 Nano Express
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Nano and mesoscale modeling of cement matrix
Nanoscale Research Letters volume 10, Article number: 173 (2015)
Abstract
Atomistic simulations of cementitious material can enrich our understanding of its structural and mechanical properties, whereas current computational capacities restrict the investigation length scale within 10 nm. In this context, coarsegrained simulations can translate the information from nanoscale to mesoscale, thus bridging the multiscale investigations. Here, we develop a coarsegrained model of cement matrix using the concept of disklike building block. The objective is to introduce a new method to construct a coarsegrained model of cement, which could contribute to the scalebridging issue from nanoscale to mesoscale.
PAC codes: 07.05.Tp, 62.25.g, 82.70.Dd
Background
Cement matrix is a colloidal gellike material that provides cohesive strength to concrete, the most widely used construction material in the world. The mechanical properties of concrete are largely related to the properties of the cement matrix [1]. The major and most important hydration product in cement matrix is the calciumsilicatehydrate (CSH) gel, which is believed to be composed of variously sized globules [2,3]. These globules are assemblies of lamellar building blocks, which are normally regarded as disklike objects approximately 1 nm in thickness and 5 to 10 nm in diameter [4]. It is recently recognized that the nanostructure of cementitious material is responsible for mechanical performance including cohesion and durability [5]. At the length scale of several nanometers, atomistic simulation is a useful investigation tool for the CSH gel building block. There are several atomistic force fields, such as the Clay force field (ClayFF) and ReaxFF, available for simulating the CSH gel [6,7]. Recently proposed atomistic models can reasonably predict the molecular behavior of the CSH gel building block at nanoscale [8,9]. Although, the feasible investigation scale of atomistic simulation is generally less than 10 nm due to the limited computational capacities. In order to extend the investigation scale, one can translate the information from atomistic simulation to the input of coarsegrained models, leading to the link between nano and mesoscale simulations of the CSH gel. Such a scalebridging process is similar to playing a jigsaw puzzle. We can construct the coarsegrained model (i.e., a mimic of the completed jigsaw puzzle) as an assemblage of many building blocks (i.e., a mimic of jigsaw puzzle pieces) after understanding the properties of a single building block. The coarsegrained model can capture the mesoscale features of the CSH gel. For example, a previously developed coarsegrained model, which assumes that the CSH gel is an aggregate of spherical particles, can reasonably describe the nucleation, packing, and rigidity of the CSH aggregates [10]. Nonetheless, the shape of the building blocks is rarely considered. In the present work, we aim to develop a coarsegrained model using the concept of a disklike building block, which can capture the shape effect. We adopt the GayBerne (GB) potential to simulate the aggregate of disklike plates [11]. The GB potential is initially developed for describing interactions between ellipsoidal particles [1214] and is used here to govern the pairwise interaction between disklike CSH gel building blocks. At the nanoscale, we run atomistic simulations of CSH gel to obtain the adhesion energy, which is later used to derive parameters for the mesoscale model. At submicroscale, we construct a series of coarsegrained samples of CSH gel using disklike building blocks. The interaction parameters are derived from both the atomistic simulation and the experimental results about the adhesion energy of the CSH gel [15,16]. In this paper, we firstly introduce the formula of GB potential and how the parameters in GB potential formula are determined; then, we describe the coarsegrained model construction and simulation and finally discuss about the properties of the model. It is envisioned that the developed coarsegrained model can help tackle the scalebridging issue from nanoscale to mesoscale.
Methods
Atomistic simulations
The adhesion energy of CSH gel is calculated using the metadynamics method. The metadynamics simulations are performed using the LAMMPS and PLUMED packages coupled with ClayFF [6,17,18], which has been successfully used for the simulations of the CSH gel system. The modeled CSH gel is constructed based on the previous study [8], which gives an equilibrated structure of glassy CSH gel. As shown by the VMD [19] snapshot in Figure 1a, this sample contains two CSH gel layers, which are composed of 360 silicon atoms, 594 calcium atoms, 1,078 oxygen atoms, 472 hydroxide (OH) groups, and 124 water molecules. These two layers of CSH gel are immersed in a water box with 3,175 water molecules. The two CSH gel layers are arranged in a facetoface manner. The cutoff distance is set to 1.25 nm for the van der Waals and Columbic interactions. The system firstly undergoes a 1ns equilibration in an isothermalisobaric ensemble (NPT ensemble), with pressure and temperature controlled by a NoseHoover barostat at 1 atm and a NoseHoover thermostat at 300 K, respectively. At the end of the 1ns simulation, the rootmeansquare displacement (RMSD) of silicon atoms becomes stable, indicating that the system has reached equilibrium state. After equilibration, metadynamics simulations are performed using the PLUMED package. The free energy is measured as a function of the centertocenter distance between two CSH layers. The metadynamics simulations last for 10 ns, permitting a full exploration of all possible states. The plot of the freeenergy against the centertocenter distance is shown in Figure 1b. The freeenergy difference between the attached state and the separated state is 1,819 kJ/mol. With the surface area being 6.89 nm^{2}, we obtain a normalized adhesion energy approximately equal to 440 mJ/m^{2}, which is in a good agreement with experimental results which are reported as 380 and 450 mJ/m^{2} [15,16].
