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Effects of polydispersity and disorder on the mechanical properties of hydrated silicate gels
Accepted Manuscript
Effects of Polydispersity and Disorder on the Mechanical Properties of Hydrated Silicate Gels Han Liu , Shiqi Dong , Longwen Tang , N. M. Anoop Krishnan , Gaurav Sant , Mathieu Bauchy PII: DOI: Reference:
S0022-5096(18)30722-1 https://doi.org/10.1016/j.jmps.2018.10.003 MPS 3470
To appear in:
Journal of the Mechanics and Physics of Solids
Received date: Accepted date:
20 August 2018 1 October 2018
Please cite this article as: Han Liu , Shiqi Dong , Longwen Tang , N. M. Anoop Krishnan , Gaurav Sant , Mathieu Bauchy , Effects of Polydispersity and Disorder on the Mechanical Properties of Hydrated Silicate Gels, Journal of the Mechanics and Physics of Solids (2018), doi: https://doi.org/10.1016/j.jmps.2018.10.003
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Effects of Polydispersity and Disorder on the Mechanical Properties of Hydrated Silicate Gels
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Han Liu a, Shiqi Dong a,b, Longwen Tang a, N. M. Anoop Krishnan a,c, Gaurav Sant b,d, Mathieu Bauchy a,* a Physics of AmoRphous and Inorganic Solids Laboratory (PARISlab), Department of Civil and Environmental Engineering, University of California, Los Angeles, California, 90095, USA b Laboratory for the Chemistry of Construction Materials (LC2), Department of Civil and Environmental Engineering, University of California, Los Angeles, California, 90095, USA c Department of Civil Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India d California Nanosystems Institute (CNSI), University of California, Los Angeles, California, 90095, USA * Corresponding author: [email protected] Keywords: Calcium–silicate–hydrate, Colloidal gels, Mechanical properties, Grand Canonical Monte Carlo, Coarse-grained simulations
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Abstract
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The colloidal calcium–silicate–hydrate (C–S–H) gel largely controls the strength of concrete. However, little remains known about how the structural features of the C–S–H gel control its
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mechanical properties. Here, based on coarse-grained mesoscale simulations, we investigate the effect of grain polydispersity and structural disorder on the nanomechanics of C–S–H. Our
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simulations offer a good agreement with nanoindentation data over a large range of packing density values. We show that, at constant packing density, stiffness and hardness are not affected
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by the polydispersity in grain sizes. In contrast, the level of disorder is found to play a critical role. We demonstrate that, in contrast to the case of ordered C–S–H models, the elastic response of disordered C–S–H gels is governed by the existence of stress heterogeneity and nanoyielding within its structure. These results highlight the intrinsically disordered nature of C–S–H and the crucial role of order and disorder in controlling gels’ mechanical properties. 1
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Graphical Abstract The extent of order and disorder in hydrated silicate gels controls mainly shear resistance through stress heterogeneity.
G: shear
2
Low
1 0
High
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-1
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Shear stress per grain
-2
Crystalline
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CSH Gel
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Introduction Colloidal gels—i.e., systems made of interacting grains with sizes ranging from nanometers to hundreds of nanometers (Dinsmore and Weitz, 2002; Lin et al., 1989)—are widely used in
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many realms of science and technology (Joshi, 2014; Masoero et al., 2012; Wang Q. et al., 2007). Colloid aggregation and gelation is controlled by a complex interplay between thermodynamics and kinetics (Anderson and Lekkerkerker, 2002; Lu et al., 2008; Zaccarelli, 2007). After gelation, the structure of colloidal gels largely depends on the formation process (Trappe and Sandkühler,
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2004; Zaccarelli, 2007). In particular, the degree of polydispersity (i.e., the distribution of grain sizes) and the level of disorder affect the mechanical properties of colloidal gels—although such linkages remain poorly understood (Masoero et al., 2014, 2012; Tong et al., 2015). Calcium–silicate–hydrate (C–S–H) gel—the glue of concrete that forms upon the hydration
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of cement—is a typical inorganic colloidal hydrogel material (Ioannidou et al., 2016b; Jennings,
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2008; Masoero et al., 2012). The C–S–H phase largely controls the strength of cement paste and concrete (Bullard et al., 2011; Ioannidou et al., 2016a). This is significant as, due to its large
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carbon impact (Bauchy, 2017; Le Quéré et al., 2012; Liu et al., 2018), there is a strong interest in improving concrete’s mechanical properties—i.e., so that less material can be used while
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achieving constant performance. In turn, the colloidal structure of C–S–H largely controls its
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mechanical response (Constantinides and Ulm, 2007; Masoero et al., 2014; Vandamme et al., 2010). At the mesoscale, C–S–H forms a polydisperse, disordered gel-like structure, with the grain size ranging from nanometers to several tens nanometers (Ioannidou et al., 2016b; Jennings, 2008; Masoero et al., 2012). However, the linkages between the structure and mechanical behavior of C–S–H are not fully elucidated.
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Recently, Masoero et al. introduced a polydisperse colloidal model of C–S–H that shows a good agreement with nanoindentation experiments (Masoero et al., 2012). Based on this model, the packing density was shown to have a first order effect on the mechanical properties of C–S– H (Masoero et al., 2014, 2012). In the same spirit, Ioannidou et al. proposed an alternative model
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to account for the early-age structure and properties of C–S–H (Ioannidou et al., 2016a). These colloidal models have been widely investigated to elucidate the relationship between C–S–H’s structure and mechanical properties (Colombo and Del Gado, 2014; Ioannidou et al., 2016b, 2014; Masoero et al., 2014). However, several questions, in this regard, remain unanswered.
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Which structural features have a first-order effect and which one do not? What is the effect of polydispersity and structural disorder on the mechanical properties of C–S–H? Indeed, although polydispersity has been shown to play a critical role in controlling the packing density (and,
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thereby, the mechanical properties) (Masoero et al., 2014, 2012), the role of polydispersity at constant packing density remains unclear. Similarly, although structural and mechanical
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heterogeneity has been pointed out to impact the nanomechanics of C–S–H gels (Ioannidou et al., 2014; Ioannidou Katerina et al., 2017; Masoero et al., 2014), the role of the extent of order and
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disorder on macroscopic properties remains poorly understood. Meanwhile, little is known about
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the role of the level of order in the mesostructure of C–S–H—which has been suggested to be higher in high-density C–S–H phases (Constantinides and Ulm, 2007; Ioannidou et al., 2016a,
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2014; Jennings, 2000).
To address these questions, we conduct some grand canonical Monte Carlo (GCMC)
simulations to investigate the effect of grain polydispersity and structural disorder on the mechanical properties of colloidal C–S–H, by relying on the model of Masoero et al (Masoero et al., 2012). Our simulations yield an excellent agreement with nanoindentation data for a wide 4
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range of packing density. We show that, at constant packing fraction, polydispersity does not affect the stiffness and hardness of C–S–H. In contrast, we demonstrate that the degree of disorder has a first-order effect on the mechanical properties of C–S–H gels. This is ascribed to the existence of some local stress heterogeneity within the disordered structure, which arises
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from the out-of-equilibrium nature of the C–S–H gels. We show that, upon loading, such stress heterogeneity induces the occurrence of some local structural nanoyielding, which, in turn, decreases the apparent macroscopic stiffness of the bulk C–S–H gel. These results highlight the critical importance of the level of order and disorder in the mechanical properties of colloidal
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systems.
This paper is organized as follows. In Sec. 2, we describe the simulation methods used to generate the C–S–H gel configurations and to calculate their mechanical properties. In Sec. 3, we
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compare the outcomes of our simulations to nanoindentation data and investigate the effect of polydispersity and disorder on C–S–H’s mechanical properties. These results are discussed in
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Methods
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Sec. 4. Finally, some conclusions are given in Sec. 5.
