Topological optimization of cementitious binders: Advances and challenges

Accepted Manuscript Topological optimization of cementitious binders: Advances and challenges Han Liu, Tao Du, N.M. Anoop Krishnan, Hui Li, Mathieu Bauchy PII:

S0958-9465(18)30128-8

DOI:

10.1016/j.cemconcomp.2018.08.002

Reference:

CECO 3113

To appear in:

Cement and Concrete Composites

Received Date: 3 February 2018 Revised Date:

26 May 2018

Accepted Date: 2 August 2018

Please cite this article as: H. Liu, T. Du, N.M.A. Krishnan, H. Li, M. Bauchy, Topological optimization of cementitious binders: Advances and challenges, Cement and Concrete Composites (2018), doi: 10.1016/j.cemconcomp.2018.08.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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, N. M. Anoop Krishnan

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, Hui Li

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, Mathieu Bauchy

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Physics of AmoRphous and Inorganic Solids Laboratory (PARISlab), Department of Civil and Environmental Engineering, University of California, Los Angeles, California, 90095, USA b Key Lab of Structures Dynamic Behavior and Control (Harbin Institute of Technology), Ministry of Education, 150090, Harbin, China c School of Civil Engineering, Harbin Institute of Technology, 150090, Harbin, China d Department of Civil Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India * Corresponding author: [email protected]

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Abstract

The properties of cementitious binders are controlled by their structure at different scales. However, the complexity of their disordered structure makes it challenging to elucidate the linkages between atomic and mesoscale structure and macroscopic properties. Recently, topological approaches—which capture the connectivity of the atoms or grains while filtering out less relevant structural details—have been shown to offer a powerful framework to guide the optimization of cementitious binders’ properties by tuning their internal topology at different scales. Here, we review recent advances in the topological optimization of cementitious binders at the atomic and mesoscopic scales and attempt to identify the present challenges that need to be overcome. Elucidating the topological genome of cementitious binders (i.e., how their macroscopic properties are encoded in their topology) could accelerate the optimization of existing binders or discovery of novel formations with unusual properties.

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Keywords: CSH, Calcium–Silicate–Hydrate, Topology, Molecular Dynamics, Mesoscale Simulations

1. Introduction

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Han Liu a , Tao Du

Various approaches have been proposed to overcome the environmental burden of cement and concrete [1–5], including the use of supplementary cementitious materials (SCM), chemical tuning of cement clinkers, or development of carbon-neutral binders [6–15]. Since the carbon impact of concrete is fairly proportional to the amount of ordinary portland cement (OPC) being produced, an alternative route is to reduce the OPC production. This can be achieved by optimizing existing binders—to enhance their properties (mechanical properties, durability, etc.)—so that less material is used to obtain constant or improved performances.

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Topological Optimization of Cementitious Binders: Advances and Challenges

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The main difficulty in optimizing cement binders comes from the very large parameter space that is accessible to these materials. Indeed, even if one restricts itself to considering CaO–Na2O– Al2O3–SiO2–H2O systems (each oxide being referred as C, N, A, S, and H in the following, respectively), cementitious binders can exhibit a virtually infinite number of possible compositions, i.e., with different oxides stoichiometry. Further, the thermodynamic conditions (temperature, pressure, relative humidity, etc.) can also affect the structure of binders at different scales [16–23]. This situation is further complicated by the fact that even small variations in the composition or structure of binders can greatly affect their properties [24]. For all these reasons, traditional optimization techniques relying on Edisonian “trial-and-error” approaches (e.g., 1

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In the following, we review recent advances in topological approaches applied to cementitious binders (with a focus on calcium–silicate–hydrate) and make an attempt to identify some challenges that are yet to be overcome.

