A multi-scale approach for percolation transition and its application to cement setting

Shortly after mixing cement grains with water, a cementitious fluid paste is formed that immediately transforms into a solid form by a phenomenon known as setting. Setting actually corresponds to the percolation of emergent network structures consisting of dissolving cement grains glued together by nanoscale hydration products, mainly calcium-silicate-hydrates. As happens in many percolation phenomena problems, the theoretical identification of the percolation threshold (i.e. the cement setting) is still challenging, since the length scale where percolation becomes apparent (typically the length of the cement grains, microns) is many times larger than the nanoscale hydrates forming the growing spanning network. Up to now, the long-lasting gap of knowledge on the establishment of a seamless handshake between both scales has been an unsurmountable obstacle for the development of a predictive theory of setting. Herein we present a true multi-scale model which concurrently provides information at the scale of cement grains (microns) and at the scale of the nano-hydrates that emerge during cement hydration. A key feature of the model is the recognition of cement setting as an off-lattice bond percolation process between cement grains. Inasmuch as this is so, the macroscopic probability of forming bonds between cement grains can be statistically analysed in smaller local observation windows containing fewer cement grains, where the nucleation and growth of the nano-hydrates can be explicitly described using a kinetic Monte Carlo Nucleation and Growth model. The most striking result of the model is the finding that only a few links (~12%) between cement grains are needed to reach setting. This directly unveils the importance of explicitly including nano-texture on the description of setting and explains why so low amount of nano-hydrates is needed for forming a spanning network. From the simulations, it becomes evident that this low amount is least affected by processing variables like the water-to-cement ratio and the presence of large quantities of nonreactive fillers. These counter-intuitive predictions were verified by ex-professo experiments that we have carried out to check the validity of our model.


DETAILS ON RELATIONS EMPLOYED
For a Random Hard Sphere (RHS) cement paste system with total volume fraction φ 0 = φ 0 g + φ 0 s , where φ 0 g denotes the initial volume fraction occupied by the cement grains and φ 0 g the initial volume fraction of the substituent material, the liquid to solid weight ratio (L/S) (or the water to cement weight ratio when the substituent is absent) is given by Where, ρ g , ρ s and ρ w denote the density of cement grain (3.15 g cm −3 ), the cement substitute (in the case of sand ρ s =2.64 g cm −3 ) and water (1 g cm −3 ), respectively. With the nuclei placed on a grain, the surface coverage (θ) is calculated as θ = π · r 2 C-S-H · N nuc 4 · π · (d/2) 2 , where r C-S-H is the radius of a C-S-H particle (here 2.5 nm). For small simulation boxes, we populate the entire grain with nuclei and then consider the nuclei within the box as valid, such that θ remains unaffected. Alternatively, the exposed grain surface area can be estimated used Monte Carlo methods. As the dissolution precipitation involving C-S-H formation takes place, we denote φ g as the grain volume fraction at a given time. Assuming uniform dissolution, the reduction in grain diameter is given by For generally observed degrees of hydration at initial set (α set ∼ 0.03), we get ∆x ∼ 4 nm, smaller than the size of a C-S-H particle. This ∆x value is negligible, and our placement of the nuclei being partially overlapping with the grain compensates for this dissolution effect.
At early ages of hydration, the law of mass conservation is valid and dissolution effects are negligible. Thus, we can write where ρ C-S-H is the density of dry colloidal unit (2.8 g cm −3 ) [1]. For non-reactive sand, no dissolution takes place and φ 0 s = φ s . With φ g known, we can calculate the degree of hydration (α) as follows For substituted cements, we choose a reference parent system and set a given amount of randomly selected grains as the substituent particles. This leads to systems that differ in their sand/cement content with relatively unchanged L/S. For the given case, for 0% to 50% substitution by sand in volume (with density ρ s = 2.64 g cm −3 ) gives L/S values ranging from 0.32 to 0.34. Our data (given in article) shows that there is no noticeable difference in α set for the experimental data and simulation data in this range.

EVOLUTION OF NUMBER OF NEIGHBORS WITH EXPLORATION RANGE (ε)
The average number of neighbors (N n ) in the exploration range increases with ε (in unit of d) (see figure  A1). A meaningful choice to study growth from a central grain should include all directly accessible grains in the close vicinity. This restriction leads us to approximate the first coordination sphere as ε (figure A1). For RHS systems, first coordination sphere corresponds to the first minima in its pair correlation function. This choice gives N n ≈ 12 for our selection of L/S from 0.3 to 0.5.

ESTIMATION OF UNCORRELATED BOND PERCOLATION THRESHOLD
In order to obtain the uncorrelated bond percolation threshold, simulations were performed on bigger RHS systems (L = 10d, 20d, 30d, 40d, etc.). A distance range d to (1+ε)·d was specified to determine the neighbors, and bonds were distributed between them with a given probability p. When all possible bonds were distributed, the system is tested for the presence of a percolating cluster in at least one direction. After many such realizations, we obtain the fraction of the total systems that has been percolated, P . This procedure was repeated for various p and system sizes (see Figure A2). For this study, the value of ε was chosen such that it is the distance corresponding to the first minimum in the g(r) curve. A minimum of 10000 samples were simulated for calculating P . The value of average bond probability p and the standard deviation SD= p 2 − p 2 0.5 were calculated by integrating over the derivative of P , where p n = 1 0 p n · d dp P (p) · dp (6) The resultant plot of p as a function of SD was extrapolated, and the p value at zero SD is taken as the system size independent percolation threshold, P c [2,3] .

CHOICE OF L OW BASED ON COARSENESS
The choice of L OW determines the distribution of observed local property or "coarseness" (C), defined as the ratio of standard deviation to the average observed value. The value of C is dependent on the nature of the parent system and the shape and size of the sampling window. For large L OW , an asymptotic limit C = K · v 0.5 exists, where K is a positive value which depends on L/S and v = L 3 OW is the volume of the sampling window [4, p. 261]. The effect of L OW on the evolution of local φ 0 g was studied using a voxel counting method using at least 1000 samples ( figure A3). When L OW ≥ d, the observed C values are close to the corresponding asymptotic values, indicating a good local representation of the parent system. Here v = π/6 · d 3 .

FINITE SIZE EFFECTS ON φC−S−H EVOLUTION
The outward growth of C-S-H continues until it reaches the C-S-H growth from a neighboring grain or the neighboring grain surface itself. Hence the optimal size of the simulation box is determined partially by L/S of the system being modeled. The sampling becomes biased for small box sizes, because samples with large inter grain distance are omitted. The evolution of φ C-S-H depends on the available free space, exposed grain surface, and the number of nuclei present in the system. Having L OW = d appears to be enough to avoid large errors for φ C-S-H values (see Figure A4) and the grain linkage fraction (see Figure A5). The final value φ C-S-H ∼ 0.46 has been observed previously for the colloidal model. The slight difference in the final values of φ C-S-H for small L OW is due to the low number of nuclei leading to relatively lower packing defects.   Setting times obtained from Vicat needle tests according to European standard EN 196-3 are given in table A1 for plain cement pastes with varying L/S and in table A2 for substituted cements with L/S = 0.30 and varying sand substitution fraction. Particle size distributions were measured for the cement (ρ g = 3.15 g cm −3 ) and sand (ρ s = 2.64 g cm −3 ) samples employed in the experiments using laser diffraction, with 2-propanol as the dispersion medium ( Figure  A6).