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Preparation of the C–S–H configurations. To establish our conclusions, we adopt here the colloidal model of C–S–H introduced by Masoero et al (Masoero et al., 2014, 2012). In this
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model, the C–S–H gel is described as an ensemble of polydisperse spherical grains that interact with each other via a generalized Lennard-Jones interaction energy potential:
(
)
(
̅
) [( )
̅
( ) ]
(Eq. 1)
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where
and
are the diameters of grains and , ̅
a given pair of atom,
(
)
is the average diameter for
is a parameter that controls the narrowness of the potential well,
distance between the centers of the grains and , and (
is
) is depth of the potential energy
(
)
̅
modulus
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well. By considering each pair of grains in contact as two springs in series, the depth is given by is a prefactor that is proportional to the bulk Young’s
, where
of a grain, wherein
(computed by the serial spring model) and
GPa (based on previous atomistic simulations of bulk C–S–H) (Manzano et al., 2013; is a correction term arising from the serial arrangement.
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̅
Masoero et al., 2014).
The potential defined in Eq. 1 shows a minimum at a grain is defined as
√
so that, by choosing
, the tensile strain at failure
√ ̅ so that the effective diameter of
. The attractive force is maximum at a distance )
̅
is close to the value
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(
√
of 5% obtained in previous atomistic simulation of bulk C–S–H (Manzano et al., 2013, 2012;
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Masoero et al., 2014). Note that orientational effects arising from the layered structure of bulk
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C–S–H are not accounted by present the two-body potential—we assume here that, overall, the grain anisotropy is lost due to the random mutual orientation of the C–S–H grains (Jönsson et al.,
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2005; Masoero et al., 2014; Pellenq and Van Damme, 2004).
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The C–S–H configurations are generated by grand canonical Monte Carlo (GCMC) simulations, as described in the following. Starting from an initially empty cubic box of size 1000 Å, some C–S–H grains are iteratively inserted, wherein the size of each grain is randomly selected from a uniform distribution between a minimum standard deviation configuration as:
and a maximum
value. The
of the distribution is then used to define the polydispersity index of the
3,10
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(
(Eq. 2)
)
Here, various polydispersity values are considered, with
ranging from 3.0 to 35 nm, and the
number of grains at saturation ranging from 4000 to 12000. In detail, each GCMC step
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comprises X attempts of grain insertions or deletions followed by Y attempts to randomly displace an existing grain. At each step, the probability of success of the attempt is given by ( Δ ⁄
) , where
is the Boltzmann constant, T the temperature, and Δ
is the
variation in potential energy caused by the trial insertion/displacement (Ioannidou et al., 2016b;
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Masoero et al., 2014, 2012). The factor R = X/Y is then qualitatively equivalent to a precipitation rate, which characterizes the time duration during which the grains are allowed to reorganize in between two successive insertions. Namely, a large R value corresponds to a high precipitation rate, wherein the grains have only limited opportunity to move during precipitation. In this work,
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we use R = 0.01. This process is iteratively repeated until the number of inserted grains reaches a
∑
is the volume of the simulation box.
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, where
of each configuration is computed as
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plateau (see Fig. 1). The packing fraction
(a)
(b)
(c)
Fig. 1: Snapshots of select monodisperse C–S–H gel configurations with increasing packing
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fraction values of (a) 0.055, (b) 0.174 and (c) 0.417.
In agreement with previous simulations (Hermes and Dijkstra, 2010; Masoero et al., 2012),
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we observe that the C–S–H models exhibiting higher degree of polydispersity eventually reach higher final packing fraction values—as small grains are able to fill the space left in between larger grains (see Fig. 2). For monodisperse configurations, the packing fraction at saturation is around 0.63, that is, close to the theoretical packing limit of random monodisperse spheres
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(Donev et al., 2004; Scott and Kilgour, 1969). Further, we note that the evolution of the packing fraction at saturation with polydispersity is in good agreement with the range of data previously observed (Hermes and Dijkstra, 2010; Masoero et al., 2012). Note that this analysis is conducted on five independent simulations of precipitation performed for each polydispersity to calculate
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the mean value and standard deviation of the final packing fraction.
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(a)
(b)
(c)
Fig. 2: Snapshots of select (a) monodisperse (polydispersity index = 0) and (b) polydisperse
C–S–H gel configurations ( = 0.49). (c) Computed packing fraction (at saturation) of the C–S– H gel configurations as a function of the polydispersity index (see Eq. 2). The grey area
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indicates the range of data previously observed (Hermes and Dijkstra, 2010; Masoero et al., 2012).
Preparation of artificial C–S–H crystalline configurations. To assess the effect of order and
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disorder on the mechanical properties of C–S–H, a selection of ―artificial‖ C–S–H crystalline configurations with varying different packing fractions are generated. This is achieved by creating a series of C–S–H configurations based on a selection of crystalline lattices and relaxing
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the configuration at zero stress to assess the stability of the crystal. Note that the same interatomic potential (Eq. 1) is used for the ordered C–S–H configurations. A series of five stable artificial C–S–H crystals is considered herein, namely, (1) a DNA-like structure ( = 0.63, = 0), (2) a CsCl-type structure (
= 0.73, = 0.15), (3) a body-centered cubic structure (HCP)
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structure ( = 0.74, = 0), (4) a face-centered cubic structure (FCC) structure ( = 0.74, = 0),
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(5) and a NaCl-type structure ( = 0.79, = 0.41)—see Fig. 3. Note that the crystals 1, 3, and 4 are monodisperse, with a grain size of 5 nm. Here, the DNA-like structure refers to a type of
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helical lattice structure where grains are aligned on the lattices—a configuration that is obtained by removing select grains from a HCP configuration (see Fig. 3a). Note that the simple cubic
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lattice is found to be an unstable configuration for C–S–H. In contrast to the previous
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configurations, the configurations 2 and 5 mimic oxide crystals, i.e., wherein small cations fill the available space left in between the O atoms. Based on the same idea, the CsCl- and NaCl-like configurations are obtained by introducing smaller grains (3.66 and 3.31 nm, respectively) in between larger grains (5 and 8 nm, respectively). This yields some polydisperse crystalline configurations exhibiting large packing fraction values.
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(a)
(b)
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(c)
Fig. 3: Snapshots of some of the artificial C–S–H crystalline configurations generated herein,
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which comprise (in order of increasing packing fraction ) (a) a DNA-like structure ( = 0.63), (b) a HCP crystal ( = 0.74), and (c) a NaCl-type crystal ( = 0.79).
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Stiffness computation. Select C–S–H configurations corresponding to different packing fractions are extracted during the GCMC process. These structures are relaxed by molecular
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dynamics simulations in the NVT ensemble, and subjected to an energy minimization prior to any
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subsequent characterization. In addition, for each degree of polydispersity considered herein, final C–S–H configurations (i.e., at saturation) are relaxed by molecular dynamics simulations in
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the NPT ensemble at zero stress and subjected to a final energy minimization. The stiffness tensor of each configuration is then computed by subjecting the simulation box to a series of
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axial and shear plane deformations along each Cartesian axis (Bauchy, 2014; Wang et al., 2015). The maximum strain for each deformation is restricted to the potential energy
and strain
. The corresponding changes in
define six stress components: (Eq. 3)
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and 36 elastic constants: (Eq. 4)
where
and
are the Cartesian direction indexes. All configurations are found to be nearly fully
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isotropic. The Young’s modulus (E), shear modulus (G), bulk modulus (K), and Poisson’s ratio (ν) are then calculated from the stiffness tensor. Finally, the indentation modulus (M) is determined as (Constantinides and Ulm, 2007; Nye, 1985):
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(Eq. 5)
Hardness computation. The indentation hardness (H) of the relaxed C–S–H configurations is
M
then computed following the method introduced by Qomi et al. (Abdolhosseini Qomi M. J. et al., 2017; Abdolhosseini Qomi et al., 2014; Bauchy et al., 2015), as described in the following. This
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method is based on the computation of the failure envelope of the system as a function of the normal and shear stresses (σ, ??). In detail, the failure envelope is determined by performing a
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series of different deformations (pure uniaxial tension and compression, pure shear, and
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combinations thereof) by incrementally increasing the strain via a series of box deformation. For each deformation, the yield stress is determined based on the 0.2% offset method. In each case,
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the normal and shear stresses at the yield point are used to draw a Mohr circle. The envelope of all the Mohr circles is then fitted by a Mohr-Coulomb failure criterion: (Eq. 6)
where C is the cohesion stress is the friction angle. The hardness H of a cohesive-frictional material such as C–S–H is then given by: 11
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(
where θ is the indenter apex angle,
)
∑
(Eq. 7)
( )
are fitting parameters for a given indenter geometry, and N
is the maximum order of the polynomial expansion. In the case of the C–S–H gels, since the is here considered negligible, and, as a result,
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friction angle is on the order of 5° or less,
the hardness is approximated as 5.8C (Abdolhosseini Qomi M. J. et al., 2017). More details about the hardness computation methodology can be found in Ref (Abdolhosseini Qomi M. J. et al., 2017).