2. Background

2.1 Structure of calcium–silicate–hydrate gels

Concrete, which comprises aggregates and the hydrated cement binder, is a complex multi-scale and multi-phase material. Concrete’s properties largely depend on the multi-scale structure of its matrix and on the nature of the interface among the different phases [27]. Nevertheless, many properties are (to some extent) controlled by its “glue”, the calcium–silicate–hydrate phase (CaO–SiO2–H2O or C–S–H), which holds the different phases together [27]. At the early age, C– S–H forms through a dissolution–precipitation mechanism in aqueous solution [28,29]. Following the dissolution of the cement clinker, the Si and Ca concentrations increase until the solution reaches a supersaturated state [30]. At this point, a poorly organized C–S–H phase starts to precipitate [28]. The structure and composition of the forming C–S–H phase then evolve over time. At a later stage, C–S–H consists in a multi-scale phase, which appears as a gel made of polydisperse grains of around 5 nm each [18,31–38]. Inside each grain, C–S–H takes the form of a layered atomic network that is poorly crystalline and of variable stoichiometry—with a Ca/Si molar ratio between around 1.0 and 2.0 [28,39–41].

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Inspired by recent progress in glass science, soft matter, and network theory, topological approaches offer a promising tool to identify optimal nano- and mesoscale structures for cementitious binders [25]. Such approaches are based on the idea that, despite the structural complexity of cementitious binders, only selected structural features control the macroscopic properties to the first-order. By capturing the important short-range connectivity of atoms or grains while filtering out less relevant structural details that only show a second-order effect, topological approaches can be used to simplify complex materials into simpler networks. These simpler networks are essentially similar to mechanical truss networks, which consist of nodes that are connected to each other by some constraints. Such a simplification often allows one to reformulate complex optimization problems into an analytical form that can be solved more easily and efficiently [26].

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adding various “random” fractions of SCMs to OPC) have a limited potential, which will likely not result in any further leapfrog in the field considering the previous decades of empirical optimization. Although data-driven approaches, e.g., machine learning, could be used, they critically rely on the existence of large databases of reliable and consistent data, which are presently lacking.

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Challenge #1: Despite recent advances, the atomic structure of C–S–H (and, a fortiori, of more complex C–N–A–S–H binders [29,42–47]) remains debated [24,28,44,48–53]—and the present contribution does not aim to review all the structural models that are available in the literature. Although it remains unclear whether a fully accurate model of the atomic structure of C–S–H is a required basis toward its optimization, it is certainly intriguing that one has not reached a consensus regarding the atomic structure of the most manufactured material in the world.

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2.2 Importance of network topology

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The idea of considering the atomic topology of cementitious binders as a first-order structural feature is mostly inspired by recent progress in glass science. Indeed, it is now well-established that many properties of glasses are primary controlled by their atomic connectivity, including hardness, toughness, viscosity, aging, relaxation, dissolution, or glass forming ability [54–69]. Similarly, the mechanical stability of granular or colloid materials is strongly correlated to the connectivity of the grains [70–72].

2.3 Introduction to topological constraint theory

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Topological constraint theory (TCT, or rigidity theory) offers a powerful, yet elegantly simple framework to characterize the topology of disordered networks [26,73–76]. TCT reduces complex networks into simpler structural trusses [26,73–75], thereby capturing the important atomic topology while filtering out less relevant structural details that only weakly impact macroscopic properties. In this framework, networks are described as nodes (atoms or grains) that are connected to each other through mechanical constraints. In granular materials, such constraints are the radial two-body bond-stretching (BS) grain contact points or inter-particle bonds [70]. Atomic networks also comprise BS constraints, namely, the radial chemical bonds that maintain the inter-atomic bond lengths fixed around their average value. In addition, atomic networks can also exhibit some directional angular three-body bond-bending (BB) bonds (e.g., which maintain O–Si–O angles fixed around their average value) [77,78]. As per Maxwell’s criterion, the mechanical stability of a network depends on the balance between the number of constraints and degrees of freedom [79]. Namely, as shown in Fig. 1, atomic networks can be flexible, stressed–rigid, or isostatic if the number of constraints per node (nc) is lower, larger, or equal to d, where d is the dimensionality of the network, that is, the number of degrees of freedom per node (NB: d = 3 for cementitious binders).

