Intermolecular Forces between Nanolayers of Crystalline Calcium

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Intermolecular Forces Between Nanolayers of Crystalline Calcium-Silicatae-Hydrates in Aqueous Medium Saeed Masoumi, Hamid Valipour, and Mohammad Javad Abdolhosseini Qomi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b10735 • Publication Date (Web): 17 Feb 2017 Downloaded from http://pubs.acs.org on February 18, 2017

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The Journal of Physical Chemistry

Intermolecular Forces Between Nanolayers of Crystalline Calcium-Silicate-Hydrates in Aqueous Medium Saeed Masoumi,† Hamid Valipour,† and Mohammad Javad Abdolhosseini Qomi∗,‡ Centre for Infrastructure Engineering and Safety (CIES), School of Civil and Environmental Engineering, UNSW Australia, UNSW Sydney, NSW 2052, Australia, and Advanced Infrastructure Materials for Sustainability Laboratory (AIMS Lab), Department of Civil and Environmental Engineering, Henry Samueli School of Engineering, E4130 Engineering Gateway, University of California, Irvine, Irvine, CA 92697-2175 USA. E-mail: [email protected]

Abstract Calcium-silicate-hydrate (C-S-H) is the major binding phase responsible for strength and durability of cementitious materials. The cohesive properties of C-S-H are directly related to the intermolecular forces between its layers at the nanoscale. Here, we employ free energy perturbation theory (FEP) to calculate intermolecular forces between crystalline C-S-H layers solvated in aqueous medium along face-to-face (FTF) and sliding reaction coordinates. Contrary to mean-field theories, we find that our counterion-only system exhibits an oscillatory behavior in FTF interaction. We correlate these oscillations with the characteristic length scale comparable to the distance between interfacial water layers at the hydrophilic surface of crystalline C-S-H. We attribute the sliding intermolecular forces to the atomic level roughness of crystalline C-S-H layers stemming from the local arrangement of nanoscale structural motifs. These intermolecular forces provide a direct access to the key mechanical properties,

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such as surface energy, cohesive pressure and elastic properties. The simulation results are in close agreement with the available experimental measurements. Furthermore, we present these intermolecular forces in a mathematical framework to facilitate coarse-grain modeling of crystalline C-S-H layers. These results provide a novel route that paves the way for developing realistic meso-scale models to explore the origins of chemophysical properties of crystalline C-S-H.

Introduction The cement liner in boreholes provides structural integrity and zonal isolation to alleviate environmental safety concerns regarding hydraulic fracturing and disposal of high-level radioactive waste (HLRW) in deep geological formations. 1,2 The durability of this protective layer is of prime importance during its service life, ranging from a few years in oil wells to million years envisioned for HLRW disposal management. Tackling durability issues at geological conditions and time scales poses a grand challenge at the interface of physics, chemistry and mechanics of materials. A promising modeling approach to address these challenges involves development of physical chemistry-based models of cementitious materials at the meso-scale. 3,4 These models operate based on potentials of mean force (PMF) between nano-scale globules of calcium-silicate-hydrates (C-S-H), 5 the main binding phase responsible for the strength, durability and thermal properties of cementitious materials. 6–10 At temperatures and pressures relevant to oil well cementing, the major product of cement hydration occurs as a C-S-H with molar calcium-to-silicon ratio (Ca/Si) close to one. 11 The molecular structure of C-S-H at such Ca/Si is akin to that of layered Tobermorite minerals. 12–16 Despite decades of research on the atomic structure of Tobermorite, the nature of inter-molecular forces between its layers remains vastly unexplored and to date lacks quantitative understanding. The cohesion between cement layers have been studied previously using dielectric continuum model approach. 17,18 Although such models are powerful, they are challenged when the nanoconfinement and local effects become significant. 19–23 In this paper, we explore intermolecular 2 ACS Paragon Plus Environment

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forces between Tobermorite layers in an aqueous solution using full atomistic simulation method. For simplicity, we assume that Tobermorite layers are infinitely long in the planar direction by considering periodic boundary conditions, see Fig. 1. Since layers are solvated in water, the entropic effects cannot be simply disregarded. Therefore, the potential energy would not be sufficient to describe the interactions in our system. On the other hand, the Helmholtz free energy considers such entropic effects and therefore is suitable to explore the interaction between the two layers. Here, we employ free energy perturbation (FEP) method to calculate the free energy difference along a pre-specified reaction coordinate. We use FEP to study the interaction between two identical Tobermorite layers either when they are approaching each other in face-to-face (FTF) direction or when they slide on top of each other. We organize the paper as follows. First, we elaborate on the molecular models, atomistic simulation and FEP calculations. We subsequently present the results of FEP calculations in FTF and sliding directions along with mechanical properties deduced from FEP results.

