Monte Carlo Molecular Modeling of Temperature and Pressure Effects

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Monte Carlo Molecular Modeling of Temperature and Pressure Effects on the Interactions between Crystalline Calcium Silicate Hydrate Layers Tulio Honorio Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b04156 • Publication Date (Web): 20 Feb 2019 Downloaded from http://pubs.acs.org on February 21, 2019

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Monte Carlo Molecular Modeling of Temperature and Pressure Effects on the Interactions between Crystalline Calcium Silicate Hydrate Layers Tulio Honorio∗ LMT/ENS-Cachan/CNRS/Université Paris Saclay, Cachan, France E-mail: [email protected] Phone: +(33) 1 47 40 68 27

Abstract The interactions of calcium silicate hydrates with water are at the heart of critical features of cement-based materials behavior such as drying and autogenous shrinkage, hysteresis, creep and thermal expansion. In this article, the interactions between nanocrystalline layers of calcium silicate hydrates are computed from Grand Canonical Monte Carlo (GCMC) molecular simulations. The effects of temperature, chemical potential and pressure on these interactions are studied. The results are confronted with simulation and experimental data found in the literature concerning surface energy, cohesive pressure and out-of-plane elastic properties. The disjoining pressure isotherms of calcium silicate hydrates are negligibly affected by changes in water pressure under saturated conditions. The surface energy decreases with the temperature, the chemical potential of water and the water pressure. Coarse-grained simulations are performed using the potential of mean force obtained at the molecular level. The mesostruc-

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ture presents hysteresis with respect to mechanical and thermal loads. The anharmonicity of the interactions identified at the molecular scale translate in an asymmetry tension/compression and thermal expansion that are also observed at the mesoscale. These results leave room to a better understanding of the multiscale origin of physical properties of calcium silicate hydrates. Keywords: Grand canonical simulations, Tobermorite, Potential of mean force, Adsorption, Mesoscale

Introduction Cement-based materials are the most widely used man-made product being responsible to about 5% of global anthropogenic CO2 emissions. 1 There is a consensus in the community that the improvement of the performance of cement-based materials, including the reduction of their CO2 footprint and the extension of their durability, require the fundamental understanding of the physical phenomena occurring at the nanoscale. 2 The main constituent phase of materials formulated with ordinary cement is the calcium silicate hydrates (C-S-H), a microporous adsorbing material. 3 The interactions of calcium silicate hydrates with water, specially at the nanoscale, are at the heart of critical features of the behavior of cement-based materials such as drying and autogenous shrinkage 4 , hysteresis 4 , creep, 5 thermal expansion, 6 cement hydration processes 7 and cohesion of early-age cement paste. 8 Also, these interactions play a critical role in the aggregation of calcium silicate hydrates nanolayers, which constitutes a promising way of developing advanced nano-architectured cement-based materials. 9,10 Cement-based materials are generally subjected to variations of temperatures during their service life. These temperature variations are critical in industrial applications of concrete involving, for example, radioactive waste disposal, early-age behavior of massive concrete structures (since cement hydration processes are exothermic) 11 and extreme solicitations in concrete structures such as explosions and fire. Thermal deformations are reported to be one 2