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Fig. 1: The three states of rigidity of a network.

Challenge #2: Although topological concepts of network stability are now well-established at the atomic scale, it remains unclear whether these ideas can be directly extrapolated to granular materials, suspensions, or colloids. Further, TCT is largely based on the assumption that all constraints are equivalent to each other, which might not be the case in mesoscale structures exhibiting a coexistence of weak and strong inter-particle interactions.

3. Atomic-scale topology 3.1 Atomic-scale topology of C–S–H upon precipitation Topological analysis (see Sec. 2.3) intrinsically require some structural inputs (coordination numbers, etc.). These inputs can be offered by experiments (e.g., nuclear magnetic resonance 3

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[52]) or atomistic simulations. In the following, we first focus on the evolution of the atomicscale topology of early-age C–S–H as it precipitates. At the early age, the kinetics of C–S–H precipitation as well as the composition, structure, and topology of the C–S–H phase that forms strongly depend on the composition of the solution (e.g., the Ca concentration) [28]. Recent advances in molecular dynamics (MD) simulations now make it possible to investigate the earlyage atomic structure of C–S–H [43,80]. In particular, reactive potentials (e.g., ReaxFF [81–83]) can now be used to simulate the chemical reactivity of silicate materials with water while remaining significantly less computationally expensive than ab initio methods [84–86]. In the following, we present some recent results regarding the role of Ca cations in controlling the kinetics of the sol-gel condensation of C–S–H in aqueous solution [87].

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First, some initial atomic structures were prepared by using the approach proposed by Cormack et al. [88]. Namely, hydrated Si(OH)4 and Ca(OH)2 complexes were periodically distributed within an initially empty cubic box with periodic boundary conditions. Water molecules were then randomly distributed in the remaining empty space. To assess the role of the Ca cations, two systems were considered: (i) a Ca-free configuration comprising 216 Si(OH)4 units and 648 water molecules and (ii) a configuration made of 216 Si(OH)4 and 108 Ca(OH)2 units, and 648 water molecules. Due to the high concentration of precursors considered herein, these initial configurations are intended to be representative of supersaturated solutions. These configurations were first relaxed in the NVT ensemble for 100 ps and the NPT ensemble for 500 ps at zero pressure. No condensation is observed at this point. To accelerate the reaction, the system is then raised to a temperature of 2000 K in the NVT ensemble for 1 ns. Note that such elevated temperatures are commonly used in atomistic simulations to accelerate the dynamics of the atoms [89]. The ReaxFF potential (using the Pitman parametrization [90,91]) and an integration timestep of 0.25 fs were used for all simulations. Fig. 2 shows a snapshot slab of the final atomic configurations obtained for the two systems.

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(a) (b) Fig. 2: Final atomic configurations of the (a) SiO2–H2O (S–H) and (b) CaO–SiO2–H2O (C–S– H) gels. Si, Ca, H, and O atoms are represented in yellow, blue, white, and red, respectively. 151 152 153 154

The time-dependent topology (i.e., connectivity) of the S–H and C–S–H gels can be assessed from the knowledge of the distribution of the Qn units—where a Qn unit is defined as a Si tetrahedral polytope that is surrounded by n bridging oxygen atom (BO, i.e., oxygen atom 4