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Stress per grain. Finally, the local stress experienced by each C–S–H grain is computed. Although stress is intrinsically a macroscopic property that is ill-defined for individual grains, a local ―stress per grain‖ can be defined based on the formalism proposed by Thompson et al
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(Thompson et al., 2009). This approach consists in expressing the contribution of each grain i to
is the local stress per grain,
⃗ ⃗⃗ ,
(Eq. 8)
, vi, and ⃗ are the volume, mass, velocity, and
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where
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the virial of the system (Yu et al., 2018):
position of the grain i, respectively, and ⃗⃗ is the resultant of the force applied on the grain i by
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all the other grains in the system. Here, we define the volume Vi of each grain based on its
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Voronoi volume. By convention, a positive stress represents here a state of tension, whereas a negative one represents a state of compression. We recently used this approach to quantify the internal stress exhibited by stressed–rigid atomic networks (Wang et al., 2017) and mixed alkali glasses (Yu et al., 2015, 2017b, 2017a, 2018). It should be noted that, in the thermodynamic sense, stress is only properly defined for a large ensemble of atoms, so the physical meaning of the ―stress per grain‖ is unclear. Nevertheless, this quantity can conveniently capture the 12
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existence of local instabilities within the gel due to competitive inter-grain forces (Ioannidou Katerina et al., 2017).
Results
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Comparison with nanoindentation experiments. We first compare the computed modulus and hardness values with nanoindentation experimental data (Vandamme et al., 2010; Vandamme and Ulm, 2009). Overall, as shown in Fig. 4, we obtain an excellent agreement between simulation and nanoindentation data over a large range of packing fractions, which establishes
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the ability of our simulations to properly describe the nanomechanics of C–S–H. We observe that both the indentation modulus and hardness increase monotonically with the packing fraction. In details, the increase is noted to be limited at low packing fraction ( < 0.5), whereas it is more
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pronounced at high packing fraction ( > 0.5)—in agreement with previous simulation data (Ioannidou et al., 2016b). The specific value of
= 0.5 corresponds to the percolation point of
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the system, which supports the idea of a granular description of C–S–H, wherein the grains interact with each other via Hertz contact points (Constantinides and Ulm, 2007; Vandamme et
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al., 2010). However, we note that a finite value of the indentation modulus is achieved even
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below the percolation point. This highlights the colloidal nature of C–S–H and the fact that the
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grains exhibit some level of attraction even when they are not in direct contact with each other.
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Fig. 4: Computed (a) indentation modulus and (b) hardness of C–S–H gels as a function of the packing fraction. The results are compared with experimental nanoindentation data
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(Vandamme et al., 2010; Vandamme and Ulm, 2009).
Effect of polydispersity. We now investigate the effect of the polydispersity in the grain sizes
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on the nanomechanics of C–S–H gels. To this end, Fig. 5 shows the computed values of the indentation modulus and hardness as a function of the packing fraction for select polydispersity
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(see Eq. 2). Overall, we observe that larger polydispersity values yield larger final packing fraction at saturation (see Fig. 2c) and, thereby, larger values of indentation modulus and
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hardness. However, we note that, at constant packing fraction, the mechanical properties of C–S–
CE
H do not depend on polydispersity since all data fall on the same ―master curve‖ (see Fig. 5). This suggests that, in the case of the present disordered C–S–H configurations, the packing
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fraction is the only order parameter that controls the stiffness and hardness of C–S–H, whereas polydispersity itself is not a relevant parameter. This result is in agreement with previous observations reporting some universal scaling relationships between density and stiffness (Ashby and Medalist, 1983; Gibson and Ashby, 1982; Greaves et al., 2011; Qin et al., 2017; Roberts and Garboczi, 2002; Zheng et al., 2014). 14
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Fig. 5: Computed (a) indentation modulus and (b) hardness of C–S–H gels with varying
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polydispersity index (, see Eq. 2) as a function of packing fraction.
Effect of disorder. Finally, we focus on elucidating the effect of disorder on the nanomechanics
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of C–S–H. To this end, Fig. 6 shows the indentation modulus and hardness of the disordered C–
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S–H configurations (i.e., obtained by GCMC simulations) and those of the artificial C–S–H crystals. First, we observe a significant shift between the indentation modulus and hardness of
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the disordered and ordered C–S–H configurations, wherein ordered configurations systematically exhibit higher values than their disordered counterparts at constant packing fraction. Second, we
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note that the indentation modulus and hardness of the crystalline C–S–H configurations significantly departs from the experimental data, which supports the intrinsically disordered
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nature of C–S–H gels. Last, we note that, in contrast to the case of the disordered C–S–H configuration, the packing fraction is not a universal order parameter for the ordered configurations. Indeed, for instance, we note that, even though the FCC and HCP configurations have the same packing fraction ( = 0.74), the HCP configuration shows significantly larger indentation modulus and hardness values than those of the FCC configuration. This demonstrates 15
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that the details of the structure have a strong impact on the mechanical properties of ordered configurations. Overall, these results highlight the critical role played by the degree of order and
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disorder in controlling the mechanical properties of C–S–H gels.
Fig. 6: Computed (a) indentation modulus and (b) hardness of disordered (back squares) and
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crystalline (blue circles) C–S–H gels as a function of the packing fraction. Experimental nanoindentation data (red diamonds) are added for comparison (Vandamme et al., 2010;
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Discussion
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Vandamme and Ulm, 2009, p. 2009).
Effect of disorder on stiffness. We now discuss the origin of the results presented in Sec. 3 and
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further investigate the effect of order and disorder on the stiffness of the C–S–H gels. Fig. 7 shows the Young’s modulus, shear modulus, bulk modulus, and Poisson’s ratio of the ordered and disordered C–S–H configurations as a function of the packing fraction. First, we note that all the moduli increase monotonically with the packing fraction (see Figs. 7a, 7b, and 7c), in agreement with the fact that lower porosity results in higher stiffness (Greaves et al., 2011). 16
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Further, we note that, similar to the indentation modulus, ordered C–S–H gels systematically exhibit higher Young’s modulus and shear modulus values than their disordered counterparts at constant packing fraction (see Figs. 7a and 7b). Again, this suggests that an increased level of order tends to enhance stiffness.
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Second, we note that the Poisson’s ratio also monotonically increases with increasing packing fractions (see Fig. 7d). Such a trend has been observed for a large variety of materials and has been suggested to arise from the fact that systems featuring higher packing fraction
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values have less ability to locally densify upon longitudinal loading and, hence, show a higher propensity for lateral deformations (Greaves et al., 2011). However, we observe a bifurcation between the Poisson’s ratios of the ordered and disordered configurations, wherein ordered gels systematically exhibit a lower Poisson’s ratio than their disordered counterparts at constant
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packing fraction. This suggests that disordered systems exhibit a pronounced propensity for
mobility of the atoms.
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lateral deformations when subjected to longitudinal loads, which may arise from an increased
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In contrast, interestingly, we note that the bulk modulus values of the ordered and disordered configurations fall on the master curve (see Fig. 7c). This suggests that, in contrast to the other
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moduli, the level of disordered and the details of the structure do not impact the bulk modulus of
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gels—and that the packing density acts as an order parameter. This can be understood from the fact that the bulk modulus is a quantity that is primarily related to the volumic density of the potential energy between grains (Philipps et al., 2017), which is similar for ordered and disordered configurations. Overall, this suggests that the level of disorder does not significantly affect the response of C–S–H when subjected to hydrostatic loads, but has a more pronounced effect in controlling its response to shear. 17
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Fig. 7: Computed (a) Young’s modulus, (b) shear modulus, (c) bulk modulus, and (d) Poisson’s ratio of the disordered (back squares) and crystalline (blue circles) C–S–H gels as a
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function of the packing fraction.