Simulation Models and Methods Tobermorite crystals are comprised of stacks of negatively-charged calcium-silicate layers. The infinitely long parallel drierketten silicate chains are tethered to both sides of pseudo-octahedral calcium oxide sheets. These layers are separated from one another via an interlayer spacing. Depending on the polymorph and stoichiometry, 19,24–26 the interlayer is filled with nano-confined water molecules, charge-balancing calcium cations or hydroxyl groups. The drierketten silicate chains are constructed of replicates of two pairing silicate tetrahedra and an immediate bridging site that protrudes outside the layers. Here, we adopt the crystal structure of Tobermorite 11 Å proposed by Hamid. 24 We build a 2×2×1 supercell of Tobermorite mineral. To reduce the tilt angle of the simulation cell, we subsequently perform a hexagonal to orthogonal transformation. This leads to the final structure of crystalline C-S-H layer with dimensions of a=26.54 Å, b=48.87 Å, c=12.20 Å, α = 82.5◦ , β = 90◦ , γ = 90◦ .



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To probe the intermolecular interactions between crystalline C-S-H layers, we duplicate the first layer along the c crystallographic direction such that the combination of the two layers reconstructs the original Tobermorite crystal, Fig. 1. The distance between the centers of mass of the layers is henceforth referred to as center-to-center distance, dcc . The FTF interaction between crystalline C-S-H layers takes place along the reaction coordinate, ξ . We vary the distance between the two layers incrementally along the reaction coordinate. We perform a series of statistically independent molecular dynamics (MD) simulations in the canonical ensemble (NVT) at room temperature to build up a catalogue of atomic trajectories and potential energies at several intermediate distances. We employ transferable CSH-FF force field to define the interatomic interactions and partial charges in our simulations. 27 (see Supporting Information). Here, we employ FEP theory to calculate mean intermolecular force along the FTF and sliding reaction coordinates. 28,29 Since the free energy calculation is sensitive to the perturbation terms, we consider several sub-stages in between the target and the reference states, such that the perturbations become reliable regarding statistical errors. This becomes exceedingly necessary where the free energy difference is expected to show a substantial variation. In this paper, we use the simple overlap sampling (SOS) method for calculations of free energy difference between sub-stages i and i + 1 (see Supporting Information). hexp(−β ∆Ui,i+1 /2)ii 1 ∆Ai,i+1 = − ln β hexp(−β ∆Ui+1,i /2)ii+1

(1)

where A is the so-called Helmholtz free energy which we can here interpret as the potential of mean force, also β = kB1T , kB is the Boltzmann constant and T is the temperature. h∆Ui,i+1 ii indicates the ensemble average of the potential energy difference between the reference state i and its perturbed configuration at location of the i + 1 state (see Supporting Information). Eq. 1 requires that the potential energy between two states varies linearly. This requirement is only met if the perturbation steps are relatively small. Considering these facts along with the excessive computational costs of FEP calculations, we adopt a perturbation step of 0.25 Å and 0.0625 Å respectively for FTF and



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sliding paths. We further refine our FEP calculations for the FTF path at dcc ≤14.7 Å. The total number of water molecules is kept constant, nw =3481, throughout the simulations to ensure that the free energy difference is only capturing the system dynamics rather than the chemical potential gradient arising from addition or removal of atoms. At each step, water molecules are removed from the interlayer spacing and placed outside in the bulk water region to make sure the water density remains equal to that of bulk water at room condition. We make sure the characteristic size of the screening bulk water region is large enough such that the interaction between layers only occurs through interlayer spacing (see Supporting Information). Once the free energy difference profile is calculated, we determine the mean force as follows:

hFi i = −

∆Ai,i+1 ∆η

(2)