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connected to two Si units). To this end, we analyze the local topology of Si and O atoms as a function of time by enumerating the number and types of neighbors present in the first coordination shell of each atom—wherein the limit of the first coordination shell is defined as the location of the minimum of the Si–O pair distribution after the first correlation peak. Note that, here, the initial configuration comprises only Q0 units, whereas glassy silica or quartz is made of Q4 units only. Fig. 3a-b shows the evolution of the distribution of the Qn units as a function of time for the two gels. Overall, for both systems, we observe that the fraction of isolated Q0 units quickly decreases with increasing time. In turn, the fractions of Q1, Q2, Q3, and—to a lesser extent—Q4 units tend to increase over time, before eventually reaching a plateau. The overall degree of polymerization of the gels can then be estimated from the molar ratio of the number of bridging oxygen per Si atom (BO/Si). Note that this ratio is initially equal to zero when all Si(OH)4 units are isolated from each other and would be equal to 2 in glassy silica or quartz, that is, when all O atoms are bridging. Interestingly, we observe here that the presence of Ca cations in solution results in an increase in the initial kinetics of the condensation reaction (see Fig. 3c), that is, the C–S–H gel initially exhibits a degree of polymerization that is higher than that of the Ca-free S–H gel. This observation is in agreement with previous simulations [88]. However, at a later stage, the trend is reversed and the degree of polymerization of the Ca-free S–H gel eventually becomes higher than that of the C–S–H gel. This is in agreement with the fact that Ca cations eventually become part of the silicate network and tend to depolymerize it by inducing the creation of non-bridging oxygen atoms [92].

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Challenge #3: The influence of the solution chemistry on the kinetics of precipitation and on the early-stage atomic structure of C–S–H remains poorly understood. Accessing this information experimentally is especially challenging since the amount of early-age C–S–H product is typically far too small for their structure to be analyzed [28]. As an alternative route, atomistic simulations can provide some valuable information in the early-age atomic topology of C–S–H and the precipitation kinetics thereof. However, due to their limited timescales, atomistic simulations are likely to be limited to the early-age precipitation of C–S–H gels. For instance, no layered structure silicate structure is observed to form within the present simulations.

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3.2 Atomic-scale structure of mature C–S–H

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In the following, we primarily establish our conclusions based on the model introduced by Pellenq et al. to describe the structure and topology of “mature” C–S–H [24,48]. Although fine structural details of this model have been criticized [28,49,51], broadly, and to the best of our knowledge, it remains the only model that is capable of describing C–S–H compositions across a wide range of Ca/Si molar ratios. In addition, we do not expect the overall atomic topology of C– S–H to significantly depend on the fine structural details of the chosen model. The C–S–H atomic models of Pellenq et al., with various compositions (different Ca/Si molar ratios), were obtained by introducing defects in an 11 Å tobermorite configuration [93] following a combinatorial approach [24]. This initial crystal consists of pseudo-octahedral calcium oxide sheets, surrounded on each side by silicate chains. These negatively charged calcium–silicate layers are separated from each other by both dissociated and undissociated interlayer water molecules and charge-balancing calcium cations. Starting from this structure, the Ca/Si ratio is gradually increased from 1.0 to 1.9 by randomly removing SiO2 groups. The introduced defects offer possible sites for the adsorption of extra water molecules, which was performed via the Grand Canonical Monte Carlo method, ensuring equilibrium with bulk water at constant volume and room temperature. Eventually, the ReaxFF potential, a reactive potential, was used to account for the chemical reaction of the interlayer water with the defective calcium–silicate sheets [91,94]. The use of a reactive potential allows us to observe the dissociation of water molecules into hydroxyl groups. This model has shown to predict realistic compositions, structure, mechanical, dynamical, and thermal properties for C–S–H [24,95–98]. The details of the methodology used for the preparation of the models, as well as multiple validations with respect to experimental data can be found in Ref. [24] and in previous works [35,40,41,55,95,96,98–101].

