Mesoscopic packing of disk-like building blocks in calcium silicate hydrate

At 100-nanometer length scale, the mesoscopic structure of calcium silicate hydrate (C-S-H) plays a critical role in determining the macroscopic material properties, such as porosity. In order to explore the mesoscopic structure of C-S-H, we employ two effective techniques, nanoindentation test and molecular dynamics simulation. Grid nanoindentation tests find different porosity of C-S-H in cement paste specimens prepared at varied water-to-cement (w/c) ratios. The w/c-ratio-induced porosity difference can be ascribed to the aspect ratio (diameter-to-thickness ratio) of disk-like C-S-H building blocks. The molecular dynamics simulation, with a mesoscopic C-S-H model, reveals 3 typical packing patterns and relates the packing density to the aspect ratio. Illustrated with disk-like C-S-H building blocks, this study provides a description of C-S-H structures in complement to spherical-particle C-S-H models at the sub-micron scale.

Scientific RepoRts | 6:36967 | DOI: 10.1038/srep36967 nanoparticles, mesoscopic models of the C-S-H aggregation can be proposed. A discrete element method has been employed in modeling nanoindentaion tests on C-S-H specimens with varied packing factors 26 . The simulated nanoindentation tests have shown a good agreement with experiments. The sub-micron structure of C-S-H has been described by a colloidal model 27 , which does not capture the lamellar nature of silica skeletons in C-S-H. As an extension, modeling of disk-like objects can reflect the morphology of the solid silica layers. The concept of C-S-H disk has been employed in the framework of continuum micromechanics to demonstrate the early strength development of C-S-H 28 . Column-like and porous polycrystalline structures have been built and investigated with the continuum models 28,29 . Here, a mesoscopic molecular dynamics model is developed using the concept of disk-like objects 11,30,31 . Simulations using this model could embody the lamellar nature of C-S-H gel at the scale of around 100 nanometers.
This study aims to link the structure of C-S-H at the sub-micron scale to the material properties at the macroscopic scale. Grid nanoindentation tests are employed to measure the material properties and molecular dynamics simulations are performed to reproduce the structures of C-S-H. The nanoindentation tests, with statistical analysis, find that C-S-H in cement pastes prepared at different w/c ratios show different phase compositions and porosities. Considering that the porosity could be related to the structure of C-S-H, we develop a mesoscopic model and perform molecular dynamics simulations to investigate how the w/c ratio induces changes to the C-S-H structure and affects the gel porosity. This study provides illustrations of the mesoscopic structure of C-S-H, links the C-S-H structure to the porosity and could enrich our understandings of cement-based materials at small scales.