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of the principal causes of cracking of concrete. 12 The prediction of cement based-materials behavior under temperature and pressure variations can benefit from a fundamental understanding of the nanoscale processes involving adsorption. In this direction, molecular simulations have been used to quantify the intermolecular forces between solids layers of calcium silicate hydrates 13,14 and other phyllosilicates such as clays 15 under drained conditions. At the nanoscale, due to the exchange of fluids between interlayer pores and gel pores, the former class of pores can be seen as in drained conditions with respect to the fluid reservoir in gel pores. Experimental evidence 12 showing instantaneous changes in water population in these pores under temperature change corroborates this assumption regarding the internal migration of water. Indeed, temperature is reported to affect the thermal expansion and cohesion of calcium silicate hydrates nanolayers with repercussions at larger scales. 16,17 In the article, the interactions between nanocrystalline calcium silicate hydrates layers are studied using classical Grand Canonical Monte Carlo (GCMC) simulations. The molecular structure of tobermorite is considered. C-S-H structure is often assumed to be a variation of defective tobermorite minerals. 18–20 Compared to previous works that investigates the interactions between calcium silicate hydrates at the nanoscale, here the effects of temperature, chemical potential of water and pressure are quantified. Temperatures and pressures of interest in industrial applications of concrete, notably the ones associated with massive concrete structures at early-age, 11,21 are considered. Inspired by previous studies on clays, 15 the thermodynamic background associated with the (meta-)stability and cohesion of the nanolayers is discussed. The results are confronted with simulation and experimental data found in the literature concerning key properties such as such as surface energy, cohesive pressure and elastic properties. The interactions obtained here are used to upscale C-S-H properties via coarse-grained simulation at the mesoscale. At the mesoscale, hysteresis is observed with respect to mechanical and thermal loads. Also, the anharmonicity of the interactions identified at the molecular scale translates in an asymmetry tension/compression and thermal expansion that are also observed at the mesoscale. These results represent a

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better understating of the nanoscale origin of the thermomechancial behavior of C-S-H.

Models and methods Atomic Structure and Force Field The atomic structure of crystalline calcium silicate hydrate is derived from Hamid 22 structure of tobermorite 11 Å. Hamid proposes three tobermorite structure with different Ca/Si ratios, namely 0.67, 0.83, and 1. Here, the structure with molar Ca/Si ratio of 1, which corresponds to the structural formula Ca6 [Si6 O18 ].2H2 O, is adopted. The average Ca/Si ratio of C-S-H phases present in commercial cements ranges from 2.3 to 0.7, with compositional heterogeneities at a scale of roughly 100 nm. 23 A molar Ca/Si ratio close to 1 is particularly relevant to applications in oil well cementing. 24 The unit cell is monoclinic with cell parameters (a,b,c)=(6.69 Å, 7.39 Å, 22.77 Å) and γ = 123.49◦ . To generate the simulation box in Fig. 1 (bottom), the original unit cell is replicate four times in each in-plane direction and duplicate along c-axis. In order to study water-tobermorite interaction, a slit pore is constructed and simulations are performed with various pore sizes according to the reaction path depicted shown in Fig. 1 (top). To construct the slit pores, the natural cleavage plane associated with the various polymorphs of tobermorite is considered. 25 The resulting surfaces of the slit pore present therefore a marked roughness due to the presence of the silicate chains. In the analysis of the influence of this roughness, hereafter tobermorite channels are defined as the space between the interfaces I 0 or I 00 (corresponding to planes passing by the out-most oxygen in the solid layers) and the silicate chains. These channels are accessible to water and counterions. Interactions between the species are described by CSHFF force field, 26 which has been extensively used in the computation of mechanical, thermal and interfacial properties of crystalline and disordered calcium silicates. 2 This force field is based only in non-bonded Lennard-Jones interactions for metal and metalloids whereas covalent bonds are only de4


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fined for water molecules and hydroxyls. With the attribution of the partial charges of the force field, the resulting system is not neutral. The surface change density of the solid layer obtained here is -0.55 C.m−2 , which is close to the experimental measurements of -0.5 C.m−2 . 27 This surface charge is compensated by calcium counterions inserted into the pore space to ensure the electroneutrality of the system. Tobermorite structure is relaxed with LAMMPS 28 in an isothermal-isobaric (NPT) simulation during 10 ns with timestep of 1 fs at 300 K and under 1 atm. Long-range interactions are treated via Ewald sum method. Periodic boundary conditions are used. The resulting simulation cell has the dimensions Lx = 26.34 Å (parallel to a-axis) Ly = 24.28 Å and Lz = 22.77 Å (parallel to c-axis). Figure 1 displays a snapshot of the tobermorite configuration after relaxation.

