## Abstract

Gelation and densification of calcium–silicate–hydrate take place during cement hydration. Both processes are crucial for the development of cement strength, and for the long-term evolution of concrete structures. However, the physicochemical environment evolves during cement formation, making it difficult to disentangle what factors are crucial for the mechanical properties. Here we use Monte Carlo and Molecular Dynamics simulations to study a coarse-grained model of cement formation, and investigate the equilibrium and arrested states. We can correlate the various structures with the time evolution of the interactions between the nano-hydrates during the preparation of cement. The novel emerging picture is that the changes of the physicochemical environment, which dictate the evolution of the effective interactions, specifically favour the early gel formation and its continuous densification. Our observations help us understand how cement attains its unique strength and may help in the rational design of the properties of cement and related materials.

## Introduction

Cement is the main binding agent of concrete, the most widely used, durable building
material. Its production is responsible for 5–8% of anthropogenic
carbon dioxide production^{1}. In spite of its great practical importance,
there is only limited understanding of the fundamental physicochemical processes that
give the material its functional properties. At present, this lack of understanding
hinders progress towards ‘rational’ cement design, which will be
needed for designing novel *green* formulations with a reduced carbon
footprint^{2}. For the control of the mechanical properties,
calcium–silicate–hydrate (C–S–H), the primary
hydration product of Portland cement paste, is of particular importance.
C–S–H is known to precipitate as nanoscale particles from the
so-called ‘pore solution’ that forms upon mixing the polydisperse
cement powder with water. The particles form a highly cohesive gel that acts as a
spontaneously densifying glue. The early-stage properties of this gel affect the
ultimate strength that cement (and hence concrete) can attain upon hardening at the end
of the hydration process^{3,4,5,6}. Hence, somewhat surprisingly, the
ultimate hardness of cement, which is important for its role in concrete, is determined
by the structural properties of the C–S–H gel, that is, of the soft
precursor of the final hardened paste.

Experiments to probe the gel formation are challenging and therefore few in number^{7,8}. More work has been carried out, in contrast, on the hardened paste
that can be probed by neutron scattering and by atomic force microscopy (AFM) or
scanning electron microscopy imaging^{9,10,11,12,13}. These studies,
mainly performed on tri-calcium silicate (the main source of C–S–H
in Portland cement and often used as a model system for cement hydration), indicate an
amorphous mesoscale organization comprising structural units of locally compact
assemblies of calcium–silicate layers with a typical size of
≃10 nm (refs 14, 15, 16, 17).
The main theoretical approaches for modelling the structural evolution of the
C–S–H gels have either assumed mesoscale nucleation and growth or
nanoscale Diffusion Limited Cluster Aggregation^{18,19,20,21,22}.
Nucleation and growth models are based on the analysis of multicomponent phase diagrams
and assume a kinetic evolution scenario to match the one observed in experiments.
diffusion-limited cluster aggregation, on the other hand, assumes a completely random,
irreversible (and hence purely kinetic) aggregation of colloidal nanoparticles.
Calorimetric studies of the (exothermic) C–S–H production show that
time evolution of this process is strongly non-monotonic^{18}, indicating
that the gel growth goes through an acceleration and a deceleration regime, a feature
that is difficult to reconcile with the growth kinetics expected for random, fractal
aggregation. As a matter of fact, there is at present no theoretical approach that
derives, rather than postulates, the scenario that links gel growth to the kinetics and
the final morphology and properties of the hardened paste. As a consequence, there is no
clear answer to the following basic questions: how can nonequilibrium aggregation, which
takes place in a complex and evolving physicochemical environment, still result robustly
in reproducible material properties? How does the low-density, fractal structure that is
typically produced by irreversible random aggregation evolve to the high density
required to give hardened cement its final strength?

One key experimental observation is that the effective interactions between the
C–S–H units depend strongly on the chemical environment, which
evolves in time because of the dissolution of the cement grains and the precipitation of
various hydration products^{23}. As the chemical composition of the solvent
changes, the effective attraction between the nanoscale units increases, mainly because
of increasing ionic concentrations in the pore solution. Owing to the presence of
multivalent ions, the effective interactions between the nanoscale units show a
combination of a short-range attraction and a longer-range electrostatic repulsion,
which progressively disappears by the end of the setting^{17,23,24,25,26,27,28}. In principle, the dependence of interparticle
interactions on the composition of the pore solution could be reproduced using fully
atomistic models^{29,30}. However, simulation of the phase diagram of such
models would be prohibitively expensive. Using a coarse-grained ‘implicit
solvent’ model that accounts for the time evolution of the interparticle
interactions can therefore help investigate how the equilibrium properties and the
aggregation of the nanoscale C–S–H clusters evolve in time during
hydration.