Nanoyielding and stress heterogeneity. Finally, we investigate the origin of the distinct
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behaviors of ordered and disordered C–S–H gels under shear. To the end, we focus on two select C–S–H gel configurations: (i) a disordered monodisperse C–S–H at saturation and (ii) a crystalline DNA-like C–S–H structure. These configurations are selected as they exhibit a different degree of order while having similar packing fraction values ( = 0.63). These two configurations are then subjected to pure shear deformation by gradually deforming the 18
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simulation box and performing an energy minimization after each incremental deformation. Figure 8 shows the corresponding stress-strain curves. First, we observe that, in both cases, the stress initially increases fairly linearly with strain until the system yields—which manifests by a plateau in the stress-strain curve (see Fig. 8). However, the slope of the stress-strain curve of the
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disordered C–S–H configuration is significantly lower than that of the ordered one, in agreement with the fact that the disordered C–S–H structure exhibits a lower shear modulus (see Fig. 7b). To assess the occurrence of any plastic deformations upon shearing, three configurations (I,
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II, and III in Fig. 8) obtained at select shear strains (0.4%, 0.8%, and 1.2%, respectively) are extracted and subsequently unloaded. We observe that, in the case of the ordered C–S–H system, the deformation is fully reversible, that is, the loading and unloading stress-strain curves are similar to each other—as expected in the case of a perfectly elastic deformation. In contrast, in
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the case of the disordered C–S–H gel, we observe a significant degree of irreversibility, that is, the unloading stress-strain curves differs from that obtained upon loading and the system remains
ED
permanently deformed after a loading/unloading cycle. Such irreversibility suggests that,
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although the whole system deforms in a linear fashion upon increasing stress, some irreversible plastic events upon loading occur in the structure. Note that, upon unloading, the stress-strain
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curve of the disordered C–S–H configurations exhibits a slope that is fairly similar to that of the ordered configuration. This suggests that the plastic events that are activated upon loading are
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solely responsible for the difference in the shear modulus of the ordered and disordered configurations.
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Shear stress (MPa)
250
Crystalline Non-crystalline
200
yield
III 150 II 100 yield I
0 0.0
0.4
0.8 1.2 1.6 Shear strain (%)
2.0
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50
Fig. 8: Computed shear stress vs. shear strain upon shearing in a disordered and crystalline C– S–H gel with similar packing fraction. In each case, three configurations (I, II, and III)
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achieved at select shear strains (0.4%, 0.8%, and 1.2%, respectively) are subsequently unloaded (dashed curves).
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To better understand the nature of these plastic events, we compute the local shear stress per grain in both the ordered and disordered C–S–H configurations before any deformation is
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applied, that is, the overall structure being relaxed to zero stress. As expected and shown in Fig. 9, we observe that all grains are experiencing a nearly zero shear stress in the crystalline C–S–H
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configuration. In contrast, we observe the existence of some significant stress variations in the
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disordered C–S–H configuration, wherein the local shear stress per grain ranges from around –1 to +1 GPa. Note that this local stress does not result in any macroscopic stress, namely, the
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grains experiencing some positive or negative shear stress mutually compensate each other so that the overall structure remains at zero stress. The presence of such eigenstress is a manifestation of the out-of-equilibrium nature of disordered C–S–H and arises from the fact that the grain precipitate in a random fashion, so that the addition of new grains in a preexisting rigid structure necessarily involves some non-optimal contact among grains and, hence, the formation 20
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of some local stress (Ioannidou Katerina et al., 2017). A visual inspection of the local stress map (see Fig. 9a) reveals that the stress distribution is highly heterogeneous, that is, most of the stress is concentrated in some local clusters of interconnected grains.
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Based on these observations, the following physical picture emerges. In crystalline C–S–H configurations, all the grains are initially at zero stress. Upon loading, the local stress experienced by each grain then increases in a homogeneous fashion. At a certain threshold stress, all the grains simultaneously reach their local yield stress, which causes the whole structure to
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yield. In contrast, in the disordered C–S–H configuration, the grains are initially pre-stressed (positively or negatively). The additional stress induced by the macroscopic deformation causes some the positively pre-stressed grains to quickly reach their yield stress and, thereby, to release their local stress through local plastic, irreversible reorganizations. Note that such ―nanoyielding‖
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is highly localized (e.g., unlike collective plastic events like shear bands (Greer et al., 2013)) and, hence, does not result in the macroscopic yielding of the gel (Colombo and Del Gado, 2014;
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Masoero et al., 2014; Schall et al., 2007). Overall, the nanoyielding events arising from the
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presence of some stress heterogeneity within the gel explain the origin of the lower apparent
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shear modulus of the disordered C–S–H configurations as well as their non-reversible behaviors.
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Non-crystalline
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Crystalline Non-crystalline
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Fig. 9: (a) Initial spatial distribution of the local shear stress per grain in a crystalline and disordered C–S–H gel with similar packing fraction before any deformation. (b) Distribution of the local shear stress per grain in the crystalline and disordered C–S–H gel configurations.
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The stress distribution of the disordered configuration is multiplied by 10× for readability.
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Disordered nature of the structure of C–S–H gels. Finally, we briefly discuss the implications of the present findings. The C–S–H gels found in hydrated cement pastes have been categorized
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into various types, namely, low-density (LD) and high-density (HD) C–S–H. Originally, an underlying assumption of this categorization was that low packing efficiency of LD C–S–H is
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due to its disordered nature, whereas the high packing efficiency of HD C–S–H arises from a
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crystal-like ordered structure (Constantinides and Ulm, 2007; Jennings, 2000; Vandamme et al., 2010). Clearly, the fact that indentation modulus of the ordered C–S–H gels does not match with nanoindentation data (see Fig. 6) suggests that the variations in the packing density observed in the C–S–H gels forming in hydrated cement pastes (i.e., from LD to HD) does not arise from some variations in the level of order, but rather from different degree of polydispersity in grain sizes. These results also suggest that the yield stress of C–S–H gels (and, thereby, concrete 22
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strength) could be greatly enhanced by controlling the level of structural order and disorder and, hence, the extent of stress heterogeneity within the structure. The existence of local eigenstress within C–S–H is also likely to control the creep relaxation behavior of cement pastes (Bauchy et
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al., 2017; Pignatelli et al., 2016).
Conclusion
Overall, these results reveal the crucial effect of structural disorder in controlling the mechanical properties of gels. This arises from the out-of-equilibrium nature of disordered gels,
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which results in the formation of some local size mismatch among neighboring grains and, thereby, the formation of some local eigenstress. In turn, the presence of such eigenstress result in the occurrence of the nanoyielding of individual grains upon loading, which reduces the effective stiffness and strength of the gel. In contrast, the mechanical properties are found to be
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independent of the extent of polydispersity in grain size. Overall, these results suggest that the
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mechanical properties of gels could be enhanced by tuning the extent of order and disorder in
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their structures.
Acknowledgments
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This work was supported by the National Science Foundation under Grant No. 1562066.