Grand Canonical Molecular Simulations Grand canonical Monte Carlo simulations are run at various temperatures (290, 300, 310, 320 and 330 K under 1 atm) and under various water pressures (-235, 0.1, 235 and 705 MPa at 300 K) for basal spacings (or center-to-center distances) ranging from 10 to 50 Å. In the simulations, the total volume V and temperature T as well as the chemical potential of the fluid µw are fixed. The number of fluid molecules Nw may fluctuate whilst the number of atoms in the solid and counterions remains fixed. Tobermorite layers and water molecules are kept rigid to gain in computational efficiency, as in previous works coping with adsorption in phyllosilicates. 14,15 The simulations are performed with Towhee 29 using configuration bias methods, namely biased insertion/deletion (with a probability p = 0.19 for water molecules), rotation about the center of mass (for water molecules, p = 0.12 with a maximum angle of π rad) and translation (p = 0.48 for both calcium counterions and water molecules, with maximum displacement of 1.5 Å), Monte Carlo moves and swaps (p = 0.14 for both water molecules and calcium counterions) as well as molecule growth (p = 0.07 for water molecules). Each simulation 5


cleavage plan

c

Tobermorite channels

c

d I’

d I’’

c

SiO4 c

a

Domain inbetween interfaces I’ and I’’

Intralayer Ca

Reaction path along c-axis

Interlayer Ca

b

Si Ca O H

plan ac

c

b plan ab

a

3D atomic structure of tobermorite with Ca/Si=1 after relaxation

Figure 1: At the top, scheme of the cleavage plan and reaction path considered in this study. The tobermorite channels, which are accessible to water and counterions, are identified as well as the interfaces I 0 and I 00 a defined in this study. At the bottom, snapshot of the the 3D atomic structure of crystalline calcium silicate hydrate according to Hamid 22 with Ca/Si = 1 after relaxation at 300 K and under 1 atm.

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is run during 30 million Monte Carlo steps comprising 10 million steps of equilibration and 20 million steps of sampling. The convergence criterion adopted is a stable energy system, with the standard deviation of the total energy below 0.02 % of the mean value. Ewald sum is used to compute long-range electrostatic interactions. Periodic boundary conditions are employed. The confining pressure is computed via the virial pressure. 29,30 The chemical potential of SPC water as a function of the temperature is obtained by ∂µw |T = means of thermodynamic integration of Gibbs-Duhem equation ( ∂P w

1 , ρw

where ρw is

the water density) from the chemical potential at the (bulk) liquid-gas coexistence point µ0w . Simulations of bulk SPC water in the canonical ensemble (NVT) are performed with LAMMPS for various densities and temperatures (inset of Fig. 2). The change in the chemical potential of water ∆µw = µw − µ0w varies linearly with the pressure Pw (Fig. 2) with an average rate of 0.017 kJ/(MPa.mol) and are indistinguishable for the temperatures considered here. Gibbs Ensemble Monte Carlo simulations of rigid SPC water 31 using continuous fractional component method yielded µ0w = -33.9 kJ/mol at 300 K,-33.5 kJ/mol at 325 K and -33.2 kJ/mol at 350 K. Other authors 32 found µ0w =-37.23 kJ/mol at 373.13 K for rigid SPC water. These values are interpolated in order to obtain µ0w at the temperatures considered here. Saturation vapor pressures of SPC water have been reported to be 0.0044 MPa at 300 K 33 and 0.120 MPa at 373.13 K 32 (the interpolated saturation curve is shown in the inset of Fig. 2 for the temperatures considered here, these values are used in the thermodynamic integration).