Here we report simulations that allow us to address these open questions and propose a consistent scenario for C–S–H gelation and densification. The computer simulations allow us to identify and characterize both stable and the kinetically arrested states that occur over a range of low to intermediate volume fractions, corresponding to the early stages of cement hydration. Surprisingly, we find that the morphology of the gel, the early development of mechanical strength and the continuous densification of the material are all intimately related to the underlying equilibrium-phase behaviour that is expected for the (evolving) effective interactions between the hydrates. This observation is far from obvious as out-of-equilibrium conditions prevail during gelation and densification. The implication is that the evolution of the material properties strongly depends on the thermodynamic conditions encountered on the path towards hardened cement, and is surprisingly less sensitive than expected to the precise timeline for traversing this path. These findings are important because they indicate that rational design of the properties of cement can be achieved by controlling the effective hydrate interactions along the densification path by a judicious choice of the physicochemical environment and/or the chemical composition of the pore liquid during densification. This would open the way to ‘nanoscale’ engineering of cement, the ultimate ‘bulk’ material.

## Results

## Model and numerical simulations

We describe the C–S–H hydrates as colloidal nanoparticles
with a diameter *σ* (ref. 5). Without
much loss of generality we assume that the interactions between the particles
are pairwise additive. The characteristic features of the pair potentials were
chosen to agree with experimental observations and atomistic simulations:
hydration experiments performed at fixed lime concentrations^{31},
nanoindentation experiments in the hardened paste^{32,33} and
atomistic simulations^{30,34,35} are consistent with a model where
the hydrate particles of typical size ≃10 nm have
short-ranged attractive interactions, with a strain at rupture of
≃5%. The hydration experiments as well as calculations of
ion correlation forces also indicate an intermediate-range residual repulsion
(typical screening length for C–S–H in water is
≃5 nm); the strength of this repulsion is controlled by the
pH, which in turn depends on the lime concentration. The strength of the
repulsion decreases in time as the lime concentration increases during the
cement hydration. However, the range of the repulsion, which depends on the
ionic strength, is roughly constant since the ionic strength is not changing
much during lime dissolution^{16,23,36}. We therefore assume a
generic model for effective interactions comprising a short-range attractive
well and a long-range Yukawa repulsion^{37}:

*r* is the interparticle distance and *κ* is the inverse
screening length. The ratio *A*_{1}/*A*_{2} gives the
relative strength of attraction and repulsion, while sets the energy scale. Inspired by the experimental observations, we
fix two parameter values to *γ*=12 and
*κ*^{−1}=0.5*σ*
and vary *A*_{1} and *A*_{2} in such a manner that the
depth of the attraction well in equation (1) is fixed to
, whereas the height of the repulsive
shoulder, *V*(*r*_{max}), varies with the ratio
*A*_{1}/*A*_{2}. We note that we have used more
parameters (*A*_{1}, *A*_{2}
*and*
) to describe the strength of the potential: two of
the three parameters would suffice. However, by defining as the depth of the potential well, it becomes easier to compare
results for different potentials with equal strength of attraction.

The choice of parameters made in the coarse-grained model allows for a reasonable
comparison with the physical systems. With the particle size
*σ*≃10 nm, the position of the minimum and
the repulsive shoulder in the interactions are consistent with the experimental
results in ref. 31 and recent atomistic simulation
studies^{35}. The reduced temperature in the simulations is
, where *T*_{0} corresponds to
the room temperature, that is, to the condition of the AFM experiments, and
is the depth of the attractive well and
the unit energy in the simulations. Using
2.43 g ml^{−1} for the density
of C–S–H, as in refs 30,
33, setting the reduced temperature to
*T*=0.15 roughly corresponds to interparticle forces measured in
experiments and simulations. In fact, when considering that the cohesive force
measured in the experiments in the early hydration is
≃0.5 nN and the contact area between two
C–S–H nanoparticles can be estimated
≃5 nm^{2}, the unit pressure in the
simulations corresponds to 20 MPa,
which makes the mechanical strength of the gels obtained here comparable to
typical values of hydrating cement paste. Moreover, the same coarse-grained
model was recently used in simulations of the hardened paste at the end of the
hydration^{38}. The morphological and mechanical properties
obtained in those simulations agree well with the experimental observations,
which further supports the choice of the coarse-grained model and interaction
parameters in this work.