where η is the perturbation’s step size. To properly sample important regions, a catalog of MD trajectories and potential energies are built in an NVT ensemble at T = 300 ◦ K using Nosé-Hoover thermostat. 30 As FEP calculations are computationally prohibitive, when possible, we freeze all atoms in the system except interlayer calcium and water molecules. We further examine the reliability of our approach by investigating free energy profiles for frozen and fully relaxed simulations. The bond length and angle in water molecules are constrained using SHAKE algorithm. 31 The MD time step is set to 1 fs to capture dynamics of nano-confined water. 19,32 We perform a series of 3 ns long MD simulations for the FTF path, leaving the initial 0.5ns as the relaxation period and sampling afterward at every 1 ps, which provides 2500 frames and perturbations for each simulation. For the sliding case, we run a 1 ns long MD simulation in which we dedicate 200 ps to the relaxation and the rest for sampling at every 0.5 ps leading to 1600 frames for the FEP computations. All MD simulations are carried out using LAMMPS software, 33 while VMD software is used for the visualization of the atomic structure. 34 The perturbation calculations are performed using a house built code.


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Results and Discussions We calculate the mean force of bringing two crystalline C-S-H layers towards each other using FEP theory. It is through FEP theory that the Hamiltonian of the target system can be expressed as a function of the reference Hamiltonian and a perturbation term (see Supporting Information for further details). 28 Fig. 2 presents the mean force for the frozen FTF scenario. In our simulations, the atomic structures of crystalline C-S-H layers are extracted from Hamid’s Tobermorite 11 Å and therefore they are already around the equilibrium point in transverse direction. This suggests that freezing layers would not significantly affect FEP results in FTF scenario. See Supporting Information for the comparison between the frozen and relaxed structures. As shown in Fig. 2, the mean force trend exhibits an oscillatory behavior with at least two distinguishable minima located at 13.5 Å and 15.9 Å. The distance between the two minima is roughly 2.5 Å, which relates to the nature of nano-confined water in the interlayer spacing. We will discuss the characteristics of the nano-confined water and its relation to the mean force oscillations later. It is also evident from Fig. 2 that due to the screening effect of polar water molecules, the interactions between layers become increasingly negligible at the limit of long distances. It is well-known that at the limit of short distances and high surface charge densities, the local effects dominate the total interaction. Therefore, mean-field theories might fail to deliver reliable predictions. Here, we seek an alternative explanation of the total interaction observed in our FEP results by examining the possible entropic effects. Both atomistic simulations 35–37 and X-ray reflectivity experiments 38–40 support the presence of strongly localized water molecules adjacent to hydrophilic surfaces. These studies show that the water ordering vanishes after a few hydration layers. To further investigate the nature of oscillatory behavior seen in the mean force graph, we plot the relative density profile of water oxygen across the interlayer spacing for different FTF distances in Fig. 3. The density profiles are symmetric with respect to the interlayer’s midplane. Therefore, we merely discuss results from one layer to the midplane. As is evident from Fig. 3.a, the water molecules populate around specific sites alongside the crystalline C-S-H layer. We illustrate the 7 ACS Paragon Plus Environment


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24

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Figure 2: The mean force arising from the interaction of two crystalline C-S-H layers approaching each other in FTF configuration. A Theoretical model presented in Eq. 3 is fitted to the results of the free energy perturbation method. The inset illustrates entropic solvation, steric contact, and total attraction contributions to the mean force in between layers. The combined attraction term takes into account the contribution of the van der Waals (for all distances), counterion condensation, and ionic correlation (for short distances). The error bars indicate the standard deviation of the mean force computation (see Supporting Information).