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Effects of Polydispersity and Disorder on the Mechanical Properties of Hydrated Silicate Gels Han Liu , Shiqi Dong , Longwen Tang , N. M. Anoop Krishnan , Gaurav Sant , Mathieu Bauchy PII: DOI: Reference:
S0022-5096(18)30722-1 https://doi.org/10.1016/j.jmps.2018.10.003 MPS 3470
To appear in:
Journal of the Mechanics and Physics of Solids
Received date: Accepted date:
20 August 2018 1 October 2018
Please cite this article as: Han Liu , Shiqi Dong , Longwen Tang , N. M. Anoop Krishnan , Gaurav Sant , Mathieu Bauchy , Effects of Polydispersity and Disorder on the Mechanical Properties of Hydrated Silicate Gels, Journal of the Mechanics and Physics of Solids (2018), doi: https://doi.org/10.1016/j.jmps.2018.10.003
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Effects of Polydispersity and Disorder on the Mechanical Properties of Hydrated Silicate Gels
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Han Liu a, Shiqi Dong a,b, Longwen Tang a, N. M. Anoop Krishnan a,c, Gaurav Sant b,d, Mathieu Bauchy a,* a Physics of AmoRphous and Inorganic Solids Laboratory (PARISlab), Department of Civil and Environmental Engineering, University of California, Los Angeles, California, 90095, USA b Laboratory for the Chemistry of Construction Materials (LC2), Department of Civil and Environmental Engineering, University of California, Los Angeles, California, 90095, USA c Department of Civil Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India d California Nanosystems Institute (CNSI), University of California, Los Angeles, California, 90095, USA * Corresponding author: [email protected] Keywords: Calcium–silicate–hydrate, Colloidal gels, Mechanical properties, Grand Canonical Monte Carlo, Coarse-grained simulations
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Abstract
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The colloidal calcium–silicate–hydrate (C–S–H) gel largely controls the strength of concrete. However, little remains known about how the structural features of the C–S–H gel control its
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mechanical properties. Here, based on coarse-grained mesoscale simulations, we investigate the effect of grain polydispersity and structural disorder on the nanomechanics of C–S–H. Our
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simulations offer a good agreement with nanoindentation data over a large range of packing density values. We show that, at constant packing density, stiffness and hardness are not affected
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by the polydispersity in grain sizes. In contrast, the level of disorder is found to play a critical role. We demonstrate that, in contrast to the case of ordered C–S–H models, the elastic response of disordered C–S–H gels is governed by the existence of stress heterogeneity and nanoyielding within its structure. These results highlight the intrinsically disordered nature of C–S–H and the crucial role of order and disorder in controlling gels’ mechanical properties. 1
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Graphical Abstract The extent of order and disorder in hydrated silicate gels controls mainly shear resistance through stress heterogeneity.
G: shear
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High
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Shear stress per grain
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CSH Gel
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Introduction Colloidal gels—i.e., systems made of interacting grains with sizes ranging from nanometers to hundreds of nanometers (Dinsmore and Weitz, 2002; Lin et al., 1989)—are widely used in
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many realms of science and technology (Joshi, 2014; Masoero et al., 2012; Wang Q. et al., 2007). Colloid aggregation and gelation is controlled by a complex interplay between thermodynamics and kinetics (Anderson and Lekkerkerker, 2002; Lu et al., 2008; Zaccarelli, 2007). After gelation, the structure of colloidal gels largely depends on the formation process (Trappe and Sandkühler,
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2004; Zaccarelli, 2007). In particular, the degree of polydispersity (i.e., the distribution of grain sizes) and the level of disorder affect the mechanical properties of colloidal gels—although such linkages remain poorly understood (Masoero et al., 2014, 2012; Tong et al., 2015). Calcium–silicate–hydrate (C–S–H) gel—the glue of concrete that forms upon the hydration
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of cement—is a typical inorganic colloidal hydrogel material (Ioannidou et al., 2016b; Jennings,
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2008; Masoero et al., 2012). The C–S–H phase largely controls the strength of cement paste and concrete (Bullard et al., 2011; Ioannidou et al., 2016a). This is significant as, due to its large
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carbon impact (Bauchy, 2017; Le Quéré et al., 2012; Liu et al., 2018), there is a strong interest in improving concrete’s mechanical properties—i.e., so that less material can be used while
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achieving constant performance. In turn, the colloidal structure of C–S–H largely controls its
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mechanical response (Constantinides and Ulm, 2007; Masoero et al., 2014; Vandamme et al., 2010). At the mesoscale, C–S–H forms a polydisperse, disordered gel-like structure, with the grain size ranging from nanometers to several tens nanometers (Ioannidou et al., 2016b; Jennings, 2008; Masoero et al., 2012). However, the linkages between the structure and mechanical behavior of C–S–H are not fully elucidated.
3
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Recently, Masoero et al. introduced a polydisperse colloidal model of C–S–H that shows a good agreement with nanoindentation experiments (Masoero et al., 2012). Based on this model, the packing density was shown to have a first order effect on the mechanical properties of C–S– H (Masoero et al., 2014, 2012). In the same spirit, Ioannidou et al. proposed an alternative model
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to account for the early-age structure and properties of C–S–H (Ioannidou et al., 2016a). These colloidal models have been widely investigated to elucidate the relationship between C–S–H’s structure and mechanical properties (Colombo and Del Gado, 2014; Ioannidou et al., 2016b, 2014; Masoero et al., 2014). However, several questions, in this regard, remain unanswered.
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Which structural features have a first-order effect and which one do not? What is the effect of polydispersity and structural disorder on the mechanical properties of C–S–H? Indeed, although polydispersity has been shown to play a critical role in controlling the packing density (and,
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thereby, the mechanical properties) (Masoero et al., 2014, 2012), the role of polydispersity at constant packing density remains unclear. Similarly, although structural and mechanical
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heterogeneity has been pointed out to impact the nanomechanics of C–S–H gels (Ioannidou et al., 2014; Ioannidou Katerina et al., 2017; Masoero et al., 2014), the role of the extent of order and
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disorder on macroscopic properties remains poorly understood. Meanwhile, little is known about
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the role of the level of order in the mesostructure of C–S–H—which has been suggested to be higher in high-density C–S–H phases (Constantinides and Ulm, 2007; Ioannidou et al., 2016a,
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2014; Jennings, 2000).
To address these questions, we conduct some grand canonical Monte Carlo (GCMC)
simulations to investigate the effect of grain polydispersity and structural disorder on the mechanical properties of colloidal C–S–H, by relying on the model of Masoero et al (Masoero et al., 2012). Our simulations yield an excellent agreement with nanoindentation data for a wide 4
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range of packing density. We show that, at constant packing fraction, polydispersity does not affect the stiffness and hardness of C–S–H. In contrast, we demonstrate that the degree of disorder has a first-order effect on the mechanical properties of C–S–H gels. This is ascribed to the existence of some local stress heterogeneity within the disordered structure, which arises
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from the out-of-equilibrium nature of the C–S–H gels. We show that, upon loading, such stress heterogeneity induces the occurrence of some local structural nanoyielding, which, in turn, decreases the apparent macroscopic stiffness of the bulk C–S–H gel. These results highlight the critical importance of the level of order and disorder in the mechanical properties of colloidal
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systems.
This paper is organized as follows. In Sec. 2, we describe the simulation methods used to generate the C–S–H gel configurations and to calculate their mechanical properties. In Sec. 3, we
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compare the outcomes of our simulations to nanoindentation data and investigate the effect of polydispersity and disorder on C–S–H’s mechanical properties. These results are discussed in
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Methods
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Sec. 4. Finally, some conclusions are given in Sec. 5.
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Preparation of the C–S–H configurations. To establish our conclusions, we adopt here the colloidal model of C–S–H introduced by Masoero et al (Masoero et al., 2014, 2012). In this
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model, the C–S–H gel is described as an ensemble of polydisperse spherical grains that interact with each other via a generalized Lennard-Jones interaction energy potential:
(
)
(
̅
) [( )
̅
( ) ]
(Eq. 1)
5
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where
and
are the diameters of grains and , ̅
a given pair of atom,
(
)
is the average diameter for
is a parameter that controls the narrowness of the potential well,
distance between the centers of the grains and , and (
is
) is depth of the potential energy
(
)
̅
modulus
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well. By considering each pair of grains in contact as two springs in series, the depth is given by is a prefactor that is proportional to the bulk Young’s
, where
of a grain, wherein
(computed by the serial spring model) and
GPa (based on previous atomistic simulations of bulk C–S–H) (Manzano et al., 2013; is a correction term arising from the serial arrangement.
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̅
Masoero et al., 2014).