We have studied the phase behaviour of the model hydrate for three typical interaction potentials (shown in Fig. 1 with the corresponding parameter values listed in Table 1) within the regime relevant to the cement hydration (roughly ): ‘high shoulder’ (HS), ‘mid shoulder’ (MS) and ‘low shoulder’ (LS).

Numerical simulations of the model allow us to characterize the equilibrium and
nonequilibrium states for different densities and values of the reduced
temperature 0.1≤*T*≤0.5.

The densities are expressed in terms of the particle volume fraction
*ϕ*≡*πNσ*^{3}/6*V*,
where *N* is the total number of particles and *V* the volume of the
simulation box (see also Methods section). We use these reduced units to
facilitate comparison with different real systems that would be characterized by
different values of and σ but by
roughly similar reduced parameters.

## Clusters and preferred local packing

At low temperatures, that is, for interaction strengths sufficiently larger than
*k*_{B}*T*, we consistently observe that the particles
aggregate into small clusters at low-volume fractions and form a complex gel
network upon increasing *ϕ*, regardless of the differences in the
effective potential.

However, the shapes and sizes of the clusters, as well as the gel morphology,
vary distinctly with the interaction potential (see Fig.
2), suggesting that the changes in the effective interactions from HS to
LS during the hydration process might drive significant morphological changes in
the C–S–H gels. In Fig. 3a we compare
the results of the Grand Canonical Monte Carlo (GCMC) and Molecular Dynamics
(MD) simulations at low-volume fractions. In GCMC simulations we evaluated the
free energy of single clusters of a fixed size (infinitely diluted limit),
whereas MD simulations where performed on larger systems
(*N*=6,912) at finite *ϕ* and included
cluster–cluster interactions. In the case of the HS potential, at
low-volume fractions both approaches reveal the existence of small, stable
clusters of a well-defined size. At higher *ϕ* a percolating
network forms through growth and aggregation of the clusters. As the height of
the repulsive shoulder is decreased (MS), the GCMC data show that the
cluster-size distribution at infinite dilution is broader and shifted towards
larger cluster sizes. In addition, the MD simulations for MS show evidence of
large, system-spanning clusters at lower *ϕ*. The trend towards
aggregation is even more pronounced for LS, where the GCMC simulations suggest a
tendency for particles to coalesce into few large clusters (at the expense of
clusters of smaller size). In fact, for LS there is no longer a sharp cutoff in
the cluster-size distribution. Hence, small clusters, once nucleated, tend to
coalesce rapidly. This is clear from the corresponding MD simulations that show
clear evidence for aggregation, even at low-volume fractions
(*ϕ*=0.052). One question is obviously whether the
cluster phase is thermodynamically stable (and, if so, whether it remains so
upon changing the potential). In particular, one might envisage a situation
where the cluster phase is metastable with respect to crystallization. To
estimate the relative stability of the crystalline and the cluster phases, we
have computed the chemical potential of the constituent hydrate particles both
in the cluster phase and in the crystalline phase. In these calculations, we
kept the temperatures and pressures of the disordered and ordered phases the
same. Hence, the relative stability of the two phases is completely determined
by the chemical potential. For the crystalline phase, we used thermodynamic
integration (Einstein crystal method) to compute the free energy per particle of
the crystal as a function of temperature, at constant pressure (the density was
chosen such that *P*≈0). From the free energy per particle, we
directly obtain the chemical potential of the particles in the crystal. Table 2 summarizes these results. For the cluster-fluid
phase, we determined the chemical potential by using the Widom particle
insertion method (see Methods) at volume fractions between
*ϕ*=0.026 and *ϕ*=0.0104 and
pressure ≃0. The results are plotted in Fig. 3b,c
for HS and LS, respectively, where they are compared with the chemical potential
of the crystalline states (black symbols *) obtained from the free
energy calculated using the Einstein crystal method and thermodynamic
integration (see Methods). For HS the clusters are clearly more stable than the
crystalline states. Going to LS instead, the crystalline states have comparable
or lower free energies, supporting the idea that in this case the clusters, even
though they persist throughout the simulation runs, are only metastable.

