PREDICTION OF THE MECHANICAL PROPERTIES OF THE MAJOR CONSTITUENT PHASES OF CEMENTITIOUS SYSTEMS BY ATOMISTIC SIMULATION

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1 PREDICTION OF THE MECHANICAL PROPERTIES OF THE MAJOR CONSTITUENT PHASES OF CEMENTITIOUS SYSTEMS BY ATOMISTIC SIMULATION H. Manzano (1), J. S. Dolado (1,2) and A. Ayuela (3) (1) NANOC-LABEIN, Spain (2) Nanostructured and Eco-efficient Materials for Construction Unit, Associated Unit LABEIN-Tecnalia/CSIC, Spain (3) Unidad de Física de Materiales, Centro Mixto CSIC-UPV/EHU, Spain Abstract Any attempt to numerically predict the mechanical performance of cementitious materials requires at certain point the knowledge of the intrinsic mechanical properties of the phases involved. Until recently there were not suitable simulation schemes to numerically calculate these nano mechanical properties, so they have been commonly obtained by experimental procedures (mainly through nanoindentation measurements). In this work we present an atomistic simulation scheme which accurately reproduces the intrinsic mechanical properties of the mayor constituent phases (either hydrated or not) present in cementitious systems. 1. INTRODUCTION The development of computational methods to describe and predict the mechanical properties of cementitious structures is of obvious practical importance, as cement-based materials are mainly used as structural materials. It is also clear that any realistic attempt to accomplish such an objective must be based on appropriate microstructure-performance relationships. It is not surprising, in fact, that our capacity to computationally estimate the elastic properties of cementitious skeletons has marched in parallel to our ability to mimic their microstructure. Up to now, and roughly speaking, two main approaches have been followed: The first one relies on writing a given elastic magnitude (i.e. bulk modulus, shear modulus, etc) in terms of pore-related variables (such as the percolation pore fraction) together with some phenomenological coefficients (the so-called structure factors) that are afterwards determined by fitting proper experimental data. As an example of this numerical procedure we can name the work of T. Voigt et al.[1] which successfully predicted the evolution of modulus of elasticity along the hydration process. Basically they simulated the development the pore 47

2 network through the 3D HYMOSTRUC code/model [2, 3] and extracted the structure factors by ultrasonic pulse velocity measurements. The second possibility is to employ finite element methods to solve the continuum linear elastic equations over Representative Elementary Volumes (REV). This approach has been employed by Haecker et al. [4] to predict the elastic moduli of Portland cement pastes as a function of degree of hydration and by S. Kamali and M. Moranville [5] to study the hydrate dissolution influence on the Young s modulus of cement pastes. In essence, both works were performed under the same two-step procedure: Firstly, the evolution of the microstructure of cement pastes was monitored by the digital approach of the CEMHYD3D code/model [6], where each voxel (the smallest division of the REV) was labelled as belonging to a single phase. Secondly, by taking as inputs the experimentally determined elastic moduli of each phase, each voxel was treated as a tri-linear finite element, and the elastic equations solved by the ELAS3D code [7]. Yet robust and efficient, the accuracy of both computational schemes depends largely on the knowledge of the intrinsic elastic properties of the cement and cement paste phases, something that, until recently, has been solely obtained by experimental procedures. Though the mechanical properties of most phases are already determined experimentally, it is fair to say that some phases (mostly the hydrated ones) are difficult to be synthesised and their experimental mechanical characterisation is clearly troublesome. Thus, for the sake of completeness, it would be interesting to count with robust computational schemes to compare to. In this scenario, atomistic simulations present themselves as a promising opportunity to provide complementary information on the intrinsic (nano?) mechanical properties of the cementitious phases. This work aims precisely to present an atomistic simulation scheme that has recently allowed us to accurately predict the structure and mechanical properties of Alite (C 3 S), Belite (C 2 S) and Portlandite (CH). We will also provide some preliminary calculations of the elastic properties of Tricalcium aluminate (C 3 A), and Ettringite (AFt). Due to its complexity, the discussion on the elastic properties of C-S-H gel is limited to those of its crystalline structural models, jennite (J) and tobermorite 14Å (T). For further details on the mechanical properties of C-S-H gel, we remit the reader to the references [8, 9], where the effect of the gel porosity and the length of the chains are thoroughly discussed. 2. COMPUTATIONAL METHODS The simulation scheme we adopt is based on the Force field [10] method. Within this method, the atoms are pictured as spheres, where their electronic nature is taken into account implicitly with some parameters, such as their charge and their radius. The interaction between atoms is described by parametrized interatomic potentials so the energy of the system can be traced back for their position. Our starting subset of interatomic potentials is the one employed in Ref. [11] to simulate a tobermorite-like structure, since these interatomic potentials account well for the interactions of Ca, Si, O and H atoms. A detailed explanation on the underlying assumptions of this scheme can be found in [8, 9, 11]. To incorporate the Aluminium and the Sulphur into the set of interatomic potentials we have employed the potentials employed in Refs. [12, 13 and 14]. More information will be presented elsewhere [15]. The simulations have been carried out with the GULP code [16]. First, the experimental data for the crystalline structures are optimized. For this purpose we allow the variation of the 48