Results
Deconvolution of nanoindentation data. The plots of indentation modulus, indentation hardness and the packing density are shown in Fig. 1a and b. In Fig. 1a, some data points lie away from the theoretical modulus line, which is a commonly observable phenomenon and it implies that the empirical formulas violate for some indentation tests 24,25 . The relative errors of reconstructed indentation hardness and indentation modulus are around − 5%, the standard deviations are around 20%, within the acceptable range 25 . A sample deconvolution result obtained from 93 tests on the cement paste with 0.3 w/c ratio is shown in Fig. 1c. The deconvolution error is in the order of 10 −4 , within an acceptable range. Deconvolution results of volume fractions of 4 characteristic  Table 1. The indentation moduli of the four characteristic phases (LP, LD, HD and UHD C-S-H) are around 10 GPa, 20 GPa 30 GPa and 50 GPa respectively, comparable to existing studies 25 . The cement sample with w/c = 0.3 contains 4% LP phase, 26% LD phase, 54% HD phase and 16% UHD phase, in good agreement with previous grid nanoindentation results 24 . As the w/c ratio increases, more low-density phases (LP + LD) are found and the porosity of the entire C-S-H composite increases, as plotted in Fig. 2. With the increase of w/c ratio, the increase of porosity is a common phenomenon observed from experiments 20,32,33 and simulations 34 . The stiffness of the mineral phase ranges from 62 GPa to 64 GPa in all the samples, comparable to previous studies 25 . The similar mineral properties in turn indicate that the grid nanoindentation results are performed on a group of materials with similar mechanical behaviors.
Change of porosity predicted by mesoscopic models with varied building block sizes. Snapshot in Fig. 3a shows the configuration of the model with aspect ratio equal to 14.6. Figure 3b shows that the total energy of the system decreases to a stable value after 1-ns simulation, indicating an equilibrated state. The equilibrated trajectory shows randomly packed circles (front surfaces of a disk) and thin films (lateral section of a disk) from the side view as shown in Fig. 3c. The schematic drawings in Fig. 3d show three typical structures formed by those objects. Volume fraction of pores in the packed system is computed and displayed by Fig. 4a. It shows that the porosity of packed disks increases as the aspect ratio of the disk increases. The trend corresponds to what we have observed from nanoindentation tests, i.e., C-S-H prepared at a higher w/c ratio contains more low-density phases and shows a higher porosity. The volume fraction of pores with different sizes is plotted in Fig. 4b. It shows that high-aspect-ratio disks tend to form large pores.   Typical packing patterns of C-S-H building blocks. The packing of the disk-like objects is analyzed in 2-dimension space because 3-dimension structures can be learned as an integration of 2-dimension slices. Three characteristic pore structures are observed. Schematic diagram in Fig. 5a presents a bunch of circles, or cross sections of cylinder columns, representing a densely packed pattern in C-S-H. We use a letter O to denote such dense packing pattern. In O-type structure, the pore (area in between the tangent circles) size is smaller than 1 nm, corresponding to intra-globular pores. The dashed lines outline the repeating unit in O-type structure, which is composed of three tangent circles. The packing density of the unit is 0.907, independent of the aspect ratio of the C-S-H building block. The second type of structure is a combination of circles and thin films. We name this  structure by a Greek letter Ω due to the resemblance in shape. As shown in Fig. 5b, this kind of structure is formed at the boundary between columns and lateral sections of disk-like objects. Size of the pores in Ω-shape ranges from 1 nm to 10 nm, corresponding to small gel pores. Packing density of the repeating unit in Ω-type structure is nearly independent of the aspect ratio of the C-S-H building block and the values range from 0.797 to 0.799, lower than HD (~0.85 in packing density) C-S-H phases. So we expect that HD C-S-H phase could be composed of O-type and Ω-type structures, while LD (~0.75 in packing density) should contain another loosely packed pattern. The third type of structure is formed by thin films, the lateral sections of the C-S-H building blocks shown in Fig. 5c. This structure is similar to a Greek letter Δ, indicating the simplest 2-dimension enclosure (i.e. a triangle) formed by pieces of thin films. The pore in the Δ-shape is larger than 10 nm, corresponding to capillary pores (i.e. 10 nm to 50 nm). The repeating unit in the Δ-shape structure is loosely packed and its packing density is a function of the diameter-to-thickness ratio. Packing density of this structure reduces from 0.190 to 0.166 among the models with increasing aspect ratio of disks (from 14.6 to 17.3 in model 1 to 7).

Discussion
At nanoscale (smaller than 10 nm), the structural water is found to be a scaffolding component enhancing mechanical properties of C-S-H nanoparticles 13 . On the contrary, the sub-micron mechanical properties (characterized by nanoindentation with a detecting window of around 500 nm) are lower when water content increases 24 . This contradiction could originate at the transient scale 1 (mesoscale, around 100 nm) between the nano-and the sub-micron scale. At the mesoscale, excess water content could influence the structure 23 of flocculated C-S-H nanoparticles and degrade the mechanical properties.
In this paper, grid nanoindentation results show increasing amounts of low-density C-S-H phases and increasing porosity as w/c ratio increases from 0.3 to 0.66. This change of porosity could be related to the mesoscopic (around 100 nm) C-S-H structure, which is assumed to be composed of disk-like C-S-H building blocks. According to a SANS experiment 23 , the aspect ratio (diameter-to-thickness ratio) of C-S-H disk increases when water content increases. Following this observation, we simulate packing behaviors of rigid disks with varied aspect ratios. The simulations embody the nature of C-S-H by setting the shape of the disks close to C-S-H disks in reference to experiments 18,21,23 . Statistical analysis on pore size shows that high-aspect-ratio disks are likely to enclose large pores. The simulation is limited by a lack of information (such as polydispersity and compaction) about the C-S-H packing in real world so it must not be an exact reproduction of real C-S-H. However, the conceptual analysis on the packing patterns should be generic for similar colloidal systems composed of disk-like objects. Conceptually, we look at 3 typical packing patterns, including i) face-to-face columns, ii) face-to-side envelopes, and iii) side-to-side polygons, to demonstrate how aspect ratio influences pore formation. The side-to-side polygon shape features a packing density adaptive to the aspect ratio of C-S-H platelets, i.e., more oblate platelets (with a higher aspect ratio) lead to a lower packing density. Combining the experimental observations and numerical simulations, we conclude that the increase of w/c ratio would increase the aspect ratio of C-S-H building blocks, enlarging the size and volume fraction of pores formed at the scale of around 100 nm.