In all cases, the shapes of the clusters change with their size: small clusters
tend to be roughly spherical, whereas larger clusters are more elongated. A
quantitative measure of the elongation is given by the normalized asphericity
(see Methods), which equals zero for a
spherically symmetric object, +1 for a needle and −0.5 for a
disk. The dependence of the asphericity on the cluster size is plotted in Fig. 4a. The data show that upon going from the HS to the LS
potential the asphericity decreases: small- and intermediate-size clusters have
a fibrillar shape in HS and MS, whereas they are much more spherical in LS,
where the asphericity emerges only for much larger clusters (see Supplementary Fig. 1). Analysis of the
particle arrangements inside the aggregates reveals that the (metastable)
clusters for the LS case are quite crystalline (see Supplementary Fig. 2). The rotational
invariants and
computed from the local bond orientational order (BOO; see Methods)^{39} allow us to distinguish face-centred cubic (fcc) or hexagonal
close-packed (hcp) crystals from the orientational order typical of Bernal
spirals (BS), which is compatible with the fibrillar growth of the clusters in
HS and MS. Figure 4b shows indeed that the nature of the
local packing in the LS clusters is quite different from that in the HS
clusters: for the LS potential, the local packing in clusters appears to fall in
the fcc–hcp range, whereas the HS clusters have a structure that is
more similar to that of a BS, typical of a fibril-like growth^{40}
(see Supplementary Figs 3 and
4).

## Arrested states and hydration kinetics

Upon increasing the volume fraction of hydrate particles, the clusters grow and
interconnect into a gel. The morphology and local packing of this gel also
depends on shape of the interaction potential. For HS potential, the fibrillar
crystals that are stable at low densities easily grow into a system-spanning
network at relatively low-volume fractions, allowing for early development of
mechanical properties. Upon decreasing the intermediate-range repulsion towards
LS, the fibrils are progressively substituted by large dense and crystalline
clusters coexisting with a gas phase. These large clusters can form
system-spanning arrested states only at higher-volume fractions (with respect to
HS) while favouring the growth of extended and homogeneous high-density regions
in the material. The observed equilibrium and arrested states are summarized in
the diagrams of Fig. 5. The solid blue line in HS marks
the onset of the geometric percolation of the fibrillar clusters (see Supplementary Fig. 5). Along the
dashed lines, energy fluctuations measured in MD simulations (NVT, i.e. at
constant number of particles N, volume V, and temperature T), display a very
weak maximum for HS, as breaking and reforming fibrillar clusters become
persistent, and a rather sharp peak in LS, as clusters form from the gas phase,
suggesting for LS some kind of coexistence (gas–solid or
gas–liquid; see Supplementary
Fig. 6). The analysis of equilibrium and arrested states discussed so
far indicates that packing, local density and morphology changes can occur in
the assembly of C–S–H nanoparticles because of changes in
the effective interactions during cement hydration (from HS to LS),
characterized by a continuous increase in the lime concentration due to the
dissolution of the cement grains. As the lime concentration increases, the
electrostatic repulsion between the nanoscale hydrates is progressively
screened^{16} (that is, going from HS to LS). In our model, we
assumed that the C–S–H nanoscale units are monodisperse.
This is of course an oversimplification: although these nanoclusters have a
characteristic size, they seem to be closer to platelet than spheres and tend to
be quite polydisperse^{11,31,41,42}. Size and shape
polydispersity is likely to decrease the stability and predominance of any
crystalline structure. However, we expect that the overall aggregation, gelation
and densification scenarios discussed above still apply. Aggregate morphologies
very similar to the ones discussed here in our equilibrium simulations were
indeed observed in nonequilibrium simulations mimicking the precipitation and
densification of C–S–H gels with the same effective
interactions^{43,44}. As a matter of fact, the anisotropic
growth characteristic of the HS regime, followed by branching, and the
densification/rigidification favoured by an evolution towards MS and LS can be
directly associated to hallmark steps in cement hydration, corresponding to
different stages of the kinetics. Figure 6a shows the
hydration curve measured via calorimetry (reprinted from ref. 6): the colour bar indicates the possible evolution of the
effective interactions over time^{16}. The early times,
corresponding to lower lime concentrations and HS interactions, are
characterized by an accelerating regime corresponding to the growth of fibrils
that rapidly lead to branching and percolation. The space-filling gel slows down
the kinetics and the deceleration regime follows: however, it corresponds to a
change towards LS, which favours further densification. Summarizing our
findings, the plot in Fig. 6b illustrates the pathways to
mechanical strength (Supplementary Fig.
7 shows a more detailed analysis). To achieve optimal mechanical
strength, cement hydrates should first form an open percolating network that can
act as a scaffold to form homogeneous dense structures through the densification
pathways. The dashed lines depict the sequences of equilibrium structures upon
densification at fixed interaction potentials (HS: blue, MS: black, LS: red),
whereas the arrow with the changing background colour indicates a possible
nonequilibrium pathway with time-varying interactions from HS to MS and LS,
consistent with the experiments in ref. 16. Such a
pathway can be manipulated by controlling the chemical composition and its
evolution before and during hydration. Building on the results obtained here,
one could quantitatively explore the consequences, for the morphology and the
mechanical properties, of a different evolution of the effective interactions by
suitably designed nonequilibrium simulations. The Supplementary Movie 1 issued from
preliminary simulations serves as a proof of concept. These ideas call for a
systematic analysis of how changing chemical composition of the dissolving
cement and/or chemicophysical conditions (that is, controlling pH, temperature
or using various additives) may change the nanoscale effective interactions
during hydration as well as their evolution in time. Such an analysis would
benefit from experimental, theoretical and simulations combined efforts. With
such an analysis at hand, our approach would allow for quantitative predictions
of which chemicophysical modifications of C–S–H and cement
hydration can enhance strength development, final mechanical performance and
durability.