GB potential parameterization
The GB potential can describe the interaction between pairwise ellipsoids, which can rotate and translate. Considering an ellipsoid with three radii a, b, and c, when c is much smaller than a and b, the ellipsoid becomes akin to a disklike plate, as schematically demonstrated in Figure 2a. Two plates can interact with each other in either a facetoface or sidetoside manner, as shown in Figure 2b. The detailed GB potential formula can be described as Equation 1:
In Equation 1, ϵ is the energy scale and is normally set to 1, γ is a constant and is set to 1, σ determines the width of the potential energy well, and h _{12} is the approximation of the closest distance between two particles, which is computed by Equations 2 and 3:
In Equation 2, let r _{12} = r _{2} − r _{1} be the vector from particle 1 to particle 2, \( {\widehat{\mathbf{r}}}_{12} \) denotes the unit vector of r _{12}, and r denotes the length. In Equation 3, S = diag (a, b, c) is the shape matrix containing three radii of the ellipsoid, and A _{ i } is a 3 × 3 transformation matrix from the simulation box frame to the particle body frame. During the parameterization, we keep the transformation matrices of all particles to diag(1,1,1) and translate the plates so as to obtain the facetoface or sidetoside interaction pattern. Next, we look at the second term in Equation 1; η _{12} is computed by Equations 4 and 5, where ν is a constant and is normally set to 1:
The last term χ _{12} in Equation 1 is computed by Equations 6 and 7:
In Equation 6, μ is a constant and is normally set to 1. In Equation 7, E = diag(ϵ _{ a }, ϵ _{ b }, ϵ _{ c }) is the energy matrix that contains the three energy well depths. After setting the constants (γ, ν, μ) to (1, 1, 1), seven parameters (σ, a, b, c, ϵ _{ a }, ϵ _{ b }, ϵ _{ c }) remain to be determined. Here, we provide a reasonable method to approximate these parameters on the basis of atomistic simulations and experiments.
Regarding parameters (a, b, c), the approximating rules include (i) a × b = 25 nm^{2} and (ii) c = 0.5 nm. The shape of the building block is a rough assumption in reference to previous studies [2,4], which have approximated the dimensions of CSH building blocks. With these settings, we can calculate that η _{12} = 5.05 in the sidetoside and facetoface cases, where the transformation matrices are diag(1,1,1). For a perfect circular plate, a, b = 5 nm are the radii of the front face and c is the thickness of the plate. Besides, we have also modeled anisotropic plates with the a/b ratio ranging from 1 to 4. The parameter σ is typically set to the minimum of the three shape diameters, which is 1 nm in the present cases.
Regarding parameters (ϵ _{ a }, ϵ _{ b }, ϵ _{ c }), the approximating rules include (i) the normalized adhesion energy of CSH gel is G = 450 mJ/m^{2} and (ii) the energy well depth is equal to the adhesion energy [11]. From the GB potential formula, we can calculate the minimum energy (energy well depth) as ϵ ⋅ η _{12}. We thereby calculate that \( {\epsilon}_c=\frac{G\cdot \pi ab}{\eta_{12}}\approx 7,000\times {10}^{21}\;\mathrm{J} \), \( {\epsilon}_{\alpha }=\frac{G\cdot \pi bc}{\eta_{12}}\approx 700\times {10}^{21}\;\mathrm{J} \), and \( {\epsilon}_b=\frac{G\cdot \pi ac}{\eta_{12}}\approx 700\times {10}^{21}\;\mathrm{J} \) when a, b = 5 nm and c = 0.5 nm.
Coarsegrained simulations
After defining the parameters for GB potential, we construct a series of samples and perform molecular dynamics simulations. The initial coarsegrained (CG) model is set up by averagely distributing the 1,000 (10 × 10 × 10) beads in the 150 × 150 × 150 nm^{3} simulation box; these beads are equally separated by 15 nm, and their orientations are randomly assigned, as shown by the BioVEC [20] snapshot in Figure 3a. The cutoff of the GB potential is 15.0 nm. With a 1fs time step, the system is equilibrated for 20 ns in an NPT ensemble with temperature and pressure controlled at 300 K and 1 atm, respectively. After a 2ns simulation, the system has reached an energyminimized state. Then, we apply quasistatic method to measure the elastic properties of the modeled cement. In the quasistatic method, the system deforms slowly so that each state is close to equilibrium during the process [11,21]. In every 100 ps, the system deforms 0.001% and the strain eventually reaches 0.01%. The stress is recorded every 0.1 ps, and the average stress over the latter 50 ps of each deformation period is calculated.
