A comprehensive review of C-S-H empirical and computational models, their applications, and practical aspects

Cement and Concrete Research 156 (2022) 106784 Available online 7 April 2022 0008-8846/© 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). A comprehensive review of C-S-H empirical and computational models, their applications, and practical aspects Eduardo Duque-Redondo a , b , * , Patrick A. Bonnaud c , Hegoi Manzano d a School of Engineering, Newcastle University, NE1 7RU Newcastle upon Tyne, UK b Physical Chemistry Department, University of the Basque Country UPV/EHU, Aptdo. 664, 48080 Bilbao, Spain c Research Organization for Information Science and Technology, Department of Computational Science and Technology, 1-18-16 Hamamatsucho, Minato 105-0013, Tokyo, Japan d Fisika Saila, University of the Basque Country UPV/EHU, Aptdo. 664, 48080 Bilbao, Spain ARTICLE INFO Keywords: Calcium silicate hydrate Molecular models Molecular simulation Atomic structure Nanostructure ABSTRACT The C-S-H gel is an elusive material. Its variable composition and disordered nature complicate a complete characterization of its atomic structure, and the elaboration of models is key to understanding it. This work aims to review those proposed models, dividing them into empirical and computational models. After a brief description of related crystalline calcium silicate hydrates, empirical C-S-H models based on interpretation of experimental data are presented. Then, we focus on the historic development of atomistic models to study the CS-H, until the current state of the art. We describe current computational C-S-H models built from the empirical models and computer simulations. We review common applications of these computational models: the aluminum incorporation, the elastic and mechanical properties, the diffusion of water and ions in nanopores, and C-S-H/organic composites. Finally, we discuss some practical aspects of the computational models and their interpretation, as well as possible future directions. 1. Introduction Ordinary Portland cement is arguably the most consumed man-made material worldwide. As such, there is a vast amount of empirical information on the material and how to control its workability, durability, or strength [1–4]. Thus, it is surprising how much remains unknown from a fundamental point of view, especially if we compare cement to other construction materials like steel, or commonly used materials like silicon. The lack of knowledge is particularly notable at the nanoscale, where the complex, multicomponent, porous, and disordered structure of cement hydrates complicates the experimental characterization of the material. The dominant phase at that scale is the Calcium Silicate Hydrate (C-S-H) gel, an X-ray amorphous product with variable chemical composition and structure. The C-S-H gel is the glue that agglomerates other crystalline hydration products and aggregates, giving cohesion to the material, and is mainly responsible for cement's mechanical properties and durability. Plausible strategies to increase cement's performance and reduce its environmental footprint include a fine control of the C-S-H gel structure, properties, or its precipitation kinetics. Traditionally an empirical trialerror approach has been used for those ends [5–7], yet most recent studies benefit from the existing information at the atomic scale to design bottom-up synthetic approaches [8–12]. To extract valuable information from the available vast amount of experimental data, the observations must be condensed into comprehensive models consistent with as much as possible data. Many different models have been proposed so far to rationalize the experimental findings and understand the atomic structure of the C-S-H gel. Ideally, those models should be able not only to explain experimental observations but also be predictive, and help to devise a material with tailored properties. In this paper, we aim to describe briefly most of the models for the CS-H gel structure proposed in the literature. We divided them into two categories: empirical models and computational models. The first group includes suggestions made on the basis of experimental facts collected over six decades. These empirical models aimed to describe the C-S-H gel structure by comparison with related crystals like portlandite or the tobermorite family and often included stoichiometric formulas. The second group covers the more detailed atomic-scale models developed in the last decade using computational simulations. These models try to define the C-S-H gel structure with the greatest possible detail, * Corresponding author at: School of Engineering, Newcastle University, NE1 7RU Newcastle upon Tyne, UK E-mail addresses: [email protected] (E. Duque-Redondo), [email protected] (P.A. Bonnaud), [email protected] (H. Manzano). Contents lists available at ScienceDirect Cement and Concrete Research journal homepage: www.elsevier.com/locate/cemconres https://doi.org/10.1016/j.cemconres.2022.106784 Received 31 May 2021; Received in revised form 15 March 2022; Accepted 17 March 2022 Cement and Concrete Research 156 (2022) 106784 2 specifying the coordinates (spatial position) of every atom in the material. Nowadays, one group cannot be understood without the other. On the one hand, empirical models are “just” schematic representations that compile experimental information, yet they set the structural conditions and stoichiometric rules that should be fulfilled in the computational model construction. On the other hand, atomic-scale computational models based on the empirical rules go beyond a mere description and can be used to compute actual C-S-H properties, to help in the interpretation of experimental data, or even contribute with new information inaccessible to experiments. Examples of how C-S-H modeling is used to get insights into and deep understanding are provided in Section 5. The remaining paper is organized as follows: first, we will introduce briefly the crystalline phases commonly used as starting points for C-S-H model construction. Second, we will survey the empirical models proposed over the years to describe the C-S-H gel and the recent computational models. Due to the increasing interest in atomistic simulation, we will review the main applications of computational models such as the study of the aluminum incorporation into the C-S-H gel, its elasticity, and mechanical properties, the nanoconfined water within the C-S-H gel layers, the ion diffusivity, and the formation of C-S-H/organic hybrid composites. We skipped other topics of great interest like C-S-H nucleation and growth or C-S-H carbonation because much less literature is available yet. Finally, we will discuss practical aspects and our perspective on future directions. 2. Crystalline structures related to C-S-H The C-S-H gel is an X-ray amorphous material and does not present long-range order like a crystal. However, it is possible to find short-range order at the atomic scale. NMR, IR-spectroscopy, synchrotron X-ray diffraction, and SANS experiments among others [13–19], reveal the CS-H gel similarities with crystalline calcium silicate hydrates [20] like wollastonite, jennite, hillebrandite, and specially tobermorite. There is a general consensus that the C-S-H structure is similar to the tobermorite family members [21], with a considerable number of structural defects and variable stoichiometry even at low Ca/Si ratios [22]. The tobermorite supergroup is the classification of a family of minerals with a layered structure similar to clays. The family includes various members of the tobermorite group plus related minerals like plombi` erite, clinotobermorite, and riversideite. They differ in the basal spacing (related to the hydration degree) and cell symmetry. Plombi` erite, clinotobermorite, and riversideite exhibit basal spacing of 1.4, 1.1, and 0.9 nm, respectively, and have Ca/Si ratios of 5/6, with compositions: Ca 5 Si 6 O 16 (OH) 2 ⋅7H 2 O, Ca 5 Si 6 O 17 ⋅5H 2 O and Ca 5 Si 6 O 16 (OH) 2 respectively. The basal spacing in the tobermorite group is ~1.1 nm and the Ca/Si ratio is not always 5/6. The general formula of the tobermorite group is Ca 4+x (Si 6 )O 15+2x (OH) 2−2x ⋅5H 2 O, with 0 ≤x ≤1. Thus, the endmember compositions correspond to tobermorite Ca 5 Si 6 O 17 ⋅5H 2 O (x =1) and to kenotobermorite, Ca 4 Si 6 O 15 (OH) 2 ⋅5H 2 O (x =0). In general, the tobermorite supergroup members are made of layers, each of them consisting of a central calcium oxide sheet, sharing oxygens with dreierketten or wollastonite-like silicate chains at each side of it. These chains run parallel to the b crystallographic axis, and dreierketten denotes an arrangement that repeats every three silicate units. In tobermorite, two consecutive units, denoted as pairing sites, are linked to the central calcium oxide layer sharing two out of their four oxygen atoms. The third silicate unit, called bridging site, is oriented towards the interlaminar space for steric reasons [23]. The Ca atoms in the central calcium oxide sheet follow a zig-zag arrangement, with poorly symmetric 7-fold coordination to oxygen atoms [24]. This structural module, characterized by a central calcium oxide sheet with 7-fold coordination sandwiched between dreierketten or wollastonite-like chains, is known as a “complex module” [25]. In the crystal, the calcium silicate layers lie on the xy or (001) plane, while the periodicity in the z-direction comes from the stacking of layers. The interlayer space is filled by water and extra calcium ions which provide cohesion via dispersive and electrostatic interactions respectively (see Fig. 1) [26]. As mentioned before, the tobermorite supergroup comprises several polytypes: the main ones being tobermorite 0.9 nm, tobermorite 1.1 nm, and tobermorite 1.4 nm (see Fig. 1). These polytypes differ mainly in the basal spacing (the distance in the z crystallographic direction between the center of two consecutive layers), which is wider as the hydration degree increases. Thus, from tobermorite 1.4 nm, it is possible to obtain tobermorite 1.1 nm, and tobermorite 0.9 nm by heating or dehydration. Another important difference between polytypes is the interlayer crosslink in tobermorite 1.1 nm. The silicate chains from adjacent layers can be linked in the c crystallographic direction by condensation of the bridging sites, forming a double wollastonite chain with “zeolitic cavities”. The protonation and Al content of silicate chains, as well as the Ca content in the cavities, define the dehydration behavior of the 1.1 nm polytype, which is sorted into normal and anomalous varieties. Thus, the presence of interlayer cross-linkages in anomalous tobermorites restrains the shrinkage during the dehydration, maintaining their characteristic basal spacing, while the absence of interlayer Si-O-Si linkages in normal tobermorites, allows the decrease of the basal spacing and the transformation of tobermorite 1.1 nm into tobermorite 0.9 nm [27,28]. Recent studies [19,29] suggested clinotobermorite as the tobermorite family member matching the best with the C-S-H gel atomic structure.The differences between clinotobermorite and tobermorite 1.1 nm are