Results and Discussion Water content and structure Figure 3 displays the density histograms and the corresponding snapshots of molecular configurations at three different basal spacings associated with zero, one and two water layers (refereed hereafter, respectively, as 1W, 2W and 3W states) out of tobermorite channels. 7


600

400 300

Pressure [MPa]

Pressure [MPa]

500

700 500 300 100 saturation

-100 0.95

200

1.05 Density [g/cm3]

290 K

1.15

310 K

1

100

300 K 320 K

0.017

0

330 K -100 -2

0

2

4

6

8

10

Δμw [kJ/mol]

Figure 2: Change in the chemical potential of bulk rigid SPC water ∆µw = µw −µ0w computed from thermodynamic integration of Gibbs-Duhem equation. The curves are superposed and show an average rate of 0.017 kJ/(MPa.mol) (or 59 MPa.mol/kJ). The inset shows the pressure isotherms (NVT) used in the thermodynamic integration. These channels can be defined by an interface placed at the position of the center of the exterior non-bridging oxygens of the bridging Si tetrahedra 34 (z = 0 in Fig. 3). For all hydration states, a fraction of water remains trapped within these channels. This fraction is reported to contribute to the anomalous diffusion observed in calcium silicate hydrates. 34,35 Within the channels, two water sub-layers can be distinguished: 34 one with the Ow at the level of the inner non-bridging oxygen of the bridging Si tetrahedra (z ≈-2.5 Å for 1W, z ≈-1.5 Å for 2W and 3W) and another at z ≈-0.5 Å for the three hydration states. Out of the channels, water ordering persist up to at least z =8 Å, as also observed in previous studies. 34 Figure 4 (a) shows the water content in tobermorite micropores at 300 K as a function of the basal spacing. In these adsorption isotherms, it is possible to identify the domains in which the number of water molecule rapidly increases with the basal spacing forming a plateau (inset of Fig. 4 (a)), which is characteristic of systems exhibiting ordering of water in discrete layers and snap-through instabilities.

8

b

I’ ≡ I’’

I’’

I’

I’ I’’

Normalized density

c

0.1

0.2 Ow Hw 0.15 Caw 0.1

0.2 Ow 0.15 Hw Caw 0.1

0.05

0.05

0.05

0.2 0.15

-5

0

z (Å)

5

10

Hw Caw

0

0

0

Ow

-5

0

5

z (Å)

10

-5

0

z (Å)

5

10

Figure 3: Density histograms and snapshots of configurations showing water ordering adjacent to calcium silicate hydrate surface. Density profiles and configurations are taken at stable basal spacings corresponding to 0W, 1W and 2W states (d= 11.97, 14.37 and 18.27 Å, respectively) at 300 K. An interface (with z =0) is placed at the position of the center of the exterior non-bridging oxygens of the bridging Si tetrahedra. H-bonds are indicated by the dashed red lines.

Disjoining Pressure and Free Energy of Swelling In nanolayered materials, the confinement pressure in drained conditions is related to the thermodynamic potential 15,37

Λ(V, T, µw ) = F − µw Nw

(1)

via the expression

1 ∂Λ P =− A ∂d T,µw

(2)

where F is the Helmholtz free energy and A is the surface area of the layer. The GCMC simulation described in the Models and Methods section minimizes the mixed thermodynamic potential Λ(V, T, µw ). 15,37 Therefore, the confinement pressure P can be directly computed from the component in the direction orthogonal to the basal plane of the virial stress tensor sampled in the GCMC simulation. Figure 4 (b) displays the disjoining pressure isotherms of 9