The potential defined in Eq. 1 shows a minimum at a grain is defined as
√
so that, by choosing
, the tensile strain at failure
√ ̅ so that the effective diameter of
. The attractive force is maximum at a distance )
̅
is close to the value
M
(
√
of 5% obtained in previous atomistic simulation of bulk C–S–H (Manzano et al., 2013, 2012;
ED
Masoero et al., 2014). Note that orientational effects arising from the layered structure of bulk
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C–S–H are not accounted by present the two-body potential—we assume here that, overall, the grain anisotropy is lost due to the random mutual orientation of the C–S–H grains (Jönsson et al.,
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2005; Masoero et al., 2014; Pellenq and Van Damme, 2004).
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The C–S–H configurations are generated by grand canonical Monte Carlo (GCMC) simulations, as described in the following. Starting from an initially empty cubic box of size 1000 Å, some C–S–H grains are iteratively inserted, wherein the size of each grain is randomly selected from a uniform distribution between a minimum standard deviation configuration as:
and a maximum
value. The
of the distribution is then used to define the polydispersity index of the
3,10
6
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(
(Eq. 2)
)
Here, various polydispersity values are considered, with
ranging from 3.0 to 35 nm, and the
number of grains at saturation ranging from 4000 to 12000. In detail, each GCMC step
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comprises X attempts of grain insertions or deletions followed by Y attempts to randomly displace an existing grain. At each step, the probability of success of the attempt is given by ( Δ ⁄
) , where
is the Boltzmann constant, T the temperature, and Δ
is the
variation in potential energy caused by the trial insertion/displacement (Ioannidou et al., 2016b;
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Masoero et al., 2014, 2012). The factor R = X/Y is then qualitatively equivalent to a precipitation rate, which characterizes the time duration during which the grains are allowed to reorganize in between two successive insertions. Namely, a large R value corresponds to a high precipitation rate, wherein the grains have only limited opportunity to move during precipitation. In this work,
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we use R = 0.01. This process is iteratively repeated until the number of inserted grains reaches a
∑
is the volume of the simulation box.
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CE
PT
, where
of each configuration is computed as
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plateau (see Fig. 1). The packing fraction
(a)
(b)
(c)
Fig. 1: Snapshots of select monodisperse C–S–H gel configurations with increasing packing
7
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fraction values of (a) 0.055, (b) 0.174 and (c) 0.417.
In agreement with previous simulations (Hermes and Dijkstra, 2010; Masoero et al., 2012),
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we observe that the C–S–H models exhibiting higher degree of polydispersity eventually reach higher final packing fraction values—as small grains are able to fill the space left in between larger grains (see Fig. 2). For monodisperse configurations, the packing fraction at saturation is around 0.63, that is, close to the theoretical packing limit of random monodisperse spheres
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(Donev et al., 2004; Scott and Kilgour, 1969). Further, we note that the evolution of the packing fraction at saturation with polydispersity is in good agreement with the range of data previously observed (Hermes and Dijkstra, 2010; Masoero et al., 2012). Note that this analysis is conducted on five independent simulations of precipitation performed for each polydispersity to calculate
CE
PT
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M
the mean value and standard deviation of the final packing fraction.
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(a)
(b)
(c)
Fig. 2: Snapshots of select (a) monodisperse (polydispersity index = 0) and (b) polydisperse
C–S–H gel configurations ( = 0.49). (c) Computed packing fraction (at saturation) of the C–S– H gel configurations as a function of the polydispersity index (see Eq. 2). The grey area
8
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indicates the range of data previously observed (Hermes and Dijkstra, 2010; Masoero et al., 2012).
Preparation of artificial C–S–H crystalline configurations. To assess the effect of order and
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disorder on the mechanical properties of C–S–H, a selection of ―artificial‖ C–S–H crystalline configurations with varying different packing fractions are generated. This is achieved by creating a series of C–S–H configurations based on a selection of crystalline lattices and relaxing
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the configuration at zero stress to assess the stability of the crystal. Note that the same interatomic potential (Eq. 1) is used for the ordered C–S–H configurations. A series of five stable artificial C–S–H crystals is considered herein, namely, (1) a DNA-like structure ( = 0.63, = 0), (2) a CsCl-type structure (
= 0.73, = 0.15), (3) a body-centered cubic structure (HCP)
M
structure ( = 0.74, = 0), (4) a face-centered cubic structure (FCC) structure ( = 0.74, = 0),
ED
(5) and a NaCl-type structure ( = 0.79, = 0.41)—see Fig. 3. Note that the crystals 1, 3, and 4 are monodisperse, with a grain size of 5 nm. Here, the DNA-like structure refers to a type of
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helical lattice structure where grains are aligned on the lattices—a configuration that is obtained by removing select grains from a HCP configuration (see Fig. 3a). Note that the simple cubic
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lattice is found to be an unstable configuration for C–S–H. In contrast to the previous
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configurations, the configurations 2 and 5 mimic oxide crystals, i.e., wherein small cations fill the available space left in between the O atoms. Based on the same idea, the CsCl- and NaCl-like configurations are obtained by introducing smaller grains (3.66 and 3.31 nm, respectively) in between larger grains (5 and 8 nm, respectively). This yields some polydisperse crystalline configurations exhibiting large packing fraction values.
9
(a)
(b)
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(c)
Fig. 3: Snapshots of some of the artificial C–S–H crystalline configurations generated herein,
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which comprise (in order of increasing packing fraction ) (a) a DNA-like structure ( = 0.63), (b) a HCP crystal ( = 0.74), and (c) a NaCl-type crystal ( = 0.79).
M
Stiffness computation. Select C–S–H configurations corresponding to different packing fractions are extracted during the GCMC process. These structures are relaxed by molecular
ED
dynamics simulations in the NVT ensemble, and subjected to an energy minimization prior to any
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subsequent characterization. In addition, for each degree of polydispersity considered herein, final C–S–H configurations (i.e., at saturation) are relaxed by molecular dynamics simulations in
CE
the NPT ensemble at zero stress and subjected to a final energy minimization. The stiffness tensor of each configuration is then computed by subjecting the simulation box to a series of
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axial and shear plane deformations along each Cartesian axis (Bauchy, 2014; Wang et al., 2015). The maximum strain for each deformation is restricted to the potential energy
and strain
. The corresponding changes in
define six stress components: (Eq. 3)
10
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and 36 elastic constants: (Eq. 4)
where
and
are the Cartesian direction indexes. All configurations are found to be nearly fully
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isotropic. The Young’s modulus (E), shear modulus (G), bulk modulus (K), and Poisson’s ratio (ν) are then calculated from the stiffness tensor. Finally, the indentation modulus (M) is determined as (Constantinides and Ulm, 2007; Nye, 1985):
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(Eq. 5)
Hardness computation. The indentation hardness (H) of the relaxed C–S–H configurations is
M
then computed following the method introduced by Qomi et al. (Abdolhosseini Qomi M. J. et al., 2017; Abdolhosseini Qomi et al., 2014; Bauchy et al., 2015), as described in the following. This
ED
method is based on the computation of the failure envelope of the system as a function of the normal and shear stresses (σ, ??). In detail, the failure envelope is determined by performing a
PT
series of different deformations (pure uniaxial tension and compression, pure shear, and
CE
combinations thereof) by incrementally increasing the strain via a series of box deformation. For each deformation, the yield stress is determined based on the 0.2% offset method. In each case,
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the normal and shear stresses at the yield point are used to draw a Mohr circle. The envelope of all the Mohr circles is then fitted by a Mohr-Coulomb failure criterion: (Eq. 6)
where C is the cohesion stress is the friction angle. The hardness H of a cohesive-frictional material such as C–S–H is then given by: 11
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(
where θ is the indenter apex angle,
)
∑
(Eq. 7)
( )
are fitting parameters for a given indenter geometry, and N
is the maximum order of the polynomial expansion. In the case of the C–S–H gels, since the is here considered negligible, and, as a result,
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friction angle is on the order of 5° or less,
the hardness is approximated as 5.8C (Abdolhosseini Qomi M. J. et al., 2017). More details about the hardness computation methodology can be found in Ref (Abdolhosseini Qomi M. J. et al., 2017).