subtle, and they tend to coexist [21]. Their composition is very similar, yet the hydroxyl groups of the bridging tetrahedra are missing in clinotobermorite, resulting in the stoichiometry Ca 5 Si 6 O 17 ×5H 2 O. Furthermore, the orientation of the top and bottom bridging tetrahedra of a given layer alternate left and right, while in tobermorite 1.1 nm they point in the same direction. This difference has been classified by naming the complex modules (see Fig. 2) as A and B for clinotobermorite and tobermorite 1.1 nm respectively. Hamid's tobermorite must be also mentioned here [30]. It was an early attempt to resolve the crystalline structure of tobermorite 1.1 nm. The basal distance is 1.117 nm, which is shorter than the one for the normal or anomalous tobermorite 1.1 nm. The stoichiometry is Ca 4.5 Si 6 O 15 (OH) 3 ×2H 2 O. In contrast to Merlino's tobermorite 1.1 nm, Hamid's model does not include interlayer cross-links between bridging silicates from adjacent layers. Like in tobermorite 0.9 nm and 1.4 nm, the silicate chains from adjacent layers are shifted in the b crystallographic direction by b/2, so the bridging sites do not face each other. The layers are also shifted in the z crystallographic axis by z/4, so they are not located in the same cleavage (see Fig. 2). Jennite description is also included here because historically it was considered as a possible model for the C-S-H gel with high Ca/Si ratios [31]. Nowadays, the models based on jennite have lost interest. Recent experiments show that high Ca/Si ratio structures can be also formed maintaining a tobermorite-like structure, and both experimental and theoretical works indicate that tobermorite structural fingerprints match with the C-S-H [32–34]. Jennite can be found combined with tobermorite in nature or can be obtained by hydrothermal synthesis [35,36]. It has a stoichiometry Ca 9 Si 6 O 18 (OH) 6 ×8H 2 O and a Ca/Si ratio of 1.55. The structures of jennite and tobermorite are related, since they both have a layered structure and wollastonite-like silicate chains. However, there are two alternated calcium oxide slabs in each layer of jennite, which allows the bridging silicates to be connected to the calcium oxide layer. In addition, the higher calcium content of jennite implies that the calcium coordination polyhedral, hexagonal in this case, are completed with hydroxyl groups. 3. Empirical C-S-H gel models In recent years there is a clear shift from empirical models to computational models, so very few additions have been done lately. Nevertheless, there are a couple of noteworthy works that should be mentioned. In this section, we will revisit the most relevant empirical models proposed to describe the C-S-H gel structure. We must mention E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 3 that there is an extensive review by Richardson in [20] discussing these models in great detail. Layered C-S-H models were first proposed in 1952 by Bernal and coworkers based on the X-ray diffraction (XRD) patterns [37,38]. First, they suggested that two types of C-S-H were obtained from cement hydration, both of them were the result of a portlandite modification via condensation of monomeric silicate anions [37]. Their general stoichiometry was Ca[SiO 2 (OH) 2 ][Ca(OH) 2 ] x , with an extra amount of water. Later, other authors also proposed models based on portlandite and monomeric silicate groups. In 1969, Shpynova et al. [39] postulated a Ca 3 (SiO 4 )(OH) 2 stoichiometry, as a result of the condensation of the 4 hydroxyl groups of an orthosilicic acid molecule with portlandite. In 1986, Grudemo et al. [40] proposed again a portlandite-silicate monomer model similar to the previous ones. However, all these models were soon deprecated due to the inconsistency with the experimental distribution of silicate groups found by 29 Si NMR measurements [41,42], in which the chains follow the rule 3n −1 with n being an integer. Also in 1952, Bernal, Jeffery, and Taylor [38] pointed for the first time to the relationship between the atomic structure of C-S-H gel and tobermorite. They analyzed by XRD the hydration product of tricalcium silicate and cement samples, concluding that “out of ten (samples) for which single crystal or oriented fiber data could be obtained, seven had one axis of 3.65 or 2 x 3.65 Å" [38], being those lengths equal to the mineral riversideite (tobermorite 0.9 nm [21]). They stressed that the marked fibrous character of the C-S-H was consistent with those minerals, yet they do not explain further details on the differences between tobermorite and the C-S-H. A few years later, in 1956, Taylor and Howison addressed the differences in composition between tobermorite and the C-S-H, especially the high Ca/Si ratio of the latter [43]. Based on the fact that the silicate chains in tobermorite have a dreierketten arrangement, they postulated that a partial omission of some bridging silicate tetrahedra and their replacement by calcium ions in the interlayer spaces could increase the Ca/Si ratio of tobermorite from 0.83 up to 1.5. [43]. This is in practice the first mention of a C-S-H model based on tobermorite with finite silicate chains, and already introduced the main motifs that nowadays are still considered the source of C-S-H disorder. In the early 60s, Kurczyk and Schwiete measured the Ca/Si ratio of the C-S-H formed in tricalcium and dicalcium silicates, obtaining values ranging from 1.80 to 1.92 [44,45]. Assuming a tobermorite-like C-S-H retaining its infinite silicate chains, they explained the discrepancy envisioning a C-S-H gel in which calcium cations and hydroxyl anions were located in the interlayer space of tobermorite. At the same time, Kantro, Brunauer, and Weiss suggested that the high Ca/Si ratio could be reached due to specific arrangements of sandwiched tobermorite layers between calcium hydroxide ones [46], while Fujii and Kondo considered that was due to a solid solution of tobermorite and portlandite [47]. However, all these models considered infinite silicate chains, which is not consistent with experimental observations. In the 1980s, Stade and Wieker designed a model based on the tobermorite structure with finite silicate chains [48,49]. They proposed a dual model: one purely dimeric, and another one mixing dimeric and polymeric chains. These models assumed full protonation of the silicate chains and were limited to the compositions studied by the authors, with Ca/Si ratios below 1.25. [48,49]. Glasser et al. [50] extended the dimeric model assuming partial deprotonation of the silicate chains and adding extra Ca ions to counterbalance the charge, reaching Ca/Si ratios up to 1.4. Based on his previous model [43], Taylor envisaged in 1986 a C-S-H gel model built by a combination of structural units of tobermorite and jennite [17,51]. According to his work, the partial deletion of some silicate bridging tetrahedra from the infinite silicate chains of tobermorite and jennite's structures results in higher Ca/Si ratios and finite silicate chains that follow the 3n −1 rule. In contrast to its previous model, the interlayer calcium remains unchanged since it is assumed that the deleted bridging tetrahedra have only one hydrogen and one of the new ends of the finite silicate chains is protonated, maintaining the electroneutrality of the system without adding extra Ca. Fig. 1. (a) Detail of the calcium silicate layer, the main structural motif in calcium silicate hydrates. (b) Detail of the dreierketten or wollastonite-like arrangement in the silicate chains, and the 7-fold coordination of Ca. (c) Atomic representation of the structure of tobermorite 0.9 nm (d) tobermorite 1.1 nm and (e) tobermorite 1.4 nm. The prisms formed by the dashed lines represent the unit cells of the polytypes of tobermorite. To facilitate the visualization of their layered structure, the unit cells have been replicated 2 times along the b-axis in all cases and 1 along the c-axis for the tobermorite 0.9 nm. Silicate atoms are shown in purple, calcium atoms in orange, and oxygen atoms in red. The interlaminar water molecules are represented by a blue ball in the molecular center of mass, and all hydrogen atoms are omitted for a better view. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 4 In the 1990s, a general model was proposed by Richardson and Groves [52–54] based on the stoichiometry of tobermorite, jennite, and portlandite. This model pictures the C-S-H gel as finite silicate chains following the 3n −1, joined to calcium hydroxide layers, to form structures that locally can resemble tobermorite, jennite, or portlandite. The C-S-H gel could be then a mixture of tobermorite-like regions and jennite-like or portlandite regions. They also introduced a variable protonation of the silicate chains and considered the presence of a variable amount of calcium and hydroxyl groups in the interlayer space to compensate for the charge imbalance [52]. The formulation of the CS-H proposed in this model includes possible substitutions or incorporations of guest ions [54,55]. They wrote a general stoichiometric formula by considering the different structural units of their model: [Ca2nHw(Si1−aRa)3n−1O9n−2]Ic+ a c(3n−1)(OH)w+n(y−2)Cany 2⋅mH2O(1) where the number of silanol groups is given by w and the degree of protonation is defined as w/n. The term in braces corresponds to the calcium and silicate layers, while the other terms are extra ions and water. R is referred to trivalent ions, such as aluminum, that substitute bridging silicon ions, I represent the monovalent or divalent cations that ensure the electroneutrality after aliovalent substitutions and c is the formal charge of these cations. This formula is extremely flexible and can virtually fit with any possible set of defects that the atomic structure of a C-S-H model based on tobermorite could have. Taylor in [17] recognized that the model of Richardson and Groves was more general than his own attempts, yet he described it as “a constitutional rather than a structural model; that is, it deals with the nature and relative proportions of the ions present but not in any detail with how these are arranged”. Nevertheless, in a series of following works Richardson focused on the discussion of structural aspects of the C-S-H (and C-A-SH) atomic structures within the original model stoichiometry for a wide range of samples and conditions [20,53,56–59]. Fig. 2. (a) Crystalline structure of the clinotobermorite and tobermorite. The arrows indicate the orientation of the bridging silicate tetrahedral. (b) Tobermorite 1.1 nm and Hamid's tobermorite. The interlayer spaces are shown to guide the eye to the differences between both structures. The colour code is the same as in Fig. 1. E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 5 Many other models and refinements have been proposed after the one of Richardson and Groves. Most of them fit within the general formulation by Richardson and Groves and describe particular structures, or structural motifs, that match with experimental observations. For instance, Cong, Kirkpatrick, and coworkers [13,41,60] proposed a CS-H model based on defective tobermorite-like structure, suggesting that jennite-like