technique hasalready been successfully applied to the Rietveld method in the re-finement of the structure of amorphous materials ab initio. In thiswork however, fragments are for the first time applied to a com-plex structure, in a way such that groups of atoms are replaced intheir original positions at an advanced stage of the refinement.
3. The refinement of crystal structures with different levels of disorderleads to different results in terms of fragment parameters and an-isotropic crystallite size. In the refinement of the ordered structure,i.e. the monoclinic one, the best fit is obtained with a strong refine-ment of fragments position and orientation, followed by the anisotrop-ic refinement of the crystallite size. On the other hand, the disorderedorthorhombic structure showed that the bestfit is obtainedwhenfirst,the anisotropic crystallite size is refined and second, a weak refine-ment of fragment parameters is done. This behavior is not surprisingif we consider that in the Hamid orthorhombic tobermorite disorderis modeled by allowing a significant overlap of atoms belonging tothe tetrahedral chains, as a result of the symmetry constraints of theorthorhombic group and the reduced cell. These results suggest thatstructural symmetry represents a constraint in the refinement offragments, howevermorework is needed to investigate the behaviorof fragments in the refinement of other complex atomic structureswith other symmetries.
4. The best fit of the diffraction pattern of C-S-H 1.0 aged 4 months isobtained by refining the atomic structure of tobermorite 11 Å withdimer tetrahedral chains in place of infinite length chain, and a raisedinterlayer spacing. The crystallite average shape is plate-like, with apreferential direction of elongation and an average (isotropic) di-mension of 54 Å.
5. The best fit of the diffraction pattern of C-S-H 1.5 aged 4 months isobtained by including a calcium atombetween the dimer tetrahedra,and a raised interlayer spacing. The crystallite average shape is again
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plate-like with a preferential direction of elongation, but the average(isotropic) dimension is 68 Å.
6. The anisotropic crystallite size refinement showed that the largestcrystallite dimension is parallel to the plane of the calcium interlayer.It is expected that the preferential direction of elongation of theplatelets corresponds to the direction of the tetrahedra chains intobermorite. This is exactlywhat happens for the orthorhombic struc-ture, whereas this is not true for the monoclinic structure. Such a re-quirement however is less strong in C-S-H since here tetrahedrachains are replaced by tetrahedra dimer.
7. In all the analyses the refinement of the atomic occupancies wasperformed in the last stage in order to improve thefit of the calculatedto experimental spectrum. In the ordered structure, this step wasinterpreted as an effective technique to simulate the structural disor-der peculiar to nanostructured and amorphous materials, like C-S-Hgel. Simulating the diffuse diffraction pattern by only decreasing thecrystallite dimension is not sufficient to model the lack of the longrange order, because this leads to a periodic repetition of perfectnanocrystallites inside the material. When the refinement ofatomic occupancy is performed instead, a repetition of disorderednanocrystallites is obtained. Note that the structures studied hereare average models and consequently they fail to describe fullythe local arrangements. The latter is the subject of investigationsby pair distribution function analyses.
8. The fit obtained with the ordered monoclinic tobermorite is betterthan that obtained with the disordered orthorhombic tobermorite,as shown by the lower values of the residual The discussion reportedin this work, however, shows that if more emphasis is given to thestructural disorder, which here is expressed mostly in terms of thelong-range order of the tetrahedra dimer, then the disordered ortho-rhombic structure may better capture the real atomic structure ofC-S-H. The novelty of our approach requires further investigationfor definitive characterization, however the good quality of fits sup-ports that both models have validity.
9. In practical applications, such as in the characterization of Portlandcement, X-ray diffractometry with Cu radiation is used. This typicallyhasmuch lower resolution comparedwith synchrotron spectra and isoften inadequate for detailed Rietveld analysis of structural features.
Acknowledgments
This publication was based on work supported in part by AwardNo. KUS-l1-004021, made by the King Abdullah University of Scienceand Technology (KAUST). The experiments at APS, beamline BESSRC11‐ID‐C was supported by the U.S. DOE, Argonne National Laboratoryunder contract number DE-AC02-06CH11357. Also thanks to Dr. ChrisBenmore for his help with the experimental measurements. We areappreciative toDr. Luca Lutterotti formodifications of theMAUD softwareand advice about the analytical procedures.
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