Result and discussion
Configuration of the coarsegrained model
In equilibrium state, the building blocks flocculate to form cylinders, as shown in Figure 3b. The evolution of the total energy is plotted in Figure 3c. The total energy decreases to a stable value at 2 ns, indicating that the ‘jamming state’ has been achieved [11]. We then probe the packing status of the system. Initially, each building block can be regarded as a single stack. When two disklike building blocks stay close enough with each other (centertocenter distance smaller than 1.5 nm) and align in a facetoface manner, they belong to the same stack. The stack size (the number of disks contained in one stack) ranges from 1 to 12 as shown in Figure 3d. Packing probability is calculated as the number of stacks divided by the total stack number. For example, the number of doublepacked stack (stack that contains two disklike building blocks) is 137 and the total stack number is 367, so the packing probability of double packed stack is 137/367 = 0.37. The probability distribution of the stack size can be fitted by a lognormal distribution, as reported in the study on clay aggregates [11,22]. Most stacks are composed of two or three building blocks, the average stack size is 2.7. Here, we demonstrate the capability of the developed coarsegrained model in investigating the packing status of the CSH system, such information could be useful in further examination on the microstructure of the CSH gel.
Properties of the CSH gel model
Considering the building block of the CSH gel as the elliptical plates, we vary the a/b ratio of the plate and study the relationship between mechanical properties and the ratio. Conformations of these variously sized building blocks are shown in Figure 4a. The packing fraction, which is the volume fraction of the CSH building block, is plotted as a function of the a/b ratio in Figure 4b. Models with anisotropic building blocks (a/b ranges from 1.1 to 2) have a higher packing fraction, compared to samples with a/b = 1, 3, or 4. According to the JT model [2], the hardening process can be described as the aggregation of globules into lowdensity (LD) CSH and finally into highdensity (HD) CSH. During this process, the packing fraction and the elastic properties of the system increase as well. Here, the increase of packing fraction is accompanied by the change of the a/b ratio, which indicates that the front surface of the disklike building block could evolve from a perfect circle to an ellipse during the hydration process. A similar trend is observed in the mechanical properties. As shown in Figure 5a,b, both Young’s modulus and shear modulus increase as the a/b ratio increases from 1 to 1.6, indicating the possible evolution of the shape of the CSH building blocks during the hardening. Afterwards, the down trend indicates that anisotropic shapes (especially when a/b is 3 and 4) could lead to degraded mechanical properties. Such phenomenon reveals that the shape of building blocks could not be too anisotropic in the realistic CSH structure. The maximum Young’s modulus is 18.6 GPa, in accordance with the LD CSH (Young’s modulus is 20 GPa). From these results, we can infer that the a/b ratio of the disklike CSH building block is not necessarily 1 but the anisotropic shapes (a/b around 1.6) are favored in hardened CSH. The present model is a monodispersed system, i.e., the model is composed of identical disklike building blocks. It is worth noting that realistic CSH gel should be a polydispersed system, i.e., the system contains variously sized particles, and the space between large particles can be filled by small ones so that both packing fraction and mechanical properties are enhanced [10]. The present model uses disklike building blocks but the polydispersity of CSH is not considered here. Because the polydispersed system should contain variously shaped building blocks, currently we do not involve the polydispersed concept and construct monodispersed models so as to solely investigate the shape effect. We envision that in future studies a more comprehensive model could be proposed by combining the concepts of disklike building blocks and polydispersed system.
Conclusions
We have performed atomistic simulations of CSH gel and developed a coarsegrained model accordingly. At nanoscale, atomistic simulations indicate that the normalized adhesion energy of the CSH gel is 440 mJ/m^{2}, in a good agreement with existing experimental results. Based on the information from atomistic simulations, we propose a coarsegrained model using the concept of disklike building blocks. GB potential is adopted to govern the interaction between pairwise building blocks. The developed model is useful in investigating the packing status of the CSH system. We vary the shape of building blocks and observe the corresponding change of mechanical properties. Results indicate that the shape of building blocks of CSH could change from uniaxial (circular plates) into anisotropic ones (elliptical plates) as the hardening proceeds. In summary, this paper proposes several rules to develop the coarsegrained model of CSH using GB potential and demonstrates a case study on the shape of CSH building blocks. Considering that the present model does not involve the polydispersity of CSH, we envision that future studies can combine the concepts of disklike building blocks and polydispersed description so as to enrich the understanding of cement matrix.
Abbreviations
 CSH:

calciumsilicatehydrate
 GB potential:

GayBerne potential
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Acknowledgements
The authors are grateful for the support from the Croucher Foundation through the Startup Allowance for Croucher Scholars with the Grant No. 9500012 and the support from the Research Grants Council (RGC) in Hong Kong through the Early Career Scheme (ECS) with the Grant No. 139113.
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The authors declare that they have no competing interests.
Authors’ contributions
ZY carried out the molecular dynamics simulations and drafted the manuscript. DL conceived of the study and participated in its design and coordination. Both authors read and approved the final manuscript.
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Yu, Z., Lau, D. Nano and mesoscale modeling of cement matrix. Nanoscale Res Lett 10, 173 (2015) doi:10.1186/s116710150862y
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Keywords
 Cement matrix
 Building block
 Coarsegrained simulation
 GB potential