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liquid ordering schematically in Fig. 3.b-e by drawing the trajectory of water oxygen atoms in the last 100 ps of MD simulations. These results indicate that some water molecules may diffuse into and remain inside the layer (next to the pairing silicate groups), forming the designated 0th layer. The relative density profile shows a peak at the level of bridging silica tetrahedra that pertains to the set of immobile water molecules adsorbed on the hydrophilic surface of C-S-H. 19,37,41 We denote these molecules as the 1st layer in Fig.3a. The ordering extends to the second solvation layer with increasing interlayer spacing. We emphasize that the shift in the position of the second solvation layer in dcc = 16.2 Å is enforced by the symmetry. With increasing interlayer distance, the first and second peaks exhibit shoulders that are associated with the molecular roughness of Tobermorite caused by drierketten silica chains. The presence of the third hydration layer is not significant at the largest interlayer spacing, dcc = 22.2 Å, and the profile density converges to the bulk value. We highlight that the distance between the first and second solvation layers equals 2.4 Å, which is exactly equal to the distance between the first and second mean force minima from FEP results. This shows that the intrinsic oscillatory nature of the mean force between crystalline C-S-H layers originates from entropic interfacial liquid ordering. Overall, the trend for the density profile of water oxygen is comparable to results reported for Tobermorite 9 Å by Kalinichev et al. 37 In addition to van der Waals attractive forces between crystalline C-S-H layers, there are other sources of attraction in the system originating from the local effects at shorter distances. Since salt is not added to the solution and also counterions are enough to charge balance the system by adsorbing to the surface, the system is analogous to the so-called counterion-only case. 42 Due to the strong charge density of Tobermorite layers, counterions adsorb to the surface within the socalled Stern or Helmholtz layer. Such binding to the surface is further reinforced by the entropic effects of water ordering and strong attraction between water molecules and hydrophilic surface of Tobermorite. At distances larger than dcc = 15Å (Fig. 3c), where at least one layer of liquid is present in the interlayer, the contribution of ionic correlation fluctuations becomes negligible. As presented in Fig. 4, the counterion’s normalized number density profile shows a sharp peak


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with limited range of fluctuations at larger distances. At distances below 15Å(absence of the designated 2nd interfacial water layer in Fig. 3.a), fluctuations of counterions are affected by the attraction from both layers, which leads to broadening of counterion density profile. Therefore, the contribution of ionic correlation forces becomes more pronounced at smaller distances. 43 At small distances, another entropic effect in the mobility of counterions becomes notable. In this range, counterions are guided to reside in the empty space adjacent to pairing sites in silica chains, Fig. 3.b. This is evident from the shift in the normalized number density profile at dcc = 12.2Å compared to other distances presented in Fig. 4. It is in this range that layers share all counterions akin to iono-covalent crystals. The effect of such condensation of multivalent counterions such as Ca2+ is a rise in total attraction. 44,45 Ionic correlation and counterion condensation effects are not two distinguishable contributions. 46 Hence, the total attractive forces can be considered as the combination of van der Waals, ionic correlation fluctuations, and counterion condensation effects. However, in our counterion-only system, the attraction mainly arises from van der Waals interactions. As layers become closer, atoms on the surface start contacting and coinciding with each other, causing a gradual increase in repulsive forces. Such interaction can be defined as "steric contact" between double-layers. 46 It is intuitive that as the layers are pushed into each other, the repulsion turns to a hard contact that diverges in accordance to Pauli repulsion rule. Taking the above discussion into consideration, we suggest the following mathematical expression for the mean forces: a1 + a3 exp(−a4 ξ ) + (a2 + ξ )3 ξ 2π (ξ − a6 ) exp(− ) a5 cos a7 a7

Fthn (ξ ) = −

(3)

where variables ai , i = 1, 2, ..., 7 are constants to be determined through fitting the model to the FEP results. The first term carries the combined attractive forces, namely van der Waals, ionic correlation, and counterion condensation. We take the ionic correlation forces to be a function of reaction coordinate, F corr ∝