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Stress per grain. Finally, the local stress experienced by each C–S–H grain is computed. Although stress is intrinsically a macroscopic property that is ill-defined for individual grains, a local ―stress per grain‖ can be defined based on the formalism proposed by Thompson et al
M
(Thompson et al., 2009). This approach consists in expressing the contribution of each grain i to
is the local stress per grain,
⃗ ⃗⃗ ,
(Eq. 8)
, vi, and ⃗ are the volume, mass, velocity, and
PT
where
ED
the virial of the system (Yu et al., 2018):
position of the grain i, respectively, and ⃗⃗ is the resultant of the force applied on the grain i by
CE
all the other grains in the system. Here, we define the volume Vi of each grain based on its
AC
Voronoi volume. By convention, a positive stress represents here a state of tension, whereas a negative one represents a state of compression. We recently used this approach to quantify the internal stress exhibited by stressed–rigid atomic networks (Wang et al., 2017) and mixed alkali glasses (Yu et al., 2015, 2017b, 2017a, 2018). It should be noted that, in the thermodynamic sense, stress is only properly defined for a large ensemble of atoms, so the physical meaning of the ―stress per grain‖ is unclear. Nevertheless, this quantity can conveniently capture the 12
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existence of local instabilities within the gel due to competitive inter-grain forces (Ioannidou Katerina et al., 2017).
Results
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Comparison with nanoindentation experiments. We first compare the computed modulus and hardness values with nanoindentation experimental data (Vandamme et al., 2010; Vandamme and Ulm, 2009). Overall, as shown in Fig. 4, we obtain an excellent agreement between simulation and nanoindentation data over a large range of packing fractions, which establishes
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the ability of our simulations to properly describe the nanomechanics of C–S–H. We observe that both the indentation modulus and hardness increase monotonically with the packing fraction. In details, the increase is noted to be limited at low packing fraction ( < 0.5), whereas it is more
M
pronounced at high packing fraction ( > 0.5)—in agreement with previous simulation data (Ioannidou et al., 2016b). The specific value of
= 0.5 corresponds to the percolation point of
ED
the system, which supports the idea of a granular description of C–S–H, wherein the grains interact with each other via Hertz contact points (Constantinides and Ulm, 2007; Vandamme et
PT
al., 2010). However, we note that a finite value of the indentation modulus is achieved even
CE
below the percolation point. This highlights the colloidal nature of C–S–H and the fact that the
AC
grains exhibit some level of attraction even when they are not in direct contact with each other.
13
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Fig. 4: Computed (a) indentation modulus and (b) hardness of C–S–H gels as a function of the packing fraction. The results are compared with experimental nanoindentation data
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(Vandamme et al., 2010; Vandamme and Ulm, 2009).
Effect of polydispersity. We now investigate the effect of the polydispersity in the grain sizes
M
on the nanomechanics of C–S–H gels. To this end, Fig. 5 shows the computed values of the indentation modulus and hardness as a function of the packing fraction for select polydispersity
ED
(see Eq. 2). Overall, we observe that larger polydispersity values yield larger final packing fraction at saturation (see Fig. 2c) and, thereby, larger values of indentation modulus and
PT
hardness. However, we note that, at constant packing fraction, the mechanical properties of C–S–
CE
H do not depend on polydispersity since all data fall on the same ―master curve‖ (see Fig. 5). This suggests that, in the case of the present disordered C–S–H configurations, the packing
AC
fraction is the only order parameter that controls the stiffness and hardness of C–S–H, whereas polydispersity itself is not a relevant parameter. This result is in agreement with previous observations reporting some universal scaling relationships between density and stiffness (Ashby and Medalist, 1983; Gibson and Ashby, 1982; Greaves et al., 2011; Qin et al., 2017; Roberts and Garboczi, 2002; Zheng et al., 2014). 14
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Fig. 5: Computed (a) indentation modulus and (b) hardness of C–S–H gels with varying
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polydispersity index (, see Eq. 2) as a function of packing fraction.
Effect of disorder. Finally, we focus on elucidating the effect of disorder on the nanomechanics
M
of C–S–H. To this end, Fig. 6 shows the indentation modulus and hardness of the disordered C–
ED
S–H configurations (i.e., obtained by GCMC simulations) and those of the artificial C–S–H crystals. First, we observe a significant shift between the indentation modulus and hardness of
PT
the disordered and ordered C–S–H configurations, wherein ordered configurations systematically exhibit higher values than their disordered counterparts at constant packing fraction. Second, we
CE
note that the indentation modulus and hardness of the crystalline C–S–H configurations significantly departs from the experimental data, which supports the intrinsically disordered
AC
nature of C–S–H gels. Last, we note that, in contrast to the case of the disordered C–S–H configuration, the packing fraction is not a universal order parameter for the ordered configurations. Indeed, for instance, we note that, even though the FCC and HCP configurations have the same packing fraction ( = 0.74), the HCP configuration shows significantly larger indentation modulus and hardness values than those of the FCC configuration. This demonstrates 15
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that the details of the structure have a strong impact on the mechanical properties of ordered configurations. Overall, these results highlight the critical role played by the degree of order and
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disorder in controlling the mechanical properties of C–S–H gels.
Fig. 6: Computed (a) indentation modulus and (b) hardness of disordered (back squares) and
M
crystalline (blue circles) C–S–H gels as a function of the packing fraction. Experimental nanoindentation data (red diamonds) are added for comparison (Vandamme et al., 2010;
CE
Discussion
PT
ED
Vandamme and Ulm, 2009, p. 2009).
Effect of disorder on stiffness. We now discuss the origin of the results presented in Sec. 3 and
AC
further investigate the effect of order and disorder on the stiffness of the C–S–H gels. Fig. 7 shows the Young’s modulus, shear modulus, bulk modulus, and Poisson’s ratio of the ordered and disordered C–S–H configurations as a function of the packing fraction. First, we note that all the moduli increase monotonically with the packing fraction (see Figs. 7a, 7b, and 7c), in agreement with the fact that lower porosity results in higher stiffness (Greaves et al., 2011). 16
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Further, we note that, similar to the indentation modulus, ordered C–S–H gels systematically exhibit higher Young’s modulus and shear modulus values than their disordered counterparts at constant packing fraction (see Figs. 7a and 7b). Again, this suggests that an increased level of order tends to enhance stiffness.
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Second, we note that the Poisson’s ratio also monotonically increases with increasing packing fractions (see Fig. 7d). Such a trend has been observed for a large variety of materials and has been suggested to arise from the fact that systems featuring higher packing fraction
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values have less ability to locally densify upon longitudinal loading and, hence, show a higher propensity for lateral deformations (Greaves et al., 2011). However, we observe a bifurcation between the Poisson’s ratios of the ordered and disordered configurations, wherein ordered gels systematically exhibit a lower Poisson’s ratio than their disordered counterparts at constant
M
packing fraction. This suggests that disordered systems exhibit a pronounced propensity for
mobility of the atoms.
ED
lateral deformations when subjected to longitudinal loads, which may arise from an increased
PT
In contrast, interestingly, we note that the bulk modulus values of the ordered and disordered configurations fall on the master curve (see Fig. 7c). This suggests that, in contrast to the other
CE
moduli, the level of disordered and the details of the structure do not impact the bulk modulus of
AC
gels—and that the packing density acts as an order parameter. This can be understood from the fact that the bulk modulus is a quantity that is primarily related to the volumic density of the potential energy between grains (Philipps et al., 2017), which is similar for ordered and disordered configurations. Overall, this suggests that the level of disorder does not significantly affect the response of C–S–H when subjected to hydrostatic loads, but has a more pronounced effect in controlling its response to shear. 17
M
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ED
Fig. 7: Computed (a) Young’s modulus, (b) shear modulus, (c) bulk modulus, and (d) Poisson’s ratio of the disordered (back squares) and crystalline (blue circles) C–S–H gels as a
CE
PT
function of the packing fraction.