features are not present or are negligible at best. Their model was therefore limited to Ca/Si ratios below 1.5. The model proposed by Nonat et al. [61,62] was also based only on tobermorite, but unlike Cong and Kirkpatrick's model, it covers a wider range of Ca/Si ratios. It stresses that the C-S-H structure at high Ca/Si ratios can be achieved in tobermorite by introducing interlaminar calcium hydroxide and that the C-S-H may not be as disordered as previously suggested. He also classified the C-S-H in three differentiated products that form at well defined Ca/Si ratios: α -C-S-H (0.66 V Ca/Si <1), β-C-S-H (1 <Ca/Si <1.5) and γ-C-S-H (1.5 <Ca/Si <2). More recently, Gartner and Maruyama [63] used a structural model similar to the previous ones, but emphasizing the importance of the interlaminar calcium ion and its solvation shell on the formation and shrinkage of the C-S-H gel. Chen et al. and Kulik [64,65] used solubility and thermodynamic data to suggest specific compositions and structures for different Ca/Si ranges, yet compatible with the general Richardson-Groves stoichiometry. Despite the general agreement in the community with Taylor's model and Richardson's stoichiometric formulation, alternatives have been also suggested. Grutzeck proposed in 1999 a sorosilicate C-S-H model, with an initial cuspidine-like structure that may evolve to dreierketten chains by diffusion-controlled phase change [66,67]. Very recently, some authors have proposed independently a model in which tobermorite-like layers and portlandite monoor bi-layers coexist [19,34]. In their studies, they calculated the pair distribution function from high-energy X-ray experiments, Grangeon et al. for C-S-H samples with Ca/Si ratio below 1.5 [34] and Cuesta et al. for samples from alite hydration [19]. Both found that, in order to match the experimental pair distribution function, they should add the signal of portlandite layers to that of clinotobermorite. Grangeon et al. specified that the coexistence must be considered for Ca/Si ratios >1.23, and both monolayers and bilayers of portlandite seem to be present. [34]. Cuesta et al. that the portlandite monolayer should be stretched for a perfect match of the PDF. This new evidences would imply a reconsideration of the accepted tobermorite-like model, bringing back the possibility of a mixed tobermorite/portlandite C-S-H. It must be kept in mind that Richardson's stoichiometric formulation still describes the tobermorite/portlandite structures. Overall, it is clear that the atomic structure of the C-S-H is still far from being settled unequivocally. 4. Computational models of C-S-H gel This section summarizes the main achievements and milestones in the development of C-S-H gel atomistic structural models. The complexity of the problem lies in the non-crystalline atomic structure. For crystalline materials, the atomic positions can be unequivocally determined using X-ray diffraction experiments. Even if crystals often include point defects like vacancies or substitutions and line defects like dislocations, the ideal atomic structure is available as starting point [68]. That is not the case for C-S-H, and the definition of the initial structure is a problem itself. We divided the models into “early” and “current” models. The former do not reproduce some of the C-S-H more important characteristics of the gel-like, yet they can be used as idealized versions or model systems for specific purposes. The latter are more complex models that include most of the C-S-H structural information. 4.1. Early C-S-H computational models In 1996, Faucon et al. [69,70] carried out the first study of the C-S-H structure employing molecular dynamics. Their work aimed to evaluate the breaking mechanisms of the C-S-H chains and identify the sources of its structural instability. For that purpose, they built two systems based on the structure of tobermorite 1.1 nm proposed by Hamid [30] that differed in the Ca/Si ratios: 0.66 and 0.83. The Ca/Si ratio of 0.83 was achieved by the substitution of two protons of the tobermorite 1.1 nm structure by a Ca ion. At Ca/Si =0.66, no break in the silicate chains was observed when the model was relaxed at 800 K. In contrast, the substitution of 2H + by Ca 2+ to reach a Ca/Si ratio of 0.83 resulted in a partial rupture of the silicate chains due to the formation of Si-O-Ca bonds. In this work, Faucon and coworkers also analyzed the stability of Al substitution in pairing and bridging sites, which is treated in Section 5.1. A few years later, in 2002, Kalinichev and Kirkpatrick [71] employed molecular dynamics to study the adsorption mechanisms of chloride anions at the surface of different minerals. In particular, they used tobermorite 0.9 nm as a model for C-S-H gel. They used the ClayFF force field [72] to describe the metal‑oxygen interactions, which enables the mobility and flexibility of hydroxyl groups from the surface. They found very low chloride sorption capacity in tobermorite 0.9 nm and the authors suggested that the chloride sorption capacity for C-S-H should be even lower, since the simulated tobermorite model contained a full set of Si-OH surface sites, while those sites are mainly deprotonated at higher Ca/Si ratios and pH. The extra negative charge due to this deprotonation should provoke high repulsion of chloride anions. Later on, these authors employed the same approach to study in detail the structure, dynamics, and energetics of water confined in C-S-H [73]. Again, the structure of non-modified tobermorite 0.9 nm was employed as a model for C-S-H, and different pore sizes were considered, constituting the first molecular dynamics models to study C-S-H pores. The molecular dynamics simulations showed high structuring of water molecules and the development of a hydrogen bond network between the water molecules and the surface. The authors suggested that the degree of depolymerization of the silicate chains and the deprotonation of pairing Si-OH in CS-H might cause distortions on the water structure and lead to lower diffusion coefficients regarding tobermorite due to hydrogen bonding with surface dangling atoms. In 2004, Gmira et al. [74] employed empirical potentials and ab initio methods to investigate the nature of the interatomic forces as well as the elasticity in the structures of tobermorite 1.1 nm defined by Hamid [30] and Merlino [27]. Changing the basal distance for a fixed interlayer water and Ca content they identified three energy minima between 1.0 and 1.4 nm and showed that the cohesion in C-S-H is essentially due to the electrostatic forces and the iono-covalent forces caused by the Ca ions and water in the interlayer. The first attempt to build a computational C-S-H model by cutting the silicate chains of tobermorite 1.4 nm and jennite minerals was done in 2007 by Manzano et al. [75,76]. This C-S-H model consisted of a mixture of dimers, pentamers, and octamers obtained by omitting bridging tetrahedra in the infinite silicate chains of tobermorite 1.4 nm and jennite minerals. The charge defects introduced in the crystalline structure of those minerals were compensated by introducing protons in the terminal oxygen atoms of the finite silicate chains, while the water and Ca contents remained unchanged. The obtained models are defective crystals rather than disordered C-S-H, as periodicity was kept in the unit cell, yet the authors showed that they were reasonable approaches to study the dependence of the elastic properties of C-S-H gel with respect to its composition and silicate chain length. The results showed that the mechanical properties increase as the chain length grows due to the higher stability of longer silicate chains. The elastic properties of the proposed C-S-H models were considerably lower than for the perfect crystals of tobermorite 1.4 nm and jennite, and in good agreement with the experimental nanoindentation measurements once nanoporosity was taken into account by poromechanical models. Also in 2007, Dolado et al. [77] published a radically different approach to model C-S-H. Instead of using tobermorite as a C-S-H model, they performed molecular dynamics study to simulate the polymerization of silicic acids in a Ca-rich environment. They employed the Litton E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 6 and Garofalini potential for amorphous glasses [78] at very high temperatures (>1500K). They observed that the polymerization was faster at lower Ca/Si ratios and that the appearance of three-dimensional structures, like branched silicate and rings, decreases as the Ca/Si ratio increases. Thus, the presence of Ca ions not only slows down the polymerization of the silicate chains, but also enforces the formation of linear structures. Likewise, they measured the silicate chain lengths, finding that the higher the Ca/Si ratio, the shorter the silicate chains. However, the resulting structure must be considered a calcium silicate glass rather than C-S-H, as they did not observe any local order or formation of calcium silicate layers. 4.2. Current C-S-H computational models Hitherto, most C-S-H models were essentially crystalline tobermorite or slightly modified variants. They were a reasonable first approach, but some of their basic features were not compatible with the characteristics of real C-S-H. For instance, the density and Ca/Si ratio are considerably higher in C-S-H ( ρ m =2.6 g⋅cm −3 and Ca/Si =1.7 [14]) than in tobermorite ( ρ m =2.23 g⋅cm −3 and Ca/Si =0.83 [79]) and jennite ( ρ m =2.32 g⋅cm −3 and Ca/Si =1.5 [80]) minerals. For that reason, in 2009, Pellenq et al. [81] employed a new approach to develop the first disordered C-SH model. They took the unit cell of dry tobermorite 1.1 nm as a starting point to build a disordered C-S-H by introducing several modifications. —In the first step, all water molecules were deleted and the specified Ca/Si ratio, 1.7, was achieved by shortening the silicate chains by removing neutral SiO 2 groups in order to obtain a defective C-S-H structure matching the experimental Q 0 , Q 1 , and Q 2 distribution given by experimental NMR analysis [60,82]. —Ca was introduced to compensate for the charge imbalance provoked by the SiO 2 deletion. — Then, the structure was relaxed, obtaining a distorted layered structure. — To reintroduce water, Grand Canonical Monte Carlo (GCMC) simulation was performed, coupling the C-S-H structure to an external water reservoir at the chemical potential of liquid water at 300 K. Water entered into the original interlaminar space, but also in new spaces created by the omission of silicate groups, reaching a density of 2.56 g⋅cm −3 , close to the experimental one (2.6 g⋅cm −3 ) obtained by small-angle neutron scattering (SANS) measurements [14]. — Finally, the structure of the system was further relaxed under constant pressure and temperature, which caused an expansion of the interlaminar space and the density was reduced up to a final value of 2.45 g⋅cm −3 . The chemical composition of the final structure was (CaO) 1.65 (SiO 2 ) (H 2 O) 1.75 , very close to the experimental average composition [14] (CaO) 1.7 (SiO 2 )(H 2 O) 1.8 . It is remarkable that the final structure had a similar amount of water