1 , ξ3

similar to van der Waals interactions. 17,47 The influence of coun-



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smooth. The location of the pairing and bridging sites in the silicate chains controls the surface roughness of these layers. Fig. 5.a provides a schematic representation of silicate chains in the C-S-H layer. Owing to the parallel alignment of the silicate chains, the sliding of layers would exhibit strong anisotropy, i.e. direction dependence. We consider two principal sliding unit vectors, − − respectively parallel (→ x1 ) and perpendicular (→ x2 ) to silicate chains. Here, we emphasize that silica − − chains are periodic along → x1 and → x2 , with the periodicity length, l p of 7.3 Å and 11 Å, respectively. To understand the origin of the resistance against sliding, we provide configurations of one periodic block from the view of a fixed window when the top silicate chain is sliding over the bottom chain. − These configurations pertain to the equilibrium, a quarter and the middle of the periodicity in → x1 − (Fig. 5.d-f) and → x2 (Fig. 5.g-i) directions. The dotted guidelines display the relative distance of two pre-specified silicon atoms in the bridging tetrahedra sites in the top and bottom layers. The maximum resistance occurs when two silicate bridging sites are forced to locate on top of each other. Since silicate chains are grinding against each other in the sliding scenario, we allow the bridging sites to freely move in FEP simulations. Fig. 5.b-c presents the mean force required to slide layers against each other in two principal directions for frozen and unfrozen cases at normal equilibrium distance (dcc = 0.4Å ). Unlike FTF case, there is a significant difference in mean force of frozen and unfrozen cases. The frozen case only captures the effect of structural motifs and therefore it is not surprising to see an unreasonably high force barrier when two bridging sites meet each other. On the other hand, for the unfrozen case, the sudden drop in the mean force after the onset of the sliding is less likely to be realistic. A closer look at the trajectory of atoms reveals that covalent Si-O bonds in bridging sites break upon induction of excessive grinding force. Since we are employing a non-reactive potential in this work, such FEP calculations would not be valid beyond a certain point (in this case x1 > 1.25Å and x2 > 0.5Å ). Such unphysical bond reconfiguration occurs up to dcc = 13 Å . These results suggests that the sliding at very short distances entails a normal jump due to strong repulsion of bridging sites’ contact. We subsequently analyze sliding at dcc = 13.5Å for the unfrozen case. Fig. 6 shows the mean force required to slide two layers of Tobermorite on top of each other at this distance. The middle of the periodicity


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Figure 6: The mean force required to slide two Tobermorite layers over each other at dcc = 13.5Å along x1 and x2 directions. Insets show the extension of sliding mean forces beyond the periodic length. represents the point where the energy barrier is the maximum and force is zero and therefore it is the unstable point along the path where two bridging sites face each other. Since the origins of sliding forces arise from structural motifs, we seek an analytical expression for such forces. One should keep in mind that such forces have a periodic nature due to the periodicity of crystalline C-S-H layers. A mathematical expression that serves such a purpose is the trigonometric Fourier series as follows:

Fths (χi ) = b0 +

8

∑ [bmcos (km χi) + cmsin (km χi)] m=1


(4)


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− where km = 2π m/l p is the wavenumber and χi is an arbitrary point along → xi . Fig. 5.b-c show the fitted Eq. 4 to FEP results. The details of fitting procedure and constants are provided in Supporting Information. The insets of Fig. 5.b-c emphasize the periodic nature of the sliding forces and explain graphically how the fitted model extends beyond the fitting domain. Generally speaking, the sliding can occur along an arbitrary path. The precise calculation of sliding forces along such − path will be a computationally prohibitive task. We expect a linear superposition of forces in → x1 and → − x2 would provide a sufficiently good estimate of forces along a random direction. Nevertheless, the sliding forces are at the molecular origins of C-S-H’s creep, 51 a long-term deformation that occurs under constant loading. 52,53 To further validate our FEP calculations in FTF and sliding directions, we calculate mechanical and physical properties of Tobermorite crystals. We calculate the elastic modulus along basal direction, Ezz , as follows: dcc E= Sint

∂ hF n i ∂ξ

(5) d0

where d0 is interlayer equilibrium distance and Sint is the surface area through which layers interact in FTF configuration. Similarly, we calculate the shear modulus using the sliding configuration at equilibrium distance as, dcc Gi = Sint

∂ hFis i ∂ xi

(6) χ0

where i =1, and 2 represents the parallel and perpendicular direction to the silicate chains and χ0 is the equilibrium position along each sliding direction which is zero. We compute the surface energy,

γs , of the FTF interaction using the amount total potential of mean force (PMF), Wtot , required to separate two interacting crystalline C-S-H layers and place them infinitely apart, − d∞0 hF n i .d ξ Wtot = γs = 2 × Sint 2 × Sint R

(7)

The results of FEP calculations are compared against available data in the literature, see Table. 1. The mean force profile in FTF configuration provides a direct access to key mechanical properties in the basal direction including elastic modulus (Ezz ), cohesive pressure (Pco ) and surface 16 ACS Paragon Plus Environment