Disjoining Pressure, P @GPaD

Nw êCaSiO18 @-D

40 30 20

6

Á ÁÁ ÁÁ Á ÁÁ ÁÁ ÁÁÁ Á Á

4 2

Á

Á

ÁÁ

Á

Á

ÁÁ

Á

10

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Á

a

Á ÁÁÁÁ ÁÁÁ Á Á 11 12 13 14 15 16 17ÁÁÁÁ Á ÁÁÁ ÁÁÁ ÁÁÁ Á Á ÁÁ ÁÁÁÁ ÁÁÁÁ ÁÁÁ Á Á Á Á Á Á

10 0

8

20

Á

Á Á ÁÁ

Á

Á ÁÁ ÁÁ

30

Á Á Á Á Á Á Á Á Á Á

Á

40

Á

Á Á

Á

GCMC

50

Á

b

6 4 2 0

Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁ ÁÁÁÁÁÁÁ ÁÁÁÁ Á Á Á Á Á Á Á Á Á Á ÁÁÁÁÁÁ ÁÁÁÁÁÁ ÁÁ Á ÁÁÁ Savitzky-Golay Filter Á ÁÁ Á

-2

Á

-4

Á Á ÁÁ

Á

10

20

2

GCMC

30

40

50

30

40

50

c

2W

0 L @kJêHmol.Å2 LD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Water moleculesêCaSiO18 @-D

Langmuir

1W

-2 0W

-4 -6 -8 -10 10

20

Basal spacing @ÅD

Figure 4: GCMC simulation of calcium silicate hydrates at 300 K and under imposed water pressure Pw (µw ) of 1 atm: (a) water content, (b) disjoining pressure isotherm and (c) free energy Λ as a function of the basal spacing d. The standard deviations of adsorption and pressure isotherms are on the order of the circle size. To compute Λ, the pressure isotherms were filtered using Savitzky-Golay smoothing algorithm 36 before the integration in Eq. 3.

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tobermorite-water system at 300 K plotted as a function of the basal spacing. The solid lines are obtained from Savitzky-Golay smoothing filter. 36 The disjoining pressure is the difference between the confining pressure and the fluid pressure Π = P − Pw . In agreement with the oscillating pressure isotherm in Fig. 4 (b), calcium silicate hydrates experimentally probed via Atomic Force Microscopy (AFM) 38 as well as previous molecular simulations 8,13,14 also exhibit oscillating force-basal spacing profiles and adhesive forces on the order of a GPa. 38 For small basal spacings, it is possible to identify the quasidehydrated state (0W), in which the confining pressure strongly increases with the decrease in d as a result of the steric repulsion between the solid layers, counterions and the remaining water molecules. The subsequent oscillations correspond to the various possible hydration states nW with water getting structured in discrete n layers (out of tobermorite channels) up to the state of pore water ∞W. The local minima associated with 0W and 1W states are, respectively, at d =12.6 and 15.7 Å. These minima are similar to the values reported by Masoumi et al. 14 of d =13.5 and 15.9 Å for 0W and 1W, respectively. The distance between the oscillations (taking, for example, the local minimum values) is approximately 3 Å, which roughly corresponds to the Lennard-Jones diameter of the oxygen in SPC water model. The portions of the curve in which the pressure increases with the basal spacing are unstable. Therefore, the system is subjected to micro- or snap-through instabilities, 39 which are reported to be at the heart of important phenomena in nanoporous materials such as shrinkage and creep. The energy profile of a system controlled by the volume can be computed from pressure isotherms by

Λ/A = Λ0 (d0 , T, µw )/A −

Z

d

P d(d)

(3)

d0

where d0 is a reference basal spacing and Λ0 is the corresponding free energy at this reference basal spacing. Figure 4 (c) displays the energy profile Λ of tobermorite-water system at 300 K plotted as a function of the basal spacing. The most stable configurations are identified by the 11