3 unit cell parameters and the free displacements of the atoms. The symmetries are also eliminated once the full unit cell was generated to allow anisotropic variations of the lattice parameters and of the atomic coordinates even if the atoms were in specific symmetry positions. The search of the local minima follows the Newton-Rapshon procedure, with the Broyden-Fletcher-Goldfarb-Shannon (BFGS) [17] scheme to update the Hessian. Then, the elastic constant tensor was obtained from the second derivative matrix at the optimized structures. The bulk (K) and the shear (G) moduli were determined from the elastic constants, according to the averaged Hill definition [18]. From them, both the average Young s modulus (E) and the Poisson s ratio ( ) were calculated through the standard relations that assume an isotropic media. 3. ANALYSED CRYSTALLINE PHASES Alite is the most important component in cement, since it represents up to the 70% of ordinary Portland cement powder. It is tricalcium silicate (C 3 S), with ionic substitutions which modified the pure crystalline structure. Tricalcium silicate exhibits seven different polymorphs. At room temperature, the pure C 3 S phase is triclinic, although this incorporation of minor trace elements, mainly the replacement of Ca +2 by Mg +2, stabilizes the monoclinic form. As it is the stable phase present in the clinker, we have chosen the monoclinic MIII superstructure for the present study even though no incorporation of Mg ions is made [19, 20]. The monoclinic superstructure presents a complex atomic arrangement due to the disorder of the silicate tetrahedral. There are several orientations of these silicate tetrahedra, which affects to the coordination of the Ca atoms. The final coordinates employed are the proposed in [20] from a Rietveld refinement of ultra-high-resolution synchrotron X-ray powder diffraction and medium-resolution neutron powder diffraction data. C 3 S is basically built of SiO 4-4, Ca +2 and O -2 ions, where the calcium ions are partially coordinated to the silicate tetrahedra and oxygen ions, and each of the last coordinated to six Ca +2 (see Fig. 1). The second component in amount in cement powder is belite (~ 30%). Analogue to alite, belite is dicalcium silicate (C 2 S) modified in composition and crystalline structure by ionic substitutions. It presents different polymorphs at different temperatures. The stable form at room temperature is -C 2 S, although the incorporation of different ions made the -C 2 S phase persist and it is the main belite form in clinker [19]. We have taken the stable polymorph at room temperature -C 2 S although no ionic incorporation was made in the pure structure. Dicalcium silicate is formed by Ca +2 ions coordinated to SiO 4-4 groups (see Fig. 1), and like in the previous case of alite, it present some disorder in its structure [21]. Tobermorite 14Å and jennite are two natural crystalline species belonging to the family of hydrated-calcium-silicate crystals. Their importance comes from the analogy between its layered structure and the proposed organization of C-S-H gel. It is nowadays accepted that the atomic arrangement of the C-S-H gel is very close related to these natural crystals. In fact, several models have been proposed that draw the gel as a mixture of tobermorite-like and jennite-like structures with multiple defects and imperfections [22, 23]. The basic ingredient Tobermorite 14Å and jennite is its layered structure. Each layer is build up by a calcium oxide layer ribbed on by silicate chains. These silicate chains follow the called dreirketten arrangement, in which the Si-O tetrahedra repeat themselves each three units. The difference between both crystals comes from the amount of calcium in the structure. In tobermorite 14Å there is a single calcium layer between silicate chains and two in three silicate units of the 49