to tobermorite 1.4 nm, but in this model, water is packed randomly in the interlayer space due to the presence of defects in the silicate chains, which reduces the basal distance. The model was validated by comparison of structural and physical properties with experimental measurements obtained by X-ray absorption fine structure (EXAFS) [83], X-ray diffraction (XRD) [84], and infrared spectroscopy (IR) [13]. It must be noticed that the experimental EXAFS and XRD C-SH samples have lower Ca/Si ratios than the model (1.4 and 1.1, respectively), and the comparisons serve to prove only qualitatively the disordered nature of the C-S-H model. Furthermore, this model was used to calculate mechanical properties, such as mechanical stiffness, strength, and hydrolytic shear response, obtaining results in good agreement with the experimental nanoindentation values [85,86]. The results given by this model suggested that the C-S-H have both glass-like short-range order and crystalline features from tobermorite [81]. This model was a great step forward in the modeling of C-S-H gel, although it was never bereft of controversy due to its inconsistencies. Indeed, some authors criticized the C-S-H model proposed by Pellenq. Thomas et al. [88] highlighted that the model density was in better agreement with experiments than crystalline tobermorite, but still quite far from real C-S-H gel. Richardson [89] also criticized it because the Ca – O distances were either shorter or longer than the measured Ca – O distances in calcium silicate hydrates and related phases by X-ray diffraction (XRD). Moreover, he also pointed out that more than half of the Ca atoms were undercoordinated, bonded to less than six oxygen atoms. The main point of disagreement was the structure of the silicate chains. The model has a Q n distribution (Q 0 =13%, Q 1 =67% and Q 2 = 20%), similar to that of experimental samples (Q 0 ~10%, Q 1 ~67% and Q 2 ~23%). However, the experiments included the Q 0 sites due to the contribution of unhydrated C 3 S, and not present in C-S-H [62,90]. Despite the criticism and inconsistencies of the C-S-H model developed by Pellenq and coworkers [81], it is possibly the biggest milestone for the description of the C-S-H using molecular scale modeling. It showed that it was possible to build more realistic models than tobermorite and lead to an increasing number of studies and model improvements. For instance, in 2012, Manzano et al. [91] refined the model enabling the dissociation of water molecules confined in C-S-H micropores. For that purpose, they used a reactive force field, ReaxFF [92,93]. They did not observe water dissociation during the energy minimization at 0 K, but as soon as the system was transferred to the canonical ensemble at 300 K, the dissociation occurs, producing the ionic pair OH − -H + . The process of dissociation is very fast, taking place within the first 0.2 ns of MD simulation. The protons react only with nonbridging oxygen atoms of the silicate chains, while the hydroxyl groups coordinate to interlaminar Ca ions, forming Si-OH and Ca-OH groups. Almost half of the original water content was dissociated, becoming a constitutive part of the C-S-H gel. It should be pointed out that no other reactions took place besides the dissociation of water molecules, so the silicate monomers did not react to form longer chains. For that reason, the authors concluded that the monomers trapped during the nucleation process may remain stable. After the water dissociation, the overall density of the system increases. The authors, based on the analysis of the arrangement structure, attributed this growth to the formation of hydroxyl groups, which induces a decrease of the long-range order, leading to a configuration more disordered and favorable energetically. In 2014, Abdolhosseini Qomi et al. [87] introduced a second improvement to the C-S-H model of Pellenq et al. (see Fig. 3), which solved some inaccuracies criticized by other authors. The building scheme is similar to the original one from Pellenq's model [81], yet the amount of Q 0 sites is zero for low Ca/Si ratios, and kept close to zero for high Ca/Si ratios. They also included a molecular dynamics step with the reactive force field ReaxFF [92,93] at 500 K, allowing the dissociation of water molecules. Then, the systems are transferred to a non-reactive environment using the CSH-FF force field [94] due to the high computational cost of ReaxFF. Using this non-reactive force field, the annealing of the sample is simulated reducing the temperature from 500 to 300 K along 3 ns at ambient pressure. In this way Abdolhosseini Qomi et al. [87] built about 1500 C-S-H gel configurations with a broad range of compositions, spanning Ca/Si ratios between 1.1 and 2.1. It should be noted that for Ca/Si ratios higher than 1.5, pairing silicates are removed and some monomers are created in the C-S-H structure, although some of them condensated during the MD simulation with ReaxFF. The aim of the authors was to build a database of atomic configurations of C-S-H validated with structural and mechanical data and classify them according to three defect attributes: Ca/Si ratios and correlation distances for medium-range Si – O and Ca – O environments [87]. In this way, it is possible to establish a structure-property correlation, enabling the screening of the database for the desired properties against the defect attribute. In 2015, Kovaˇ cevi´ c et al. [95] proposed three models for the structure of C-S-H gel based on tobermorite 1.1 nm. They developed a large E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 7 Fig. 3. The refinement of Pellenq's defective tobermorite models by Qomi et al. (a) Structure of tobermorite 1.1 nm was used as a baseline to build the C-S-H models. (b) Components of tobermorite: silicate chains (above), intralayer calcium (medium), and interlayer calcium (below). Atomic structure of C-S-H models with Ca/Si ratios of (c) 1.1, (d) 1.5, and (e) 1.8. Adapted by permission from SpringerNature, [87], Copyright 2014. Fig. 4. Proposed models by Kovaˇ cevi´ c and coworkers for the structure of C-S-H gel with Ca/Si ratio 1.68, corresponding to (a) model 1, (b) model 2, and (c) model 3 (see text). The colored regions show examples of which type of site was removed in each model: blue for bridging sites, orange for dimers, and red for pairing sites. Adapted from [95], Copyright 2021, with permission from Elsevier. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 8 number of configurations with a random distribution of variable-size silica oligomers for each model. As a starting point, they employed the structure of tobermorite 1.1 nm described by Merlino et al. [27], while Pellenq-type models [81,87] employed Hamid's tobermorite 1.1 nm [30]. The three different models arise from the strategy followed to increase the Ca/Si ratio up to 1.68. This can be achieved by the addition of Ca ions and/or the removal of SiO 2 groups. In short, the three models were done by: — Model 1: Removal of bridging SiO 2 sites, controlling the length of the silicate chains to follow the 3n-1 rule and adding Ca in the interlayer space. In this way, the C-S-H model is composed of dimers and pentamers, without the presence of monomers (Fig. 4a). — Model 2: Removal of bridging SiO 2 sites +SiO 2 dimers and adding Ca in the interlayer space. This combination obeys the 3n1 rule, yet leads to areas of the calcium oxide layer exposed to the interlaminar space (Fig. 4b). —Model 3: Random deletion of bridging and pairing silicate tetrahedrons (Fig. 4c). Therefore, there are monomeric silicates in the structure of C-S-H, and the 3n-1 rule is not fulfilled. Following this procedure, they generated a great number of C-S-H structures, more than a thousand structures in total, at the same Ca/Si ratio (1.68) and for H/Si ratios from 1.3 to 2.3. All those configurations were relaxed and then, molecular dynamics simulations were performed using the ReaxFF force field in order to estimate their bulk densities and total energies. The models and their configurations do not have the same stoichiometry and they also contain different amounts of water, inserted by Grand Canonical Monte Carlo, which complicates the interpretation of the results. The calculated total energies are rescaled to make them comparable, resulting in significant energy differences between the three models. The first model exhibits the lowest energy, suggesting that it is the most probable one. Furthermore, the analysis of the bulk density of the samples shows that the first model has the highest corrected density (2.76 g⋅cm −3 for H/Si ratio of 1.8), very close to the experimental value (2.6 g⋅cm −3 ). In this way, the first model, based on the combination of the incorporation of Ca ions and random deletion of bridging silicates that forms dimers and pentamers, results in the most stable model and with a density value that matches the experimental one. The building strategy and resulting structure could be considered as the same one achieved by Pellenq-type models [81,87], correcting the presence of silicate monomers, whose presence is not supported by experimental studies [62,90]. Despite the clear conclusions of this work, other researchers have continued using models with missing pairing sites to explore the effect of the silicate chain structure on the C-S-H properties [96]. The C-S-H computational model proposed by Mohamed et al. in 2018 [97] is a radically different approach to C-S-H construction. It is based on the experimental synthesis and characterization of C-S-H samples at a wide range of Ca/Si ratios reported in [8]. From the point of view of the C-S-H atomic structure and stoichiometry, the model is not that different from the previous ones. Its main characteristic is that Ca ions are incorporated in well defined positions as bridges of terminal silicates Q 1 of the chains, rather than randomly in the interlayer space. The authors claimed that the presence of those bridging Ca, together with the hydrogen bond network, contributes to the stabilization of the C-S-H structure, which is particularly important at high Ca/Si ratios. The process to evolve from tobermorite to a more realistic structure of C-S-H starts with the deprotonation of the silanol groups of the bridging silicates, which is compensated by the incorporation of Ca(OH) + ions. Then, the bridging silicate tetrahedrons are deleted and the Fig. 5. (a) Definition and notation of motifs employed in the brick C-S-H model. (b) Example of a brick built from the motifs in (a). In this case, the brick represents the primitive unit cell of 1.4 nm tobermorite, using the motifs corresponding to the code <S ′<C7 >S′>. Adapted from [97], Copyright (2018), with permission from Elsevier. C-S-H structures proposed by Kumar et al. for the C-S-H structure with Ca/Si ratios of (c) 1.25, (d) 1.75, and (e) 2.0 viewed along the [100] direction. Adapted from [8] Copyright (2018) with permission from American Chemical Society. E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 9 electroneutrality is maintained by adding two protons, one proton and one Ca(OH) + , or one Ca +2 ion, coordinated to the bridging sites. Extra Ca(OH) 2 