Nanoyielding and stress heterogeneity. Finally, we investigate the origin of the distinct
AC
behaviors of ordered and disordered C–S–H gels under shear. To the end, we focus on two select C–S–H gel configurations: (i) a disordered monodisperse C–S–H at saturation and (ii) a crystalline DNA-like C–S–H structure. These configurations are selected as they exhibit a different degree of order while having similar packing fraction values ( = 0.63). These two configurations are then subjected to pure shear deformation by gradually deforming the 18
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simulation box and performing an energy minimization after each incremental deformation. Figure 8 shows the corresponding stress-strain curves. First, we observe that, in both cases, the stress initially increases fairly linearly with strain until the system yields—which manifests by a plateau in the stress-strain curve (see Fig. 8). However, the slope of the stress-strain curve of the
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disordered C–S–H configuration is significantly lower than that of the ordered one, in agreement with the fact that the disordered C–S–H structure exhibits a lower shear modulus (see Fig. 7b). To assess the occurrence of any plastic deformations upon shearing, three configurations (I,
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II, and III in Fig. 8) obtained at select shear strains (0.4%, 0.8%, and 1.2%, respectively) are extracted and subsequently unloaded. We observe that, in the case of the ordered C–S–H system, the deformation is fully reversible, that is, the loading and unloading stress-strain curves are similar to each other—as expected in the case of a perfectly elastic deformation. In contrast, in
M
the case of the disordered C–S–H gel, we observe a significant degree of irreversibility, that is, the unloading stress-strain curves differs from that obtained upon loading and the system remains
ED
permanently deformed after a loading/unloading cycle. Such irreversibility suggests that,
PT
although the whole system deforms in a linear fashion upon increasing stress, some irreversible plastic events upon loading occur in the structure. Note that, upon unloading, the stress-strain
CE
curve of the disordered C–S–H configurations exhibits a slope that is fairly similar to that of the ordered configuration. This suggests that the plastic events that are activated upon loading are
AC
solely responsible for the difference in the shear modulus of the ordered and disordered configurations.
19
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Shear stress (MPa)
250
Crystalline Non-crystalline
200
yield
III 150 II 100 yield I
0 0.0
0.4
0.8 1.2 1.6 Shear strain (%)
2.0
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50
Fig. 8: Computed shear stress vs. shear strain upon shearing in a disordered and crystalline C– S–H gel with similar packing fraction. In each case, three configurations (I, II, and III)
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achieved at select shear strains (0.4%, 0.8%, and 1.2%, respectively) are subsequently unloaded (dashed curves).
M
To better understand the nature of these plastic events, we compute the local shear stress per grain in both the ordered and disordered C–S–H configurations before any deformation is
ED
applied, that is, the overall structure being relaxed to zero stress. As expected and shown in Fig. 9, we observe that all grains are experiencing a nearly zero shear stress in the crystalline C–S–H
PT
configuration. In contrast, we observe the existence of some significant stress variations in the
CE
disordered C–S–H configuration, wherein the local shear stress per grain ranges from around –1 to +1 GPa. Note that this local stress does not result in any macroscopic stress, namely, the
AC
grains experiencing some positive or negative shear stress mutually compensate each other so that the overall structure remains at zero stress. The presence of such eigenstress is a manifestation of the out-of-equilibrium nature of disordered C–S–H and arises from the fact that the grain precipitate in a random fashion, so that the addition of new grains in a preexisting rigid structure necessarily involves some non-optimal contact among grains and, hence, the formation 20
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of some local stress (Ioannidou Katerina et al., 2017). A visual inspection of the local stress map (see Fig. 9a) reveals that the stress distribution is highly heterogeneous, that is, most of the stress is concentrated in some local clusters of interconnected grains.
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Based on these observations, the following physical picture emerges. In crystalline C–S–H configurations, all the grains are initially at zero stress. Upon loading, the local stress experienced by each grain then increases in a homogeneous fashion. At a certain threshold stress, all the grains simultaneously reach their local yield stress, which causes the whole structure to
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yield. In contrast, in the disordered C–S–H configuration, the grains are initially pre-stressed (positively or negatively). The additional stress induced by the macroscopic deformation causes some the positively pre-stressed grains to quickly reach their yield stress and, thereby, to release their local stress through local plastic, irreversible reorganizations. Note that such ―nanoyielding‖
M
is highly localized (e.g., unlike collective plastic events like shear bands (Greer et al., 2013)) and, hence, does not result in the macroscopic yielding of the gel (Colombo and Del Gado, 2014;
ED
Masoero et al., 2014; Schall et al., 2007). Overall, the nanoyielding events arising from the
PT
presence of some stress heterogeneity within the gel explain the origin of the lower apparent
AC
CE
shear modulus of the disordered C–S–H configurations as well as their non-reversible behaviors.
21
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Non-crystalline
0.8
Shear stress per grain /GPa
2 1 0
Probability density
Crystalline
Crystalline Non-crystalline
0.6
0.4
0.2
-1 -2
(a)
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0.0
-2
-1 0 1 2 Initial shear stress per grain (GPa)
(b)
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Fig. 9: (a) Initial spatial distribution of the local shear stress per grain in a crystalline and disordered C–S–H gel with similar packing fraction before any deformation. (b) Distribution of the local shear stress per grain in the crystalline and disordered C–S–H gel configurations.
M
The stress distribution of the disordered configuration is multiplied by 10× for readability.
ED
Disordered nature of the structure of C–S–H gels. Finally, we briefly discuss the implications of the present findings. The C–S–H gels found in hydrated cement pastes have been categorized
PT
into various types, namely, low-density (LD) and high-density (HD) C–S–H. Originally, an underlying assumption of this categorization was that low packing efficiency of LD C–S–H is
CE
due to its disordered nature, whereas the high packing efficiency of HD C–S–H arises from a
AC
crystal-like ordered structure (Constantinides and Ulm, 2007; Jennings, 2000; Vandamme et al., 2010). Clearly, the fact that indentation modulus of the ordered C–S–H gels does not match with nanoindentation data (see Fig. 6) suggests that the variations in the packing density observed in the C–S–H gels forming in hydrated cement pastes (i.e., from LD to HD) does not arise from some variations in the level of order, but rather from different degree of polydispersity in grain sizes. These results also suggest that the yield stress of C–S–H gels (and, thereby, concrete 22
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strength) could be greatly enhanced by controlling the level of structural order and disorder and, hence, the extent of stress heterogeneity within the structure. The existence of local eigenstress within C–S–H is also likely to control the creep relaxation behavior of cement pastes (Bauchy et
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al., 2017; Pignatelli et al., 2016).
Conclusion
Overall, these results reveal the crucial effect of structural disorder in controlling the mechanical properties of gels. This arises from the out-of-equilibrium nature of disordered gels,
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which results in the formation of some local size mismatch among neighboring grains and, thereby, the formation of some local eigenstress. In turn, the presence of such eigenstress result in the occurrence of the nanoyielding of individual grains upon loading, which reduces the effective stiffness and strength of the gel. In contrast, the mechanical properties are found to be
M
independent of the extent of polydispersity in grain size. Overall, these results suggest that the
ED
mechanical properties of gels could be enhanced by tuning the extent of order and disorder in
PT
their structures.
Acknowledgments
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This work was supported by the National Science Foundation under Grant No. 1562066.
AC
References
Abdolhosseini Qomi M. J., Ebrahimi D., Bauchy M., Pellenq R., Ulm F.-J., 2017. Methodology for Estimation of Nanoscale Hardness via Atomistic Simulations. Journal of Nanomechanics and Micromechanics 7, 04017011. https://doi.org/10.1061/(ASCE)NM.2153-5477.0000127 Abdolhosseini Qomi, M.J., Krakowiak, K.J., Bauchy, M., Stewart, K.L., Shahsavari, R., Jagannathan, D., Brommer, D.B., Baronnet, A., Buehler, M.J., Yip, S., Ulm, F.-J., Van Vliet, K.J., Pellenq, R.J.-M., 2014. Combinatorial molecular optimization of cement hydrates. Nat Commun 5, 4960. https://doi.org/10.1038/ncomms5960 23
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