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minima in this energy profile. The global minimum corresponds to the minimum associated with 0W state at d = 11.8 Å. The basal spacings corresponding to 1W and 2W do not present local minima in Λ profiles at 300 K and under imposed water pressure Pw (µw ) of 1 atm. The Λ profile is non-convex, i.e. the first derivative of Λ (c.f. pressure isotherm) is not a non-monotonically decreasing function, as can be clearly observed in the 0W/1W transition. In systems with many stacked layers, which can be observed in the mesoscale, the mechanical response will conform to the convex hull of the energy profile Λ. Research on other phyllosilicates such as clays shows that under saturated conditions (i.e. in systems in which the chemical potential and pressure of water exceed the ones associated with liquid saturation) the confining pressure P under different imposed fluid pressure Pw can be computed from a single disjoining pressure isotherm Π. 15,40 The GCMC simulations of tobermorite exhibit a similar behavior, in which the disjoining pressure can be approximated as a function of only the basal distance and the temperature. This feature translates in the apparent Biot coefficient 41 b = −

∂P , ∂Pw d

which quantifies the transmission

of stresses between the fluid and the solid layers, being close to 1. Layered materials with confined fluids may exhibit apparent Biot coefficients much larger than the unit or even negative. 41,42 Interfacial effects may modify the way the pressure of fluids is transferred to the solid phases in materials with nanometric pores, even if the solid phases remains poorly compressible. 42 Figure 5 shows the apparent Biot coefficient computed from the confining pressure isotherms obtained at four different water pressures Pw (µw ). The apparent Biot coefficient b is computed from the least square fit of the linear regression of P (using the values filtered with Savitzky-Golay algorithm) plotted against Pw for a fixed basal spacing d. The average apparent Biot coefficient hbi computed from basal spacings ranging from 10.5 to 40 Å is 1.3. At the global minimum of Λ (d = 11.8 Å) b is 1.1. These values of b close to 1 shows that the transmission of stresses between the solid and fluid phases in tobermorite micropores is not strongly affected by interfacial effects. Figure 6 (a) displays the energy profile Λ according to four temperatures of interest in

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10 Apparent Biot Coefficient, b @-D

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Á Á

Á

=1.3

5

Á Á Á Á Á Á Á ÁÁ ÁÁ Á Á ÁÁ Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁ Á Á Á Á Á Á Á Á Á Á Á ÁÁ Á Á Á Á ÁÁ Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁ Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁ Á Á ÁÁ Á ÁÁ Á Á Á Á Á Á Á Á ÁÁ Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁ Á Á Á ÁÁ Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á ÁÁ Á Á Á Á Á Á Á ÁÁ

Á

0

Á Á Á

-5 10

15

20

25

30

Basal Spacing @ÅD

35

40

∂P Figure 5: Apparent Biot coefficient b = − ∂P of tobermorite at 300 K as a function of the w d basal spacing. The gray zone represents the typical values of Biot coefficient in the range [0,1]. The red dashed line hbi=1.3. is the average value of b sampled on basal spacings ranging from 10.5 to 40 Å.

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industrial applications of concrete, notably regarding temperature rise at early-age due to cement hydration exothermy. 21 As observed in other layered systems such as clays 15 and Lennard-Jones fluids confined between 9-3 Lennard-Jones walls, 37 the depth of the wells in energy profile Λ decreases with the temperature. In the inset of Fig. 6 (a), this decreasing is linear with the temperature with a rate of 1.32 kT/(nm2 .K) obtained from least-square fitting of the points. In the Fig. 6 (b), the energy profile Λ is plotted as a function of the basal spacing for four chemical potentials of water. As shown in Fig. 2 the chemical potential of water varies linearly with the water pressure at the range of temperatures considered here with a rate of 59 MPa/(kJ/mol). Therefore, the results in Fig. 6 (b) can be also interpreted as energy profiles according to four different water pressure Pw with pressure increments of ∆Pw =-235, 0, 235 and 705 MPa, corresponding to increments of chemical potential of water ∆µw of, respectively, -4.0, 0.0, 4.0 and 12.0 kJ/mol. . Approximating the evolution of the well depth as a function of the chemical potential or water pressure by a linear function, the corresponding slopes of the fitted lines are, respectively, 2.7 kT mol/(kJ.nm2 ) and 0.046 kT/(nm2 .MPa) for the the chemical potential and water pressure plotted versus Λ. The depth of the wells in energy profile Λ decreases with the increase of the chemical potential of water. Thus, the total energy needed to disjoin two calcium silicate hydrate layers decrease with either with the temperature or with the chemical potential and pressure of water. For further validation, some key physical properties can be directly computed from the pressure isotherms. 14 Table 1 gathers the cohesive pressure Pco (i.e. the pressure needed to disjoin two solid layers, which corresponds to the global minimum in the pressure isotherms), out-of-plane elastic modulus Ezz = −deq /A