groups can be added to increase the Ca/Si ratio if necessary. From a computational point of view, the construction method is completely different from the previous works. Starting from tobermorite, Mohamed et al. defined a collection of basic building blocks covering a range of possible irreducible motifs within the C-S-H (Fig. 5 a). Those building blocks are then combined to form different unit cells or “bricks” (Fig. 5 b), which are checked after a DFT energy minimization, evaluating the Ca – O distances, the coordination number of Ca ions, and local charge neutrality after relaxation. The bricks that satisfy the structural checking are then combined to form a variety of C-S-H structures with different structures and/or compositions (Fig. 5 c). A variable amount of interlayer Ca(OH) 2 and water molecules can also be specified as additional bricks, but it is not very clear how these motifs are placed. The authors show how the brick-by-brick building scheme can reproduce tobermorite and jennite crystals, as well as and previous C-S-H models. It is flexible enough to create multiple structures with Ca/Si ratios from 1.0 to 2.0, with point defects, stacking faults, different protonation degrees of the silicate chains, etc. They also propose specific blocks to reproduce C-S-H surfaces, an innovative approach to study systematically C-S-H interfacial properties. In a recent paper [98] the authors extended further the collection of building blocks adding Al, expanding the available chemical space to include C-A-S-H. Since the development of the previous models, most works have focused on their application to compute nanoscale properties, yet recent studies related to structural refinements have also been done. For instance, comparing DFT simulated and experimental XRD patterns of synthetic C-S-H with a Ca/Si =1, Basquiroto de Souza et al. suggested that the C-S-H structure is less disordered than the proposed by other models, and that the stacking disorder is a key factor to take into account [99]. Although the basic pillars of the C-S-H models are well established, there is still room for improvement and refinement. 4.3. C-S-H surfaces and pore models In Section 5, we will see that a large part of the applications of atomistic C-S-H models is related to interfacial properties and nanopores. Therefore, it is relevant to review how C-S-H surface models are built. The C-S-H surface and pore models are built following the standard practices in atomistic simulations: a two dimensional slab is created by “splitting” a bulk system through a given crystallographic plane, introducing a “vacuum layer”. The plane must be chosen carefully to avoid, when possible, breaking covalent bonds, maintaining the highest possible coordination numbers, and avoiding charged slabs (see the Tasker surface classification for more details [100]). In the case of the CS-H gel, given the tobermorite-like layered structure, the researchers have adopted the practices from the clay community, introduced mainly by Kalinichev et al. [71]. Clay particles are formed by the stacking of independent layers, with variable water content and ionic composition in the interlayer space. The direction perpendicular to the layers is therefore the most obvious choice to create the surface: it is the main surface in terms of exposed area, and chemically stable due to the lack of dangling bonds. The “vacuum layer” can then be saturated with water, organic molecules, ionic solutions, etc. [101–106]. Nevertheless, there are important differences between clays and the C-S-H, due to the non-crystalline nature of the latter. In clays, the surfaces are unequivocally determined by the crystallographic structure, with no dangling bonds. In the case of the C-S-H gel, surfaces will have a random composition and structure, and will have dangling bonds from the finite silicate chains, just like the bulk. The surface (or interfacial) structure will vary depending on the pore solution chemistry and especially with the pH. However, the vast majority of reported works do not take into account this point, and the surfaces have the same structure as the bulk layers. The impact of such an approximation is difficult to evaluate, and considerably more work will be necessary. Among the few attempts done so far, Jamil et al. computed using MD simulations the surface and solid-water interfacial energies of different tobermorite planes. In agreement with the previous discussion, the (001) direction with the surface coinciding with the tobermorite layers is the most favorable one, with negligible interfacial energy [107]. However, other surfaces also present very low surface energy, especially the (100), making them relevant. In addition, the authors explore the hydroxyl group density on the surfaces and the associated approximate pH. Yet the system is tobermorite and not a C-S-H model, the concepts introduced to show that other surfaces than the (001) cannot be discarded and that the surface chemistry should be taken into account. Another interesting approach has been proposed, although not explored in detail, in the framework of the brick model [97]. The authors suggested that specific bricks could be constructed to reproduce surface states. That opens the possibility of creating surface compositions and hydroxylation states independent of the bulk composition. 5. Applications of the C-S-H computational models In this section, we review some of the most prominent applications of the C-S-H computational models. In many cases, we include paper “early models” to show the evolution of the field, even though they cannot be strictly considered C-S-H models, as we will discuss later. 5.1. Al-incorporation into the C-S-H gel The determination of aluminum preferential incorporation sites into the C-S-H gel has been probably the most successful application of atomistic simulations in cement research, as atomistic simulation has reproduced the experimental observations in most cases. It is also an interesting example of how the atomistic models have evolved from very simple cluster models to the current complex models. Briefly, experimental studies using Si-NMR and Al-NMR proved that tetra-coordinated aluminum, Al(IV), is incorporated into the C-S-H gel silicate chains in the bridging site [108–113]. As a consequence, dimers merge into pentamers and longer aluminosilicate mean chain lengths (MCL) are observed. For low Ca/(Si +Al) ratios, Al(IV) enhances interlaminar chain cross-links via condensation of bridging sites, forming zeolite-like cavities as in tobermorite 1.1 nm and clinotobermorite. The first attempt to investigate the aluminum incorporation into tobermorite was done in 1994 by Kashihara et al. [114]. They aimed to study the tobermorite 1.1 nm case using a cluster model and ab initio simulations. Likely, some methodological flaws (like charge imbalance due to the substitution) and computational power limitations did not allow a proper energy minimization, and their simulations predict the lowest substitution energy for pairing sites. Manzano et al. used an even simpler model, testing with Density Functional Theory (DFT) the Si for Al substitution energy in a 5membered silicate chain isolated from tobermorite 1.4 nm [115]. The heavy atoms (Si and Al) were frozen to maintain a dreierketten arrangement, while O and H were relaxed. The calculated aluminosilicate condensation energies, Al for Si substitution energies, and cluster stability index indicated that the bridging site is preferred over pairing and ending sites. The suggested reason was a better stabilization of the electronic structure and charge localization when Al was located in the bridging site. The pairing site and ending sites were energetically less favorable than the bridging by 4 and 21 kcal⋅mol −1 respectively. Pegado et al. also computed the substitution energies with DFT methodology, yet in a more complex model consisting of a single tobermorite layer isolated from the crystal 116. Some bridging sites from the silicate chains were removed, creating finite chains, and multiple possible Al for Si substitutions were analyzed for good statistical analysis. Essentially, they reach the same conclusion as [115], with the bridging site as the most favorable from a thermodynamic point of view, followed by the pairing and the ending sites. However, the energy E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 16 [162] investigated the mobility in the interlayer of Ca 2+ , Mg 2+ , Al 3+ , and Na + ions using ReaxFF. They found residence times for water in the hydration shell of Al 3+ and Mg 2+ ions much greater than the ones for Na + and Ca 2+ ions and diffusion coefficients that follow the order: D Na >D Ca >D Al >D Mg . Nevertheless, the results from this paper should be taken with care, since the ReaxFF parameters for Mg, Al, and Na in combination with the Ca/Si/O/H set have not been tested properly. 5.5. C-S-H hybrids and composites In pursuit of developing more sustainable building materials and their use in innovative applications, a wide variety of additives or chemical admixtures have been used. [179–185]. The intercalation of organic molecules or low dimensional materials into the C-S-H gel at the nanoscale could have a synergistic effect that results in enhanced performance of the hybrid C-S-H or composite material [186–192]. Atomistic simulations are an excellent tool to provide detailed information about the nanoscale interaction between the C-S-H gel and the different additives. As a matter of fact, MD simulations have already been used to investigate the organic-inorganic interactions in other layered materials, like clays [101,102,107,193–196] and layered double hydroxides [197]. In 2008, Sanchez and Zhang were pioneers in the atomistic modeling of C-S-H composites [193]. They studied the interfacial interactions between C-S-H, using as a model the structure of tobermorite 0.9 nm [198], with graphite and graphite functionalized with hydroxyls, carboxyl, carboxylate, carbonyl, and amine groups. They found that the binding affinity of the graphitic structures to the surface of tobermorite increases with the polarity of the functional group, enhancing the stability of the interphase and the overall strength and durability of the resulting composite. Subsequent MD studies using Pellenq-type models confirmed the significant improvement in Young's modulus and strength for functionalized graphene/C-S-H composites and, to a much lesser extent, in non-funcionalized/C-S-H composites regarding pure C-S-H [199–201]. These studies also reported that higher Ca/Si ratios may contribute to increase the tensile strength due to the formation of more Ca – O ionic bonds between the Ca ions of the C-S-H and oxygen atoms from the graphene oxides (GO) [201]. Atomistic modeling was used to study the mechanical properties of C-S-H reinforced with carbon nanotubes (CNT) inserted along the weakest direction, perpendicular to the C-S-H sheets [202]. The authors reported an increase in the ductility and significantly higher tensile strength in the perpendicular direction of silicate layers, while the compressive behavior was not substantively altered. However, they found a worsening of the