4 chains are intimately connected to this layer whereas the remaining one is oriented to the interlaminar space. In the case of jennite, there are two calcium layers placed between the silicate chains, with half of the calcium atoms linked to chains and the other half bonded to OH groups. The arrangement of the silicate chains is therefore different, and all the silicate tetrahedra are connected to the calcium layers (see Fig. 1). The stoichometryc formula are Ca 5 Si 6 O 16 (OH) 2 7H 2 O [24] with a Ca/Si ratio of 0.83 for tobermorite 14Å, and Ca 9 Si 6 O 18 (OH) 6 8H 2 O with Ca/Si=1.5 for jennite [25]. Tricalcium aluminate (C3A) is present in clinker in lower quantities than C3S and C2S. It represents something between 5% and 10% of the clinker [19]. Tricalcium aluminate pure structure is cubic, and it appears in clinker in admixture with ferrite crystals. Ionic incorporations may occur, mainly Na+ cations, forming orthorombic polymorphs, although we have taken the pure cubic structure. The cubic form is build of rings of six AlO4 tetrahedra, and Ca+2 cations in positions with different coordinations, some of them regular octahedric and some others irregular coordinated to 5 or 6 oxygen atoms in a range of 2.8 Å as can be appreciated in figure 1 [26]. Ettringite, C6A 3H26, is the most important AFt phase. AFt phases are the result of the hydration processes of aluminate and ferrite clinker phases in presence of gypsum. In cement paste it has a rod-like morphology and reaches length up to 10 m [19]. Ettringite structure is based on columns arranged parallel to the z axis, formed by aluminium atoms octahedrally coordinated to hydroxyl groups and alternate with three calcium atoms in a plane, sharing hydroxyl groups with aluminium and coordinated to four water molecules placed in the void space between columns. The stoichometry of the columns lead to a charge excess, neutralized by sulphate ions placed in the channels formed by these columns [27] (see Fig.1). a) b) c) d) e) f) Figure 1: Schematic representations of the crystalline structures under study. a) C 3 S view in the ac plane, b) C 2 S in the bc plane, d) tobermorite 14Å in the ac plane, d) jennite in the ac plane, e) C 3 A in the ab plane and f) ettringite in the ab plane. The same colour code has been employed for all the species. Ca is represented in orange, Si in purple, Al in blue, O in red, H in white and S in yellow. 50

5 4. RESULTS AND DISCUSSION. The cell parameters of the optimized structures are presented in the Table 1, together with the experimental values and the error percentage between both values. From this table it can be seen that there is a great agreement between the computed values and the experimental ones, with error percentages smaller than the 3% in all the cases except from the c axis of C 2 S tobermorite 14Å and ettringite. In the case of tobermorite, the expansion is attributed to the shielding effect of water, which reduces the dispersive force between silicate layers. Nevertheless, the internal structure and atomic arrangement of all the species has been checked finding no changes respect from the experimental data. Table 1: Calculated unit cell parameters of the studied crystalline species. The experimental values and the error percent are included. Alite (C 3 S) Belite ( - C 2 S) Exp this work %error Exp this work %error a (Å) b (Å) c (Å) (º) (º) (º) Tobermorite 14 Å Jennite Exp this work %error Exp this work %error a (Å) b (Å) c (Å) (º) (º) (º) Tricalcium aluminate (C 3 A) Ettringite Exp this work %error Exp this work %error a (Å) b (Å) c (Å) (º) (º) (º) Our results for the elastic properties are showed in the Table 2, as well as the experimental data when available. The results for the clinker phases C 3 S and C 2 S are in great agreement 51