∂Λ ∂d deq

and surface energy γs = −1/(2A)

R∞ deq

Λd(d),

where deq is the equilibrium basal spacing corresponding to the minimum in Λ energy profiles computed at various temperatures. The estimates are compared against results from experiments and simulations reported in the literature and a fair agreement is observed. As observed in other layered materials such as clays, 15 the surface energy γs in Fig. 6 (a) de-

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100

100

a

290 K 300 K

0

0

Ûmw =0 kJêmol Ûmw =4 kJêmol

320 K

-200

-200

minHL-L0 LêA

-240

-300 -400

-260

Á

-280 -300 -320

Á

12

14

Á

-300

Á Á

R2 =0.996

-400

-340 290

-500 10

Ûmw =12 kJêmol

-100

330 K

16

18

310

330

20

22

T @KD

Basal spacing @ÅD

minHL-L0 LêA

-100

b

Ûmw =-4 kJêmol

310 K

HL-L0 LêA @k Tênm2 D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

-260

Á

-280

Á Á

-300 -320

Á

-340 -5

24

-500 10

12

14

R2 =0.919

16

18

0

5

10

Ûmw HkJêmolL

20

Basal spacing @ÅD

22

15

24

Figure 6: Energy profile Λ for a tobermorite-water system according to (a) the temperature and (b) chemical potential of water.

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creases with the temperature. The fluctuations associated with the pressure computations in GCMC (notably at small basal spacings - see the Supporting Information) make it difficult to identify a clear tendency in Pco : these fluctuations are on the order few hundreds of MPa while the differences due to temperature are on the order of tens of MPa. In the previous study on clays, 15 in which the range of temperature are much larger (from 300 to 500 K), a clear tendency was also not observed. In another study dealing with Lennard-Jones (LJ) fluids confined in-between 9-3 LJ walls, 37 the Pco clearly tends to decrease with the temperature for all hydration states but the differences in terms of pressure are very small compared to the oscillations due to water ordering. Prohibitive long simulation must be required to achieve the degree of precision necessary to identify a clear tendency in system more complex than confined LJ fluids. Table 1: Physical properties computed from the confining pressure isotherms: validation against experimental and simulation results from the literature.

Current work

Data from the literature

T (K) 290 300 310 320 330 -

Out-of-plane elastic modulus Ezz (GPa) 60.9 61.5 60.9 77.4 75.7 14,43 77.6; 61 44 46 89; 68.4 47

Surface energy γs (J/m2 ) 1.22 1.20 1.17 1.16 1.14 43 0.42; 0.67; 14 1.15 45 0.32-0.4 48,49

Cohesive pressure Pco (GPa) 4.19 4.17 4.23 4.19 4.15 6.5 14 5 43 5.0 50

Stability under pressure control and Upscaling data The energy profile for a system controlled by the confining pressure is the Legendre transform of Λ:

Λ∗ /A = Λ/A + P d

(4)

which is also called swelling free energy 51–53 in clay science. Figure 7 reports the free energy Λ∗ per unit area according to various (compressive and tensile) confining pressures P . 16


Page 17 of 33

Micromechanics-based studies 54 show that the standard deviation of stresses in C-S-H are on the order of the stresses applied at macroscopic scale (generally on the order of few tens of MPa in concrete structures). Therefore, the curves at P = ±0.3 GPa encompass the major part of industrial applications of cement-based materials. In all cases, the global minimum is related to 0W state with a basal spacing that slightly decrease with P . For confining pressures P < −0.6 GPa, the 1W state exhibits a local minimum. The stable zones (i.e. where

∂Λ ∂d

< 0) related to each hydration state are depicted in the figure by the yellow zones.