## Discussion

The simulations presented in this paper provide new insight into how the underlying chemistry of cement formation allows this material to attain its unique mechanical strength, and explain how the natural time evolution of the interactions between hydrate nanoparticles is crucial for attaining the mechanical strength. As the hydrate concentration increases, the repulsion between the nanoparticles gets weaker and densification sets in through thickening of the strands of the gel. This thickening results in increased mechanical strength. Importantly, as gel formation already takes place at low-volume fractions with HS, the densification happens uniformly throughout the system. This is important for the development of mechanical strength: if the densification were to take place through phase separation involving large droplets or crystallites, the resulting material would lack the cohesion that is crucial for its mechanical strength. Moreover, the evolution of the chemical environment that progressively screens electrostatic repulsion as in LS ends up favouring further the continuous densification of the material and hence the development of the mechanical strength needed for construction materials.

Understanding the mechanism by which cement is able to attain its unique mechanical strength is important for the rational design of novel, cement-like construction materials. The implication is a novel perspective and opportunity to start designing the hydration process and material properties. By providing new fundamental understanding, our study gives a proof of principles of how controlling and changing the chemical composition and the time evolution of the chemical environment can become the handles for a scientifically guided material design of cements. While indicating a novel investigation path for nanoscale research in cement, our work provides a new opportunity to bridge the microscopic features of the material with the engineering-scale description as attempted in refs 45, 46, 47. The same insights could also be used profitably to achieve high mechanical strength in other (for example, polymeric) materials that can be formed by nanoparticle aggregation.

## Methods

## MD and Monte Carlo simulations

We have investigated the three cases (LS, MS and HS) by MD simulations in the
microcanonical (NVE, i.e., at constant number of particles N, volume V and
energy E) and canonical (NVT) ensemble using a Nosé–Hoover
thermostat^{48}. Initial configurations of 2,048 and 6,912
particles were prepared at different fixed-volume fractions by varying the
simulation box size (from
*L*_{box}=13.679*σ* to
*L*_{box}=51.706*σ*), at high
relative temperature (*T*=1), and slowly cooled to carefully
equilibrate them at *T*=0.15. We used a time step
*δt*=0.0025 for the MD, with up to
10^{7} steps for thermalization. All the results presented from
the MD simulations are averaged over five independent samples. From the particle
number density *ρ* we estimate the fraction *ϕ* of
the total volume occupied by particles as approximately
*ϕ*≃(*π*/6)*ρσ*^{3}.
Most of the simulations were performed by LAMMPS^{49}. We have
monitored internal energy, kinetic energy and different time-correlation
functions, which did not show any significant aging up to
*ϕ*≃0.15. For *ϕ*>0.15 the
thermalization requires increasingly longer times and in the simulations we
obtained arrested states that depend on the thermalization protocol and on the
system size.