shear strength in all directions for the C-S-H/ CNT composites. More recently, the use of different surfactants to disperse the CNTs into the cement matrix was also investigated [203]. Based on the interactions between CNTs and the dispersing agents in bulk water, the Gum Arabic (GA) was identified as the most suitable surfactant since only GA was able to interact with both the CNTs and the C-S-H surface. The mechanical characterization revealed that the incorporation of the CNT along with GA in a drilled hole perpendicular to the silicate chains doubles the overall strength, increases considerably the elastic, shear, and bulk moduli, and switches the nature of the material from brittle to ductile. Besides graphene and CNTs, other organic compounds have been used to improve the mechanical performance of the C-S-H gel (Fig. 9). Hou et al. investigated the intercalation of poly(acrylic acid) (PAA), poly (vinyl alcohol) (PVA), and poly(ethylene glycol) (PEG) in the interlayer space of a C-S-H model with a Ca/Si ratio of 1.3 [204–207]. They used different force fields with different results. When they used a combination of ClayFF [72] or CSH-FF [94] with CVFF [208], they observed an increase in Young's modulus, tensile strength, and failure strain, while improving the ductility. In contrast, the polymer intercalation barely affected the Young's modulus and tensile strength when ReaxFF is used [207]. The authors attributed these differences to the rupture of C – C bonds from the polymers to form C-O-Si. However, this unrealistic result can be due to a methodological flaw since the ReaxFF Ca/Si/OH set of elements is incompatible with organic parameters. In any case, further research is needed to evaluate the quality of the different FFs. The elastic response of C-S-H/PEG composites was also studied by Zhou and coworkers comparing simulations and experiments [209]. Experimental HP-XRD measurements revealed a significant increase in the bulk modulus of the C-S-H/PEG composite (K =52.7 GPa) with respect to a pure C-S-H sample (K =45.6 GPa). MD simulations reproduced the pure C-S-H value (K =45.9 GPa), yet predicted a considerable decrease for the C-S-H/PEG composite (K =34.7 GPa). The authors suggested that in the experimental samples, PEG is located in large pores within C-S-H particles and not sandwiched in the interlayer space as was assumed for the simulation model. The result points out the importance of a correct model. Continuing with polymers and organic molecules, a variety of studies have investigated the adhesion or adsorption and the interaction of organic moieties with C-S-H substrates. Overall, the results show the same qualitative results: highly-polar polymers, with carboxyl and Fig. 9. Two examples of C-S-H organic interactions. (a) C-S-H/PEG composite. A PEG chain of 40 monomers is inserted into the C-S-H interlayer space. (b)C-S-H/ APTES hybrid. The APTES forms covalent bonding with the C-S-H silicate chains. E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 17 amide groups, adsorb better in C-S-H as the Ca/Si ratio increases due to the interaction with Ca and formation of hydrogen bonds [210–212]. On the contrary, low adhesion energies were reported for polymers like polypropylene, which does not have polar groups on its structure. Polymers are also used to improve the adhesion between the C-S-H surface and other phases. For instance, Han et al. used PVA to enhance the adhesion of rubber hydrocarbon (RH) to the C-S-H [213]. According to their calculations, the adhesion energies of the PVA/C-S-H and RH/ PVA interfaces were much higher than that of the RH/C-S-H interface, suggesting that PVA is a good candidate as a bridge between the RH and the C-S-H. Thus, the presence of PVA enhances their adhesiveness and mechanical performance or rubberize cements, in agreement with the experiments, which show a significant increase in the adhesive and compressive strength for RH/PVA/C-S-H composites. Sulfur-black carbon (SBC) polymers have been used to improve the poor adhesion of concrete layers created with concrete 3D printing technology. Hosseini et al. created CSH-SBC-CSH models by sandwiching SBC polymers between C-S-H gel layers and subjected the composite to shear and tensile strains to analyze the mechanical performance [214]. The simulations show that the formation of electrostatic interactions between the SBC polymers and the Ca ions from the C-S-H surface increase the interfacial shear and tensile strengths of 50% and 88% regarding the pristine C-S-H, in the line with the experimental results obtained by three-point bending loading tests. The affinity of certain organic molecules, as gluconate, for Ca ions was exploited by Androniuk, Kalinichev, et al. to enhance the uptake of uranyl ions in C-S-H at Ca/Si ratios between 0.83 and 1.4 [215,216]. They found that gluconate is not directly bound to the C-S-H, but it creates complexes with the Ca ions from the C-S-H surface. In this way, the complexation of gluconate with Ca reduces the binding strength of these cations to the C-S-H and leaves vacant sorption sites on the surface that can be occupied by uranyl, enhancing indirectly the uranyl uptake. 6. Discussion and perspectives In this section, we will discuss several practical aspects of the current computational models, as well as future directions. 6.1. Which available C-S-H computational model is the best? First of all, we must stress that the discussed C-S-H models belong to the same family, which could be classified as “imperfect tobermorite” models. They have differences (missing pairing sites, interlayer Ca positions, disorder degree) but the very basic foundations are the same. That said, we can classify the C-S-H model construction into two main groups: the defective-tobermorite models and the brick model [97]. The first group includes Pellenq-type models [81,87] and Kovaˇ cevi´ c's model [95], which can be seen as a specific formulation for a Ca/Si =1.68. The original models as presented by the authors had important differences, but, in principle, one could reach the same C-S-H final structure using either construction method. It must be noted that we will discuss the construction method rather than the specific models presented in the original papers. For instance, Pellenq's original model had some structural features, such as silicate monomers, necessary to reach Ca/Si ratios >1.5. However, the 3n-1 rule can be fulfilled by adding Ca(OH) 2 in the interlayer space as proposed later [87,95,97] and Ca/Si up to 2 have been reported [156]. The brick construction method [97] is an elegant solution to build a vast number of random models with the desired composition by mixing the type and position of the predefined building blocks (see Table 3). The structure of a given model is easy to track by using the bricks' codes, which make them reproducible. It is possible to do a direct computation of the C-S-H model properties from those brick codes, like the Ca/Si ratio, MCL, formal charge densities, protonation degree, etc., with little effort. In addition, having a clear alphanumeric code to define structures sets the ground for machine learning applications [217–219]. Furthermore, it is possible to extend the model by adding additional and/or specific bricks for C-S-H surfaces [97] or with Al for C-A-S-H [98]. As limitations of the brick model (as it has been presented so far), we could mention that the resulting structures have a considerable structural order. The interlayer species (Ca 2+ , OH − , and water) are positioned in similar initial positions in each block, inducing some correlation and ordering in the structures. This may not be an issue in the context of C-S-H synthesized under controlled conditions [8], but may not be a general case. To encircle this possible problem a larger collection of bricks could be defined, or the bricks could be used to build the dried skeleton and water introduced randomly with a packing algorithm. Unfortunately, a practical implementation of the brick model needs a considerable programming effort, and so far it has been used only by its developers [97,98]. The defective-tobermorite model is the one generally used by researchers. Overall, this construction method is more general, as the C-SH model does not need to rely on predefined bricks. It is relatively simple and fast to build 1–5 different C-S-H samples for a particular study without coding, just using visualization software like CrystalMaker [220], VESTA [221], or Avogadro [222], and build C-S-H models for specific applications, broadening the studies to less common structural features. For instance, tobermorite 1.1 nm can be used to build C-A-S-H models with Q 3 sites bridging adjacent layers (not developed yet for the brick model). The construction method described in the original references [81,87] did not include any silanol or hydroxyl groups, which were formed later during a molecular dynamics simulation from the dissociation of water molecules [91]. This would result in a Si-OH +CaOH pair of hydroxyl groups, which is not consistent with the experimental trend [60,223]. Nevertheless, it is possible to set their number during the construction to avoid that problem. The obtained C-S-H gel structures are generally more disordered than those of the brick model, especially at high Ca/Si ratios. This is considered to be more representative of C-S-H obtained from cement or C 3 S hydration, but in some cases can lead to instabilities in the models, losing their layered structure. 6.2. How do I build a defective tobermorite model? It is relatively simple to build a defective tobermorite model following the recipes given in [81,87,157]. Tobermorite minerals are used as a bedrock for C-S-H model development. But then, which member of the family should be used to rationalize the experiments and build computational models? In principle, there is no limitation. Any of them could be a valid starting point, provided that the structural modifications needed to match the C-S-H experimental stoichiometry and structure are introduced correctly. More rigorously, the thermodynamic properties of a system in a given stable state are independent of the path followed to reach that state. As an example, we can consider a common Table 3 Summary of the discussed practical aspects and differences between the defective-tobermorite and brick-based C-S-H building methodologies. Defective tobermorite Brick model Description Usability ✓ Easy and fast to build up to 10 models without coding Mass production ✓ ✓ Capacity to build hundreds of models with a code Rigurosity ✓ Confidence in correct chemical coordination, hydroxylation degree, charge balance... Order/ Disorder Disorder Order Degree of structural order of the initially generated structures Transferability ✓ Possibility to reproduce the exact same model by other authors Flexibility ✓ Possibility to create models with a wider range of compositions and structures (for instance Q 3 sites) E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 18 step in the construction scheme of the C-S-H: the removal of the interlayer water from the initial crystalline structure and reinsertion after creating defects. In the original