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energy (γ ). High-pressure x-ray diffraction (HPXRD) studies on incompressibility of crystalline C-S-H phases yield values of 89.0 GPa and 61 GPa for 14Å Tobermorite and C-S-H(I), respectively. Density functional theory (DFT) calculations and atomistic simulations of 11Å Tobermorite yield respectively 68.4 GPa and 72.0 GPa for elastic modulus in the basal direction. These values are in close agreement with FEP results. The cohesive pressure is the maximum pressure that must be applied to disjoin C-S-H layers. Previously, Pellenq et al. 54 estimated the cohesion of 5 GPa for Hamid’s Tobermorite model, 24 which is close to 6.5 GPa extracted from current FEP results. Brunauer 55,56 measured experimentally the surface energy of Tobermorites, with Ca/Si ratio of 1.5, more than half a century ago. Although his reported Ca/Si ratio differs from our contemporary understanding of Tobermorite minerals, he measured surface energies ranging between 0.32 and 0.40 J/m2 . Here, we propose a value of 0.67 J/m2 for 11Å Tobermorite through FEP, which is lower than the estimated 1.15 J/m2 of surface energy from atomistic fracture toughness calculations. Furthermore, we can deduce the shear modulus of the system via the sliding force profile for the unfrozen case at the equilibrium normal distance dcc = 0.4Å . Here, we measure values of 18 and 73 GPa along x1 and x2 , respectively. The arithmetic average of the shear modulus become 45.5 GPa, which is in good agreement with previously reported DFT 57 and atomistic simulation 58 results. Table 1: Mechanical and physical properties calculated via FEP approach and compared against other experimental and simulation methods. Property Ezz Pco γs G1 , G2 G¯

(GPa) (GPa) (J/m2 ) (GPa) (GPa)

Current Work

Others

77.6 6.5 0.67 18, 73 45.5

61, 59 89, 12 72, 58 68.4 57 5.0 54 0.32-0.4, 55,56 1.15 60,61 -, 19 34, 36, 57



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Conclusion In this work, we apply the free energy perturbation method to calculate the intermolecular mean forces acting on nanolayers of crystalline C-S-H along face-to-face (FTF) and sliding reaction coordinates. For our counterion-only system, we observe an oscillatory behavior along FTF direction, contrary to mean-field theory predictions, indicating the importance of local effects in the total interaction between the layers. We investigate the nature of such oscillatory behavior and find it to be perfectly correlated to interfacial liquid ordering phenomenon. We provide an analytical expression of the FTF intermolecular mean force as the summation of attraction forces (including van der Waals forces at all distances, also the ionic correlation and counterion condensation at short distances), the steric contact, and entropic solvation forces. Seeking the molecular origins of creep, we measured the intermolecular forces required to slide crystalline C-S-H layers over each other. We find these forces to be mainly influenced by the atomic scale surface roughness, which stems from structural motifs. At small interlayer distances, we observe unphysical bond breakage during the sliding mode. This suggests that the sliding at very small distances would certainly involve relative normal displacement of layers. We study sliding forces profile at a normal distance above the equilibrium point where the displacements are within the acceptable range. Since the atomic structure of Tobermorite is periodic along and perpendicular to the silicate chains, we fit the sliding forces with a periodic function. We express results of computationally demanding FEP calculations for FTF and sliding interactions in a easy-to-implement mathematical expressions. These expressions can facilitate computationally efficient course-grain modeling of C-S-H materials at the mesoscale. Furthermore, we calculate surface energy, cohesive pressure and elastic properties and compared them against available measurements, e.g. nanoindentation, and high pressure XRD experiments. The present results provide a novel route that paves the way for developing realistic meso-scale models to explore the physical chemistry origins of long-standing durability issues such as creep, fracture toughness and strength of cementitious materials.


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Acknowledgement We would like to thank Dr. D. Ebrahimi for fruitful discussions. Computational resources used in this work were provided by Intersect Australia Ltd, the National Computational Infrastructure (NCI), which is supported by the Australian Government, high performance computing center at the UNSW Australia and University of California, Irvine, and Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575.

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Intermolecular Forces between Nanolayers of Crystalline Calcium

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