3.3 Atomic topology of C–S–H

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Rather than relying on unproven guesses regarding the connectivity of the atoms, the atomic topology of mature C–S–H can be assessed from the previously presented MD simulations, following a well-established methodology [60–63,77,78,102–107]. This method is based on the idea that topological constraints prevent any significant relative motion between atoms. In turn, a large relative motion is indicative of the absence of any underlying constraints (see Fig. 4a-b). In details, the number of radial BS constraints acting on a central atom can be assessed by calculating the radial excursion of each neighbor. As shown in Fig. 4c, we observe a clear gap between intact (low radial excursion of the 4 O atoms around each Si) and broken constraints (high radial excursions of the atoms of the second coordination shell). The limit between these two cases was found to be at around 7% of relative motion [77,100], which is fairly close to the Lindemann criterion [108]. Similarly, the number of BB constraints created by a central atom can be assessed by computing the excursion of all the angles formed by the central atom 0 and its neighbors 1, 2, 3, etc. (the neighbors are here ranked based on their respective distance from the central atom. Fig. 4d shows the excursion of the angles 102, 103, 104, 105, 106, 203, 204, etc. (with a total of 15 angles, i.e., the 15 independent angles that can be formed by the six nearest O neighbors of the central Si atom). We observe that only select angle exhibit a low excursion (namely, the 6 angles formed by the four O atoms surrounding the Si atoms). Based on previous work, the limit between intact (low angular excursion) and broken (high angular excursion) BB constraints was found to be around 13-to-15° [77,100].

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The state of rigidity (i.e., as quantified by the number of constraints per atom) can be determined via mean-field averaging. As shown in Fig. 5a, the number of constraints per atom (nc) in C–S– H decreases with the Ca/Si molar ratio, effectively separating a stressed-rigid domain (nc > 3, Sirich) from a flexible one (nc < 3, Ca-rich) [55]. Specifically, C–S–H shows a rigidity transition around Ca/Si = 1.5 [55]. This transition is associated to some structural and mechanical transitions, as C–S–H is found to be largely crystalline and transversely isotropic at low Ca/Si and fairly amorphous and isotropic at high Ca/Si [24]. One of the main interest of this method is that, once the number of topological constraints that are created by each atom is known (e.g., Si atoms create 4 BS and 5 BB constraints), nc can be analytically expressed in terms of composition without the need of any further MD simulation. That is, nc can be explicitly calculated for any CaO, SiO2, and H2O stoichiometry in C–S–H. This is depicted in Fig. 5b, which shows the predicted ternary rigidity phase diagram of C–S–H [55]. Since nc is correlated to the macroscopic properties of C–S–H (e.g., its harness [55]), this illustrates how TCT can be used to efficiently explore a given compositional envelope and pinpoint promising composition featuring optimal properties.

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Fig. 4: Illustration of the MD-based methodology used to enumerate the number of (a) BS and (b) BB constraints in C–S–H. Small and large radial (angular) excursions arise from intact and broken BS (BB) constraints, respectively. (c) Radial and (d) angular excursions of the neighbors of Si atoms in C–S–H as a function of the neighbor number (ranked based on the distance from the central Si atom. In this case, Si atoms exhibit four intact BS constraints (i.e., formed with its four O nearest neighbors).

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Challenge #4: The extrapolation of the rigidity results to a wide range of compositions (as illustrated in Fig. 5b) relies on the assumption that the number of constraints created by each atomic species does not depend on the composition of the binder. Although this is true in C–S– H, this assumption might not be valid in more complex C–A–S–H binders, as the coordination state of Al atoms is likely to change (4-, 5-, or 6-fold coordinated) depending on the presence, or not, of a charge-compensating Ca atom at its vicinity [42,92]. More generally, this highlights the fact that further atomistic simulations validated by high-resolution experiments are required to determine the atomic topology of complex cementitious binders.

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Challenge #5: More generally, although cementitious binders with targeted compositions can be studied by simulations and TCT, it remains, in practice, very challenging to finely control the composition of the binders that are experimentally synthesized. Clearly, novel protocols or manufacturing processes are required to synthesize the promising compositions that can be pinpointed by topological approaches.

3.4 Hardness optimization

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(a) (b) Fig. 5: (a) Number of constraints per atom nc in C–S–H as a function of the Ca/Si molar ratio. The inset shows the contributions of the radial (BS) and angular (BB) constraints [55]. (b) Predicted rigidity phase diagram of C–S–H [55]. The red, blue, and green circles indicate the composition of three selected C–S–H samples that are stressed-rigid, isostatic, and flexible, respectively.