scheme from Pellenq [81], the defective structure without water was relaxed, with the consequent shrinkage of the basal space, and then water was introduced at the fixed volume by Grand Canonical Monte Carlo. This procedure results in the “correct” water content for a given basal distance. But it is computationally expensive. Alternatively, and given that we roughly know the experimental water content [14,41], one could directly introduce the necessary amount of water molecules using a random packing algorithm [224], and then relax the volume of the system. In both cases, the final thermodynamic state should be the same, reaching the correct water density by different paths: fixing the volume and changing the number of molecules in the first case, or fixing the number of water molecules and changing the volume in the second case. As general boundaries, the structures must fulfill Richardson's stoichiometry, the 3n-1 rule for the silicate chains, and other aspects shown in the next section. Of course, we assume that the selected computational method provides a realistic description of the chemistry, i.e. local coordination around Ca and Si, water structure, etc). Besides those points, some hints are given below: — If high Ca/Si ratios are desired, when calcium silicate layers are independent, both tobermorite 1.4 nm and Hamid's tobermorite are good choices. Tobermorite 1.4 nm is more similar to the C-S-H in terms of water content, yet its density, 2.23 g⋅cm −3 [79], is considerably lower than the experimental one, 2.60 g⋅cm −3 [14], and the basal space too large. Hamid's tobermorite is often preferred as a starting point, as the smaller basal distance and lower water content imply a higher starting density, 2.39 g⋅cm −3 [30]. Nevertheless, after modifications to reach a given stoichiometry and silicate chain structure, both Hamid's tobermorite and tobermorite 1.4 nm should ideally converge. — If C-S-H with the presence of Q 3 sites is desired (low Ca/Si or high aluminum contain), tobermorite 1.1 nm or clinotobermorite should be used to model C-S-H gel due to [225]. The differences between normal or anomalous tobermorite 1.1 nm, located at the interlayer space [28], should vanish after the necessary modifications to achieve disorder. In contrast, the differences between the complex modules A and B for clinotobermorite and tobermorite 1.1 nm (see Section 2) may persist when the defects are introduced. In this case, the structural differences are subtle, and the preference of one structure over the other is difficult to predict. 6.3. Model validation and prediction of C-S-H gel properties The reader may have noticed that we have discussed the “quality” of the models without much quantitative validation or comparison with experimental data. Despite the large number of recent publications, the lack of connection between the C-S-H atomic scale modeling and the experimental properties is still an open issue in the field. The difficulty arises from the diverse time and length scales involved in computation and experiments. Most of the experimental measurements involve large sample volumes that include important heterogeneities in the composition and structure of the C-S-H gel, pores filled (or not) with a pore solution in equilibrium with the C-S-H, etc. Therefore, the link between the experimental data about the C-S-H gel and the computational atomistic models is indirect at best. However, following our current knowledge, there are several characteristics based on experiments that can be considered as a checklist for C-S-H model validation. Some reference values are given in Fig. 10: — XRD. Any C-S-H model must be X-ray amorphous, i.e. the lack of long range crystalline order is a necessary characteristic of any CS-H model. The exception is the (002) reflection at small angles corresponding to basal distances between adjacent layers of ~10–14 Å, indicative of the well defined layers. A realistic model should have a layered structure [16,17,60]. In addition, synchrotron XRD experiments give access to Pair Distribution Functions (PDFs) which describe the short range order of the structure [19,34]. The PDFs are collected for the whole sample, and may include features that do not correspond to the bulk. Nevertheless, a general agreement with the experimental PDFs is desirable. — NMR measurements. From Si-NMR experiments it has been possible to quantify the mean silicate chain length as a function of the Ca/Si ratio, and the structure of those chains (Dreierketten arrangement and 3n-1 rule, see Section 2). A C-S-H model should in principle reflect that structure, although the silicate chain structure is still the subject of recent studies [96]. Similarly, AlNMR data must also be observed when Al is incorporated into the models. — IR and Raman spectroscopy. Spectroscopic techniques have been used to determine the relative concentration of silanol groups, hydroxyl groups linked to Ca, and water within the C-S-H gel [13,41,60]. As in the previous cases, the technique will give a global value for the whole sample, including surface groups and species dissolved in the pore solution. Nevertheless, the concentration of hydroxyl groups follows a significant trend with the Ca/ Si ratio that should be preferably followed. The same holds for the water content within bulk C-S-H as a function of the Ca/Si ratio: despite an indirect measurement in dried sample, the values were validated by estimations from SANS [14] for Ca/Si ratios of ~1.65, and should be taken into account. — Neutron Scattering. Small Angle Neutron Scattering (SANS) was used to investigate the composition, density, and size of C-S-H nanoparticles [14]. An important characteristic that C-S-H models must follow is the higher density compared to tobermorite crystals, between 2.5 and 2.65 g⋅cm 3 for the C-S-H gel compared to the ~2.2 g⋅cm 3 of tobermorite 14 Å [14]. This value was later on corroborated by 1 H-NMR relaxometry [228]. Therefore, C-S-H atomistic models would be denser than their Fig. 10. (a) Silicate Mean Chain Length as a function of the Ca/Si from [8,64,87,226,227] (b) Si-OH/Si and Ca-OH/Ca ratio as a function of the Ca/Si [13,41,60,223] (c) Water to silicon ratio as a function of the Ca/Si ratio [41]. E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 19 crystalline counterparts. On the other hand, Inelastic Neutron Scattering experiments have been used to quantify the Ca-OH/Ca ratio as a function of the Ca/Si ratio, being complementary to spectroscopic techniques [223]. 6.4. Current problems and limitations with C-S-H computational models First of all, it is important to make a clear distinction between what is a valid C-S-H atomistic description and what is not. A non-negligible number of publications name as C-S-H a fully crystalline tobermorite structure [229], or even a fully disordered glass without layered structure. Those systems, together with cluster models, may be used as oversimplified models to investigate very specific aspects of the C-S-H gel, but are not valid descriptions in most cases. To be labeled as “C-S-H model” the atomic structure should have finite silicate chains, higher Ca/Si ratios and density than tobermorite minerals, and certain disorder to avoid at least the long-range order characteristic of crystals. A second concern is that nowadays atomistic simulation studies on the C-S-H are essentially irreproducible. Usually, scarce details of the building procedure are given, and the absence of an actual file in a readable format (like the Crystallographic Information File [230] or the Protein Data Bank [231]) makes it virtually impossible to reproduce an exact model. Few works include in their S.I. fully reproducible atomic structure [8,81,95,97,98]. Therefore, the standardization of the C-S-H models used for simulations is urgent. The brick model has an intrinsic advantage in this sense. The use of predefined building blocks makes it simple to identify a C-S-H gel structure based on a unique code that specifies the constituent blocks and their order. Therefore, this unique code allows two different authors to build exactly the same model. Unfortunately, the code used to combine the bricks in the original paper [97] is not available for the community nowadays. This standardization is more complex in the case of the defective-tobermorite models since their construction involves random steps, i.e. the removal of silicate groups from the silicate chains and the insertion of counterbalancing ions, Ca, hydroxyl groups, and water molecules. Therefore, they cannot be duplicated exactly. The best option for a good research reproducibility would be to collect them in a “C-S-H structure database”. Existing models and new ones could be labeled with an ID number, so they could be used for reproducible purposes as well as new investigations. In addition, we should always be aware of the variable structure of the C-S-H when we compute its properties. Even for a specific Ca/Si ratio, there is a humongous number of possible configurations, and particular models could have large deviations from the average properties. As a fast example: if we aim to build a C-S-H model made by replicating the unit cell of tobermorite 3x3x2, and then we remove just 4 out of the available 72 bridging sites, we have more than 24 ×10 6 possible combinations! Most of the 24 million configurations above mentioned will have the point defects homogeneously distributed along the silicate chains, resulting in similar energies. But in the particular case of the 4 defects in neighboring sites, or in the same silicate chain, the structure may be energetically unfavorable. Therefore, it is necessary to understand the sensitivity of the different properties to the particular configuration, yet most publications report properties collected for a single structure. Another interesting case was discussed by Mohamed and coworkers in [97]. They revisited the C-S-H models in the S.I. of reference [95], and they found regions with considerable charge excesses within an overall neutral system. From a statistical point of view, we could expect a considerable local charge homogeneity in the C-S-H gel, with narrow deviations, and that model would represent a conformation with less weight in the final average properties. Therefore, using results from a single C-S-H model is not correct from the statistical point of view, and a compromise between computational cost and good configurational sampling should be sought. 