6 with the experimental values. The Young moduli (E) of these phases have been determined experimentally by nanoindentation [28,29] and resonancy frecuency [28] finding values ranging from 135 to 147 and 130 to 130 GPa for C 3 S and C 2 S respectively. Our values are within those ranges with E=139 and 137 GPa. Moreover, in the case of C 3 S also the bulk (K) and shear (G) moduli have been determined by transmission of ultrasonic waves [30].The documented bulk modulus (K=105.2 GPa) is also in great agreement with our result of K=103 GPa, whereas the computed shear moduli is ~10 GPa higher than the experimental (G= 54.5 and 44.8 GPa respectively). Unlike the already commented calcium silicate structures, there is a lack of experimental data for the mechanical properties of the calcium silicate hydrate species tobermorite 14Å and jennite. However, some atomistic calculations have been performed in tobermorite-like structures with different Ca/Si ratios and water content [31, 32, 11]. For a Ca/Si ratio of 0.83, four water molecules per unit cell and 1.4 nm of basal distance, they found a bulk modulus (K) of and shear modulus (G) of 3.93 GPa. This structure is comparable to the tobermorite 14Å proposed by Bonacorsi et al [24] with the difference of the water content. The structure proposed by Bonacorsi contains 10 water molecules more per unit cell. As can be seen in Table 2 our computed values are significantly superior, with K=44.8 and G=19. These disagreements can be explained in terms of the water content. The absence of 10 water molecules per unit cell implies an empty space between the calcium silicate layers, decreasing the mechanical properties of the material. The effect of this nanoporosity is particularly strong in the z axis, this one perpendicular to the layers. The Young modulus in that direction calculated by Pellenq et al. is 5 GPa [32], in opposition to the x and y axis, which Young modulus is considerably higher (E ~ 90 GPa). In our calculations for tobermorite 14Å there is also a anisotropy between the parallel and perpendicular directions but the difference is smaller (Ex=68.8, Ey=68.9 and Ez=52.8 GPa) due to the higher content of water. The polycristalline Young modulus in our case is 49.9 GPa. From this picture it gets clear the structural character of water in the crystalline structure of tobermorite 14Å. This is supported by the self-diffusion coefficient of water in the interlaminar space calculate by Pellenq et al [32] by means of molecular dynamics simulation. They found a value about 1000 times smaller for water in tobermorite than the value for bulk water. To the best of our knowledge, there is not experimental information of the elastic properties of Jennite crystal. The results obtained in this work are showed in Table 2. The bulk and shear moduli are of the same order of magnitude for tobermorite 14Å and jennite, though a bit higher in the case of Jennite. The difference in the elastic properties between both species can be explained in terms of density and nanoporosity (see Fig. 1). Jennite is denser than tobermorite 14Å (2.24 g/cc and 2.15 g/cc respectively for the optimized structures) and the space between consecutive lamellas in the z direction is smaller. Therefore, despite their similar arrangement and composition, the more compact structure of jennite gives rise to higher mechanical properties. Finally we have displayed in Table 2 some preliminary results for the elastic properties of C3A and Aft. The computed mechanical properties of C 3 A show a promising agreement with the experimental measurements. Our calculated value for the Young modulus (E) is GPa whereas the values obtained by frequency resonance and nanoindentation are 160 GPa and 145 GPa respectively [28, 29]. Similarly, the computed poisson ratio (0.29 GP) accords well with the value (0.30 GPa) estimated in Ref. [33]. 52

7 Table 2: Bulk (K), shear (G), Young (E) moduli and Poisson ratio ( ) in GPa computed in this work. K G E exp this work exp this work exp this work exp this work Alite , , ,28 Belite , , ,30 Tobermorite 14Å ,31 Jennite ,27 Triclacium Aluminate , , , ,29 Ettringite The mechanical properties of ettringite have been recently determined by Brioullin spectroscopy at ambient conditions [34]. Our calculated poisson ratio coincides pretty well with the experimental value, ( = 0.34), but our preliminary results for the bulk, shear and Young s moduli are smaller than the experimental ones. Though they are of the same order of magnitude, it is clear that much more work is needed to explain the origin of such slight mismatch. 5. CONCLUSIONS Atomistic simulations present themselves as a promising tool to complement the information gained experimentally on the mechanical properties of cementitious phases. We have presented one computational scheme that reproduces quite well the mechanical properties of C 3 S, C 2 S, CH (and even C-S-H gel is more things are accounted for!). This simulation scheme is currently being enlarged to describe the interactions of other species such as SO 4 ions or Al and/or Fe atoms. Promising results are found for the C3A, whereas the case of Ettringite would require further analysis. ACKNOWLEDGEMENTS H. Manzano acknowledges the grant received from the Fundación Centros Tecnológicos Iñaki Goenaga. Thanks are due to the Basque Government for funding NANOMATERIALES project, under the ETORTEK programme. We also wish to acknowledge The Spanish Government for the concession of the project MONACEM (Ref- MAT ) within the framework of the Plan Nacional de Ciencia y Tecnología. Finally, the computing resources from the Supercomputation Center of Galicia (CESGA) are gratefully acknowledged. 53

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