The knowledge of the rigidity of C–S–H can be used as a basis to establish some composition– property predictive models. In particular, the hardness (H), which characterizes the resistance to permanent deformations upon indentation, is an important mechanical property for various materials. Although hardness is usually not directly relevant to cementitious applications, it has received much interest as the hardness of C–S–H is one of the few properties that is accessible (by nanoindentation [109]) despite the multiscale nature of cement binders. As such, this information can be used to ensure the handshake between atomistic simulations and experiments [24]. As initially proposed by Smedskjaer and Mauro [54], atomic networks first require a critical number of constraints to be cohesive, after which each additional constraint contributes to increasing hardness. This model was later refined by Bauchy et al. based on the observation that hardness primarily depends on the angular BB constraints rather than on the total number of 8

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constraints nc [55], which was later confirmed for a broad selection of glasses by Yue and Mauro [110]. The relationship between hardness and the number of angular constraints per atom was used to analytically predict the compositional dependence of the hardness of C–S–H, as shown in Fig. 6.

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Fig. 6: Predicted hardness ternary diagram of C–S–H [55].

3.5 Fracture toughness optimization

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Developing novel materials with enhanced toughness (KIc, which captures the resistance to fracture of materials) remains a “grand challenge” in materials science [111]. However, the relationship between composition and fracture toughness remains poorly understood. Fig. 7 shows computed values of KIc for C–S–H, which were obtained by subjecting pre-notched C–S– H configurations to a gradual tensile deformation (mode I fracture) and integrating the resulting stress-strain curves (see more details in Refs. [56,96]). The fracture toughness of C–S–H feature a maximum inside a window located near the isostatic composition (nc = 3) [96,112]. Note that such a window appears analogous to the “Boolchand intermediate phase” observed in many chalcogenide glasses [113–115]. As shown in Fig. 7, the maximum of KIc exhibited by isostatic systems appears to be a general feature of disordered materials, which is also observed in sodium silicate and Ge–Se glasses [112,116,117]. This has been explained by the fact that flexible networks feature the ability to resist fracture through ductile deformations, thanks to the existence of internal floppy modes. However, these systems exhibit low surface energies. In contrast, stressed-rigid networks break in a brittle way due to the existence of internal eigenstress, but have higher surface energies. Eventually, isostatic systems, which are simultaneously free of eigenstress and floppy modes, feature the best balance between surface energy and ductility [112]. As such, TCT offers a powerful framework to pinpoint optimal compositions featuring maximum fracture toughness values.

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Among many other degradation mechanisms (e.g., steel corrosion), concrete creep, i.e., the delayed deformations that can occur under constant load, is one of the main limitations of cementitious materials [118]. However, the role of the atomic structure in controlling the propensity of materials for creep remains poorly understood. When subjected to a constant load, C–S–H has been noted to show a logarithmic creep, whose propensity is captured by the creep compliance (J) [119,120]. Fig. 8 shows the computed creep compliance of C–S–H with respect to the number of constraints per atom [58,59]. Interestingly, the C–S–H compositions that are located at the vicinity of the isostatic threshold feature the lowest propensity for creep [58,59]. Again, this suggests that the compositional window (3 < nc < 3.2) in the C–S–H system is analogous to a Boolchand intermediate phase, as isostatic glasses have been shown to feature weak aging over time [57]. As shown in Fig. 8, the minimum of J was found to be correlated to a minimum in the dissolution rate of C–S–H, which suggests that the creep of C–S–H could occur through a dissolution-precipitation mechanism [59] or that the resistance to creep and dissolution could have a common topological origin. Creep compliance J (TPa )

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Challenge #6: As shown in Fig. 7 and Fig. 8, optimal values of fracture toughness and creep compliance are achieved within an intermediate phase located between the flexible and stressedrigid regions rather than at a fixed composition threshold. The origin of such intermediate phases has been attributed to the fact that, at the vicinity of nc = 3, disordered networks are able to selforganize to become rigid without exhibiting any internal stress, that is, to become isostatic. However, although water has been noted to play a key role in the origin of the intermediate phase in C–S–H [121], the factors that control the extent (i.e., width) of the compositional window over which self-organization occurs are unknown. Controlling the extent of the intermediate phase would allow one to extend the range of C–S–H compositions exhibiting optimal values of toughness and creep compliance.