Methods
Sample preparation. Cement pastes were prepared using ordinary Type I Portland cement. The water-tocement (mass) ratio ranges from 0.3 to 0.7. It should be mentioned that for the 0.7 w/c ratio case, bleed water was accumulated and segregation was observed in cubic mold. In order to calculate the effective w/c ratio, we ( 3 ) 2 is highly related to the diameter to thickness ratio. measured the density of cement after unmolding. The effective w/c ratio is 0.66 in the designed 0.7 w/c ratio case. Details of the sample preparation are provided in the supporting information.
Nanoindentation test. The Triboindenter with a Berkovich tip (angle of 65.03°, tip radius of 0.2 μ m) was used. A total of 100 indentations which were distributed as 10 × 10 grid were performed with 10 μ m spacing length in each specimen. The designed loading program for C-S-H structure is as follows: 10 s for loading, 5 s for holding and 10 s for unloading stage. The 5 s holding time for peak load is designed to minimize the creep effect on the unloading 35 . Through applying continuum scale model to the P-h curve, two important quantities, i.e. hardness H and indentation modulus E r can be calculated using equations (1) and (2) 36 : where P max is the measured maximum indentation force; A is the projected contact area; S is the stiffness of unloading curve that can be evaluated based on = = ( ) A full set of nanoindentation data is provided in supporting information.
Statistical analysis on nanoindentation data. After obtaining a matrix of (H, M) data points, we adopt the following statistical approaches to extract information about packing density (η). For details of these formulas of the two functions in equation (3), the reader is kindly referred to the supporting information as well as literature 25 . Now we have a collection of (H, M, η) data. Each (H, M, η) data point falls into one of four characteristic compositions, namely loosely-packed (LP), low-density (LD), high-density (HD) and ultra-high-density (UHD) C-S-H. As a result, the entire (H, M, η) data collection, represented by a cumulative distribution function (CDF), is a combination of 4 sub-functions. Experimental CDF is obtained by counting data points as defined by equation (5). Theoretically, the CDF can be deconvoluted into 4 sub-functions as shown in equation (6). These sub-functions are in form of Gaussian CDF with mean value μ, standard deviation s and phase fraction f.
Molecular dynamics simulation. The fundamental assumption for developing coarse-grained C-S-H models is the concept of disk-like building blocks 11,30 , which have been identified in recent experimental works 18,23,37 . It has been found that disk-like objects with high aspect ratio (diameter-to-thickness ratio) exist in C-S-H. The C-S-H in cement paste prepared at higher w/c ratios consists of larger disk-like building blocks. Hence, we set up a series of models with increasing aspect ratios for representing the increasing trend of w/c ratio. The diameter is from 3.25 nm to 4.75 nm, according to colloidal model-II (CM-II) 18,21 . The aspect ratio increases from 14.6 to 17.3, corresponding to the SANS experiment 23 . The interaction between building blocks is described by the Gay-Berne potential 38 , which calculates potential energy between pairwise spheroids, with considerations of shape, rotation and position of each particle. It has been employed to simulate clay minerals with an oblate shape 39 and disk-like C-S-H building blocks as well 11,30,31 . The key idea is that when the aspect ratio of an ellipsoid is large (above 10), the ellipsoid resembles a disk. The potential energy of a pair of platelets is calculated by equation (9). Detailed formula of the function in equation (9) can be found in literature 39 and supporting information. Unknown parameters {∈ a , ∈ b , ∈ c }, which determine the values of energy well depth, can be derived from a minimization process, as shown by equation (10). The U exp is the adhesion energy of C-S-H, calculated by multiplying the normalized surface energy by the surface area, e.g., in the face-to-face case π = = ⋅ U r G bc ( (1 0 0)) exp 12 , where G~450 mj/m 2 is characterized by both experiments and atomistic simulations 11,40,41 . After defining the parameters for GB potential, we perform molecular dynamics simulations. The initial coarse-grained (CG) model is set up by averagely distributing the 8000 (20 × 20 × 20) beads in a simulation box, with a 6-nm spacing. The cutoff of GB potential is set to be 6.25 nm. With a 1 fs time step, the system is equilibrated for 20 ns in NPT ensemble with temperature and pressure controlled at 300 K and 1 atm respectively. The equilibrated trajectory is analyzed statistically to obtain porosity and pore size distribution. Details about the minimization process, the parameters and the statistical analysis can be found in supporting information.