In order to sample more efficiently the equilibrium phases, we have performed
parallel tempering Monte Carlo (MC) simulations^{50}. We simulated
*M*=40 replicas of the same system at different temperatures
*T*_{1},…,*T*_{M}. The
replicas’ temperatures range from *T*_{1}=0.15
to *T*_{M}=0.7 in such a way that the two adjacent
temperatures differ by a factor of
*T*_{i+1}/*T*_{i}=1.04.
Replicas are either randomly initialized or initialized from the MD
configurations equilibrated at the corresponding temperatures. For each replica
we perform the same number of MC steps at a given temperature using MC moves of
individual particles as well as cluster moves, that is, translation moves of all
the particles that belong to the same cluster. Cluster moves that result in
cluster fusion are rejected.

We attempt to swap particle configurations between two adjacent replicas *i*
and *i*+1 with the prescribed Metropolis-based probability^{50},

where *T*_{i} and *T*_{i+1} are
the temperatures and *U*_{i} and
*U*_{i+1} are the total potential energies of
the two replicas. In order to achieve sufficient swapping acceptance between
replicas, the average energy difference between the adjacent replicas
*U*_{i}−*U*_{i+1}
need to be comparable to energy fluctuations. Since expected energy
*U*_{i} scales linearly with the number of particles
*N*, whereas the corresponding fluctuations as , the larger the number of particles the smaller the temperature
difference between the adjacent replicas needed to be, that is, more replicas
are needed to cover the desired temperature range. Therefore, for MC we have
used a smaller number of particles (*N*≃500). This enables for a
successful replica-swapping with acceptance probabilities of
∼0.10–0.3, which is considered an optimal choice^{51}. We performed simulations with 20–80 ×
10^{6} MC steps, using the first half for equilibration and the
second half for sampling.

The percolation thresholds just reported depend on the temperature and also on
the cooling protocol used for the systems at different *T* and
*ϕ*. They have been evaluated for a fixed system size; hence,
they are not intended to be exact by any means, since this would require a
finite-size scaling analysis. The arrested states discussed here depend of
course on the specific kinetic path selected by the MD simulations, that is,
they are specific of the NVE or NVT MD simulations performed and of the cooling
protocol chosen. We have verified that, for sufficiently slow cooling, the gel
morphology and properties discussed here are not significantly affected by the
cooling rate. They could change if MD simulations were performed at constant
pressure (rather than at constant volume), although we have monitored the
pressure in our simulations and we could not detect any steep or abrupt change
in pressure upon gel formation. On this basis, we do not expect too large
differences. With respect to experimental conditions, one could expect gelation
to result into the development of tensile internal stresses because of the net
attractive interactions that drive the gelation and to the formation of a poorly
connected structure. Hence, the periodic boundary conditions and the imposed
fixed volume used for the MD simulations could share some similarities with the
experimental conditions under which gels are forming.

## Free energy calculations

We have used a reversible path between the solid obtained in the simulations and
an ideal Einstein crystal with the same structure^{52,53}. The
ideal Einstein crystal consists of particles attached to their lattice positions
via harmonic springs of constant Λ_{E}, and its free energy
*A*_{0}(*T*,*ρ*) can be analytically
evaluated. The free energy difference,
Δ*A*_{1}(*T*,*ρ*), between the
ideal Einstein crystal and the interacting Einstein crystal (in which particles
interact through the interparticle potential used in the simulations) can be
expressed as

where *U*_{lattice} is the potential energy of the perfect lattice
and *U*_{s} the potential energy of the solid obtained in the
simulations with the given interparticle interactions. The free energy
difference between the interacting Einstein crystal and this solid
Δ*A*_{2}(*T*,*ρ*) can be computed
as

where the integrand is the mean square displacement of particles from their
lattice positions. To calculate
Δ*A*_{1}(*T*,*ρ*) and
Δ*A*_{2}(*T*,*ρ*), we performed
canonical (NPT, i.e. at constant number of particles N, pressure P and
temperature T) MD simulations of 2,048 particles at zero pressure and different
temperatures using LAMMPS^{49,54}. From
Δ*A*_{1}(*T*,*ρ*) and
Δ*A*_{2}(*T*,*ρ*), we get the free
energy of the solid obtained in the simulations as