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4.1 Mesoscale structure of C–S–H

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Challenge #7: More generally, additional studies are clearly needed to determine which properties are controlled by the topology of the atomic network of C–N–A–S–H binders and, hence, can be predicted within the framework of TCT—and which ones cannot.

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In the following, we primarily establish our conclusions based on the mesoscale model of C–S– H introduced by Masoero et al. [34,36]. Although other mesoscale models have been proposed [29,33], our goal is here to illustrate how topological analysis can be used to extract relevant structural features from complex configurations rather than to review all the mesoscale models that are available in the literature. In Masoero’s model, the C–S–H gel is represented as dense aggregates of polydisperse nanoparticles (with diameters ranging from 3 to 35 nm) that interact with each other through strong cohesive forces, which were parametrized to reproduce the cohesion of the continuous molecular structure of C–S–H described in Sec. 3.2 [34]. Within this framework, C–S–H gel models were generated by inserting grains of random sizes within a simulation box via Grand Canonical Monte Carlo (GCMC) simulations, until the number of inserted grains reaches a maximum value. C–S–H models exhibiting a higher degree of polydispersity eventually reach higher final packing density as small grains are able to fill the space left between larger grains (see Fig. 9). The model was found to yield stiffness, hardness, and creep modulus values that are in excellent agreement with nanoindentation data [34,36,120]. More details, validations, and limitations of this model can be found in Refs. [34,36].

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In details, each GCMC step comprises (i) N attempts of grain insertion or removal and (ii) M attempts to randomly displace an existing grain. In each case, the probability of success of the attempt is given by exp(−Δ ⁄ B ), where B is the Boltzmann constant, T the temperature, and Δ is the variation in potential energy caused by the trial insertion/displacement [34]. The factor R = N/M is then qualitatively equivalent to a precipitation rate, which characterizes the time duration during which the grains are allowed to reorganize between two successive insertions. Namely, a large R value corresponds to a high precipitation rate, wherein the grains have only limited opportunity to move during precipitation, whereas a low R value corresponds to a low precipitation rate, wherein the grains can extensively reorganize to form a more stable structure. In the following, R is referred to as the kinetic rate.

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4.2 Optimization of the mesostructure at percolation

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At the mesoscale, the topology of C–S–H is captured by the average number of contact points between C–S–H nanoparticles (or grains). Fig. 10a shows the evolution of the average grain– grain coordination number as a function of the packing fraction for three different kinetic rates R in the case of a monodisperse C–S–H model—i.e., wherein all grains have the same size. It is observed that the kinetics of precipitation significantly affects the topology of the C–S–H mesostructures that are formed. Namely, higher kinetic rates result in a lower inter-grain connectivity at a given packing fraction. As shown in Fig. 10b, the kinetics of precipitation controls the topology of the C–S–H gel at percolation, that is, when a cluster of interconnected grains reaches the length of the simulation box. Note that the percolation point corresponds to the stage at which the gel starts to have a finite modulus/strength (i.e., qualitatively equivalent to the setting of the paste). Average grain coordination number

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(a) (b) (c) Fig. 9: Snapshots of various C–S–H gel configurations exhibiting increasing degrees of polydispersity from (a) to (c). Different colors indicate different grain diameters.

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Specifically, we observe that, at percolation, high kinetic rates yield poorly-coordinated mesostructures made of elongated fiber-like clusters (see Fig. 11a). In contrasts, low kinetic rates result in highly-coordinated mesostructures comprising large, roughly spherical, and wellorganized clusters (see Fig. 11b). This can be understood as a “Tetris effect,” in the sense that, at high precipitation rate (i.e., high “falling rate of bricks”), the fast kinetics results in a C–S–H gel that is not efficiently packed, with significant internal porosity in between the grains (i.e., with many “empty boxes in between poorly organized bricks”). Hence, a higher precipitation kinetics results in a more disordered mesostructure exhibiting a lower packing fraction.