6.5. Practical computational hints We have discussed the statistical validity of the results and the compromise between good sampling and computational cost, we think that it is interesting to give some practical hints. There are many different factors that may have influence both on the computational performance and on the simulation results. Due to the typical number of atoms in C-S-H models, above a thousand in the best case scenario, the use of ab initio methods is usually prohibited. The application of these methods is limited to compute properties inaccessible to empirical potentials in small model systems, such as NMR shifts [98,234], or for benchmarking purposes [116,234,235]. Regarding empirical potentials, there are several implementations available for the atomistic simulation of C-S-H gel, being the most common ones ClayFF [72], CSH-FF [94], CementFF [233], the Interface FF [232], and ReaxFF [236]. Despite an existing review of their capabilities and limitations [237], we lack a real exhaustive benchmark of the impact of the force field choice when C-S-H properties are computed. In addition, there is a considerable number of papers that report doubtful force field choices. For instance, ReaxFF [91,236,238] is a very flexible potential capable of studying both disordered and ordered phases and is the only force field that can reproduce chemical reactions in C-S-H. But it also has some limitations, such as its high computational cost, even 40 times slower than other FF (see Table 4). More importantly, it has not been specifically parametrized for the interaction of common elements in cement like Cl, Mg, Zn, etc. with C-S-H, and, in its current form, the C-S-H potentials are incompatible with organic compounds. In contrast, IFF [232] has been specifically parametrized to be compatible with common potentials for organic compounds and should be the natural choice for hybrid systems. ClayFF [72] is also a very common choice for modeling C-S-H, compatible with organic force fields and a wide variety of aqueous ions. However, it has limitations to reproduce some of the C-S-H properties like elasticity, as it was initially developed for clay and clay-related materials. The CSH-FF [94] potential was a reparametrization of ClayFF which makes it, in theory, more suitable to reproduce C-S-H systems. On the bad side, the IFF, ClayFF, and CSH-FF use partial charges on the atoms that were developed for particular cases, usually tobermorite, and they need to be modified from the reported values to reach electroneutrality in each C-S-H model, introducing new uncertainties in the results. The CementFF solves the issue using formal charges on atoms. Despite formal charges being a less realistic representation than partial charges, they can be used consistently over a wide range of structures. In addition, the CementFF uses a core-shell or Drude polarization scheme which proves to be necessary to reproduce accurately certain properties [239]. The cost to be paid is computational Table 4 Summary of the most employed force fields and their main characteristics. Force field Speed Charge scheme timestep (fs) Key aspects Ref ClayFF ref Partial, fixed, nonpolarizable 1 Developed for clays aqueous ions compatible with CS-H [72] CSH-FF x1 Partial, fixed, nonpolarizable 1 ClayFF reparametrized for C-S-H [94] IFF x1 Partial, fixed, nonpolarizable 1 Special focus on compatibility with organic FFs [232] ERICAFF x0.25 Formal, fixed, polarizable 0.2 C-S-H specific and polarizable [233] ReaxFF x0.025 Partial, variable, polarizable 0.1 Reactive FF [91] E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 20 performance, due to the increase in the number of particles and decrease of the integration time step. Machine learning Potentials (MLP) may be an interesting alternative in a close future. Some initial efforts have been started in the field [240,241], yet still focused on tobermorite and with a high computational cost. The computational resources is another parameter to be considered. Some tests for illustration are given in Table 4. The simulation time strongly depends on the force field, number of atoms, number of processors, and hardware architecture. The tests were made with a C-S-H model with a Ca/Si ratio of 1.65 and w/Si ratio of 1.6. The initial system had 749 atoms, and subsequent sizes were built by replicating the simulation box, up to 13,482 atoms. The simulations were done in the supercomputer of the UPV/EHU, using Xeon 2680v4 cores at 2.8 GHz. In Fig. 11 it can be seen the different performance of the FFs, and the expected simulation times with the system size using 8 cores. Considering a simulation box of ~4.5x3x5 nm with ~6500 atoms, we can simulate per day nearly 5 ns with ClayFF, CSHFF, and IFF, 1.25 ns with ERICAFF, and 0.125 ns with ReaxFF. Longer times can be achieved by using a larger number of processors, as shown in 11. The speed up with the number of cores is not linear, and depends on the FF used. For the previous example, an increase from 8 to 32 cores will translate into ~ 12 ns/day with ClayFF, CSHFF, and IFF, 3.75 ns/day with ERICAFF, and just 0.2 ns/day with ReaxFF. Those numbers are just a guidance, as the speed up depends also on the system size and on other computational details. But the orders of magnitude are clear, and they are important for the study of some properties. There is no such a thing as an “ideal simulation time”, yet the experience advises us that simulations below ns might not be properly equilibrated and are of dubious quality. Furthermore, longer simulations could be necessary for some properties. For instance, transport properties like diffusivity or viscosity, are very sensitive to the system size and to the simulation time [242], and even hundreds of ns could be necessary. Similarly, when mechanical properties are computed, the effect of the strain rate must be carefully checked to avoid unrealistic responses from the material. 6.6. What is next? Do we need more/new/better C-S-H (computational) models? If we accept the consensus on Taylor's model, with a C-S-H gel made of tobermorite-like layers with defective silicate chains and additional calcium ions in the interlayer space, current computational models are enough to cover all the possible structures and stoichiometries if they are built with care. Thus, we consider that more/better models to describe tobermorite-like C-S-H are not necessary. If on the contrary, we do not assume that Taylor's model describes the C-S-H gel, or at least not all the possible varieties of the gel, we would need to build and test new models. For instance, the C-S-H structure for low Ca/Si ratios can be easily described using Taylor's model, but high Ca/Si ratios are clearly a problem. By removing all the bridging silicate groups and compensating all the dangling siloxane groups with Ca, the maximum possible Ca/Si ratio is 1.5. In fact, recent studies based on pair distribution functions from high-energy X-ray experiments have proposed that clinotobermorite-like layers and portlandite monoor bi-layers should coexist in the C-S-H gel [19,34], setting the Ca/Si threshold to 1.23 [34]. However, Kumar and coworkers [8] used a very fine-tuned synthetic method to obtain C-S-H with Ca/Si ratios up to 2 without detecting the presence of portlandite. The difference between those structures has great implications on the C-S-H properties, especially for long-term durability, decalcification, etc. [243–245]. Computational models could be a great tool to further investigate differences between suggested C-S-H models and determine their stability [246]. Another interesting point that could be explored with computational models is the stacking faults and orientation of the C-S-H layers and silicate chains. In tobermorite, both the layers and the chains are perfectly aligned. A key aspect for the stability of a disordered material is the maximization of its entropy, or in other words, the number of possible configurations for a given thermodynamic state, and those defects would increase dramatically the entropy of the system and hence its (meta)stability. In fact, it has been already suggested that stacking faults could be necessary to reproduce the C-S-H XRD patterns. Furthermore, atomistic simulation is an appropriate tool to explore the configurational space, looking for new or exotic structures, like C-S-H nanotubes [247], C-S-H nanoparticles, and clusters based on different crystalline phases like xonotlite, etc. The next frontier in the C-S-H gel atomic structure determination must be linked to investigations of its nucleation and growth mechanism. If we understand how the solid C-S-H “particles” are formed, we would implicitly obtain very detailed information of their structure. From an experimental point of view, recent papers have explored in great detail the structure development during the heterogeneous nucleation of C-S-H and portlandite [248–250]. Atomistic simulations could be used as a complementary technique to understand the process. Despite unraveling nucleation and growth mechanisms of crystals and nanoparticles is a complex problem for atomistic simulations, recent advances in the field are opening new possibilities [251–254]. 7. Conclusions After decades of research, the C-S-H gel continues to be a thorn in the flesh in the field of cement science. Its structure, composition, nucleation and growth, and durability are recurrent research topics, and different models are used to rationalize the results. In this review, we have made a tour of the most relevant empirical and computational C-SH models proposed over the years. Taylor's tobermorite-like structure and Richardson's stoichiometry are the accepted empirical C-S-H descriptions by the community. However, many important details remain unsettled, as the presence of Fig. 11. (a) Simulated time (ns) in 1 h of real time using 8 cores as a function of the system size for different FFs. The ClayFF and CSHFF times overlap and are shown together. (b) Speed up as a function of the number of processors used for the simulation for the selected FFs. E. Duque-Redondo et al. Cement and Concrete Research 156 (2022) 106784 21 nanoportlandite at high Ca/Si ratios. Computational C-S-H models and atomistic simulations have burst onto in the last years, aiming to bring some light to these uncertainties, and explore nanoscale properties hardly accessible to experiments. We have reviewed the historic development of C-S-H computational models, from the early attempts with simple models to the current realistic disordered tobermorite-like structures. Then, we have inspected the main application of these C-SH models to investigate Aluminum incorporation to the silicate chains, C-S-H elasticity and mechanics, water and ion mobility within the C-S-H nanopores, and recent studies of C-S-H/organic systems. Finally, we discussed practical aspects of the main computational models, as well as give some hints on possible future directions. Overall, we hope that this paper will serve as a good reference guide for researchers in the field, giving a global vision of the empirical and computational C-S-H models and their common applications. CRediT authorship contribution statement Eduardo Duque-Redondo: Conceptualization, Investigation, Writing – original draft, Writing – review & editing, Supervision. Patrick A. Bonnaud: Investigation, Writing – original draft, Writing – review & editing. Hegoi Manzano: Conceptualization, Investigation, Writing – original draft, Writing – review & editing, Supervision, Project administration. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors would like to acknowledge funding from “Departamento de Educaci´ on, Política Lingüística y Cultura del Gobierno Vasco” (Grant No. IT912-16 and IT1639-22) and the technical and human support provided by the Scientific Computing Service of SGIker (UPV/EHU/ ERDF, EU). 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