Additional thermodynamic information has been obtained by measuring the chemical
potential using the Widom insertion method^{55}. The total chemical
potential *μ* of interacting particles is composed of two parts,

with the first term corresponding to an ideal gas contribution of density
*ρ*, and *μ*_{exc} being the excess part
due to the presence of particle–particle interactions. For the purpose
of test particle insertions, we used particle configurations obtained by the MC
simulations. From the resulting potential energies, *U*_{i}
of the inserted particle at random positions (2 × 10^{6}
per sampled snapshot), the excess chemical potential can be evaluated from

where 〈…〉 denotes the average over all particle positions over the whole sampled trajectories.

Owing to the lower repulsion, LS crystals have much lower free energy than the HS ones (The free energies of the fcc and hcp crystals are the same within the error bar of our calculation; therefore, we cannot distinguish which crystal is more stable.).

## Grand canonical simulations

As already mentioned, in cases of low *ϕ* we obtain persistent
clusters whose nature, stable or metastable, is difficult to discriminate. To
elucidate this point, we have also evaluated the cluster-size distribution by
free energy calculation using GCMC simulations. We followed the same procedure
introduced to investigate micelle formation in refs 56, 57.

In an approximation of a dilute gas of clusters, such that interactions between the clusters are negligible, we can decompose the total partition function of the system into contributions of individual clusters

where *N*_{s} is the number of clusters with size *s*,
*V* is the volume of the system and *Q*_{s} is the
single cluster (of size *s*) partition function. We have ignored the
(irrelevant) factor because of the thermal de Broglie wavelength. Upon
minimizing the total free energy −*k*_{B}*T* log
*Q* with respect to cluster size *N*_{s}, with the
constraint of total number of particles , one
obtains the number of clusters of size *s* in equilibrium,

The related cluster-size distribution is therefore
*n*_{s}=*N*_{s}/*N*_{cl},
where corresponds to the total number of clusters
in the system. This yields

with *ρ*=*N*/*V* being the particle number
density. The parameter plays the role of the
chemical potential in a diluted system with
*β*=1/*k*_{B}*T*, and the
parameters
*n*_{1}=*N*_{1}/*N*_{cl}
and are fixed by the two constraints

The ratio of partition functions *Q*_{s}/*Q*_{1}
is computed in the GCMC simulations. As in the MC simulations, we implemented a
parallel tempering scheme, but limited the simulations to one single cluster by
rejecting all the moves that would break the cluster. That is, besides particle
translation moves as in usual MC, we also performed insertions and deletions of
particles according to the standard GCMC scheme with the restriction to preserve
a single cluster in the system, that is, only insertions within the attractive
wells of particle interactions were attempted. Here the volume in the standard
GCMC acceptance rule for the particle insertion corresponds to the total volume
of all attractive wells. In order to preserve detailed balance, in case of
overlap of *k* attractive wells of neighbouring particles at the location
of an insertion attempt, the standard insertion acceptance probability needs to
be corrected (multiplied) by a factor 1/*k*. During the simulation, we were
sampling the probability distribution of the single cluster size
*P*(*s*), which is related to the ratio of the partition functions
as

where *μ*_{GC} is an arbitrary value used in the simulations
for the chemical potential, which does not affect the ratio
*Q*_{s}/*Q*_{1} but influences the
sampling efficiency. The chemical potential *μ*_{GC} used in
GCMC should not be confused with in equation (10). In order to further increase sampling
efficiency, we have implemented the standard Umbrella Sampling scheme where we
sample at once a cluster of only two possible sizes, *s* and
*s*+1. The results for all cluster sizes are obtained by stitching
together the simulation results of different cluster sizes^{57}.

## Observables

The radial distribution function *g*(*r*) is defined as

where *N* is the number of particles and *ρ* their density. We
define a cluster as a group of those particles that are mutually separated by
*r*≤*r*_{c}, where *r*_{c} is distance
of the minimum followed by the first peak in *g*(*r*). As the
aggregation proceeds, we sample the cluster-size distribution as the number of
clusters *n*_{s} composed of *s* particles. To
characterize the morphology of such clusters, we determine the gyration
tensor

where *N*_{s} is the total number of cluster of *s*
particles, is the *a*_{th}
coordinate of the position vector **r**^{i} of the
*i*_{th} particle belonging to the cluster *j* and
is the *a*_{th} coordinate
of the position of the centre of mass of cluster *j*. The gyration radius
is defined as the sum of the eigenvalues of the gyration tensor:

where the eigenvalues of *S*_{ab}(*s*) are sorted in
descending order, that is,
*λ*_{1}≥*λ*_{2}≥*λ*_{3}.
In addition, the asphericity parameter

measures the deviation from the spherical symmetry (*b*=0).