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Challenge #8: Besides the precipitation kinetics, little is known regarding how the inter-grain interaction energy controls the topology of the C–S–H gel. Understanding such linkages could be key to design cement hydrate matrices with tailored mesostructures. This is important as, from early-age slurries to mature cement pastes, the mesostructure of C–S–H has a major influence on many properties, including early-age rheology and setting [37,38], mechanical properties [33], and long-term viscoelastic relaxation [120].

4.3 Optimization of stiffness

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(a) (b) Fig. 11: (a) Snapshots of C–S–H gel mesostructures at the percolation threshold obtained with a kinetic rates R of precipitation of (a) 0.01 (high precipitation rate) and (b) 0.0001 (low precipitation rate). In each case, the largest percolating cluster is highlighted in red while smaller clusters are in blue. Note that periodic boundary conditions are applied.

One of the major assumptions of the topological approaches presented in Sec. 3 is that all the constraints are equivalent to each other—with the exception of the distinction between BS and BB constraints for hardness predictions [55]. Although such an assumption is fairly justified in atomic networks as, for example, the Si–O and Al–O bonds exhibit similar interaction energies [122], it cannot apply to highly heterogeneous networks wherein each constraint is associated to a different energy. In the case of the mesoscale models of C–S–H, the inter-grain potential energy is proportional to the volume of the grains [36], that is, larger grains are more strongly connected to each other than smaller grains. In addition, the connectivity of each grain largely depends on their size. As shown in Fig. 12a, the average coordination number of the C–S–H grains increase with their diameter (for a fixed degree of polydispersity). Note that this curve is not universal and depends on the degree of polydispersity of the system. This implies that, upon 13

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loading of the structure, the resulting internal stress gets concentrated around the larger particles rather than being equally distributed within the structure. As a result, the stiffness of the gel does not depend on the average connectivity of the network, but rather on that of the larger grains only, which constitute the rigid skeleton of the network. This is illustrated in Fig. 12b, which shows a poor correlation between the indentation modulus of the gel and the average coordination number calculated from the entire network. In contrast, Fig. 12c shows that the indentation modulus increases fairly linearly with the average coordination number of the rigid skeleton network (defined here by excluding the 25% smallest grains). Overall, these results illustrate how topological descriptors (here, the particle–particle coordination number) can be used as reduced order parameters to understand how structure (here, degree of polydispersity) controls macroscopic properties (here, stiffness).

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Challenge #9: More generally, elucidating the effect of the mesoscale topology on the properties of C–S–H is complicated by the fact that its mesoscale structure is highly heterogeneous. As such, mean-field approximations—as commonly used to describe the average atomic topology of disordered networks—might not be applicable. Novel frameworks capturing the effect of topological heterogeneity are needed [123]. (a) Average CN

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5. Conclusions

These examples highlight how topological constraint theory can be used as a rational tool to predict the properties of disordered materials or guide the design of optimized systems with tailored functionalities. It is especially well-suited for disordered systems with no fixed stoichiometry, for which traditional trial-and-error approaches are rendered inefficient due to the virtually infinite number of possible compositions.

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Fig. 12: (a) Average coordination number (CN) of the C–S–H grains as a function of their diameter for a fixed degree of polydispersity. Computed indentation modulus as a function of (b) the average coordination within the entire network and (c) the average coordination number of the skeleton rigid network.

By capturing the relevant structural information of each scale (atomic and mesoscale) while filtering out less relevant “2nd order” details, topological approaches can simplify complex disordered structures and, thereby, seamlessly connect one scale to the next—by capturing the essential features of a given scale and modeling it as a reduced-dimensionality input for the next scale. These approaches can, therefore, facilitate the development of upscaling approaches capturing the combined effects of binder composition and mesostructure on macroscopic 14

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This work was supported by the National Science Foundation under Grant No. 1562066.

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7. References

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