Finally, the BOO parameters enable us to
characterize the local order^{39}. The BOO are defined as

where is the number of the bonded neighbours of
particle *i* (using the cutoff distance *r*_{c}). The unit
vector specifies the orientation of the bond
between particles *i* and *j*. The are
the corresponding spherical harmonics. The second-order rotational
invariants

and the normalized third-order invariants

with

(where the coefficients in the sum are the Wigner 3-j coefficients) can be used
to identify specific orientational order. In fact, the first non-zero
(apart from
*l*=*m*=0) as found in systems with cubic
symmetry for *l*=4 and in systems with icosahedral symmetry for
*l*=6. The BOO parameters ()
are generally sufficient to characterize orientational order typical of
crystalline solids, colloidal gels and glasses. We use and , as they are sufficient to
characterize the local order of interest and distinguish between fcc and hcp
crystals. We compute the BOO locally for each particle within the cutoff
distance *r*_{c}.

## Data availability

The authors declare that all data supporting the findings of this study are available within the article and its Supplementary Information Files.

## Additional information

**How to cite this article:** Ioannidou, K. *et al*. The crucial effect of
early-stage gelation on the mechanical properties of cement hydrates. *Nat.
Commun.* 7:12106 doi: 10.1038/ncomms12106 (2016).

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## Acknowledgements

The authors thank Roland J.-M. Pellenq for insightful discussions on effective interactions in cement hydrates. This work was supported by the SNSF (Grants No. PP00P2_126483/1 and PP00P2_150738) and Georgetown University, by the Fundamental Research Funds for the Central Universities of China, ERC Advanced Grant 227758 (COLSTRUCTION), ITN grant 234810 (COMPLOIDS) and by EPSRC Programme Grant EP/I001352/1. K.I. thanks the French National Research Agency (ICoME2 Labex Project ANR-11-LABX-0053 and A*MIDEX Project ANR-11-IDEX-0001-02) for support.

## Author information

## Affiliations

### Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

- Katerina Ioannidou

### MIT-CNRS Joint Laboratory at Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

- Katerina Ioannidou

### Helmholtz-Zentrum Berlin für Materialien und Energie, D-14109 Berlin, Germany

- Matej Kanduč

### Department of Theoretical Physics, J. Stefan Institute, SI-1000 Ljubljana, Slovenia

- Matej Kanduč

### Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, UK

- Lunna Li
- & Daan Frenkel

### International Research Center for Soft Matter, Beijing University of Chemical Technology, Beijing 100029, China

- Jure Dobnikar

### Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

- Jure Dobnikar

### Department of Physics, Institute for Soft Matter Synthesis and Metrology, Georgetown University, Washington, District of Columbia 20057, USA

- Emanuela Del Gado

## Authors

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## Contributions

K.I., D.F., J.D. and E.D.G. designed research; K.I., M.K. and L.L. performed research; K.I., M.K. and L.L. developed analytical tools and ran the analysis; K.I., M.K., D.F., J.D. and E.D.G. wrote the paper.

## Competing interests

The authors declare no competing financial interests.

## Corresponding authors

Correspondence to Jure Dobnikar or Emanuela Del Gado.

## Supplementary information

## PDF files

- 1.
### Supplementary Figures

Supplementary Figures 1-7

## Videos

- 1.
### Supplementary Movie 1

Animation of the densification of cement hydrates when changing the interactions from HS to LS. The initial configuration was a slowly equilibrated gel structure of HS at φ = 0.15 and T = 0.15. The precipitation of cement hydrates in μVT ensemble with μ = 3, V = 13986.024σ3 and T = 0.15 was performed according to the method introduced in Ref.[44] of the paper and after changing the interaction for all particles (existing and precipitated) to LS. In the video, the bonds among the particles are colored depending on the number of nearest neighbor, using the same color code as in Fig. 2.

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