Ó

Ó 0W Ó Ó

600

200 0 -200

-600 -800 11

Min .

Ô

Ma x .

>

Sp i. Te n s .

traction

-400

Ó

Ó0W Ó GPa P = 1.5 Ó(meta) Ó Ó Ó Ó Ó Ó Ó P = 0.6 GPa Ó Ó Ó Ó Ó Ó P=0 Ó Ó Ó Ó Ó Ô Ó Ô Ó Ó Ô Ó Ó P = -0.6 GPa Ô Ó Ó Ó Ô Ó Ó Ô Ó Ó Ô Ó Ô Ó Ô Ó Ô Ó 1W P = -1.5 GPa Ô Ó Ó Ô Ó 14 15 16 Ô 13 Ó Ó Ô Ó Ó Ô Ó Basal spacing @ÅD Ô Ó Ô Ó on

HL* -L* 0 LêA @k Tênm2 D

400

> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > 12 > > > >

compressi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Figure 7: Energy profiles Λ∗ as a function of the basal spacing d for various control pressures P (from -1.5 to 1.5 GPa with increment of 0.3 GPa). The local minima and maxima are displayed according to the hydration states as well as the spinodal decomposition points for a tensile load (1W→2W or ∞W). The points of spinodal decomposition define the limit of the metastable zones. These 17


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

points (ds , Λs ) satisfy the condition

∂ 2 Λs ∂d2s

= 0. For a given hydration transition xW→yW,

the free energy Λ∗ is monotonic for confining pressures P exceeding the pressure at the spinodal points (i.e. there is no energy barrier associated with the transition xW→yW). In the Fig. 7, the curves linking the minima of 0W state, the spinodal points (0W→1W or ∞W) and the maxima of 1W state converges to a single point in a energy profile corresponding to control pressure equivalent to the cohesive pressure P = −Pco =-4.2 GPa. The analysis of the curves in Fig. 7 shows that the difference of energy between the minimum corresponding to 1W state and the energy at the point of spinodal decomposition (associated with a tensile load applied on the system) is on the order of a hundreds of kT /nm2 . For comparison, in other phyllosilicates such as smectites, the energy barriers accompanying hydration state transitions are on the order of a few kT /nm2 only. 15,51 The energy barriers associated with calcium silicate hydrates are an indication that the system is prone to exhibit metastable states since the thermal fluctuations of the system may not be sufficient to overcome such energy barriers in timescales of interest in industrial processes involving cement-based materials. Indeed, the hysteresis observed in cement-based materials have been associated with metastability at the nanoscale. 39 Additionally, considering the dimensions of calcium silicate hydrates layers (in-plane dimensions ranging from few tens of nanometers up to several micrometers 23 ), these energy barriers are an indication that hydration transition may involve bending of solid layers at specific domains, which further favors metastability. 55 The free energy Λ∗ as a function of the basal spacing at P = 1 atm is shown in Fig. 8. This free energy is a potential of mean force (PMF) representing in an effective way the interactions 56 between calcium silicate layers at drained conditions. This profile is fitted using by the following mathematical expression, for all distances larger tha a inner cut-ff of di =9.5 Å and smaller than the cut-off distance dc =60 Å:

18


Page 18 of 33

Page 19 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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"

#

(d − a6 ) d a1 exp Uf it (d) = − 4 + a3 exp (−a4 d) + a5 cos a7 a8 (a2 + d)

!

∀di < d < dc

(5)

where the fitting parameters ai (i = 1..8) are obtained by least square fitting of the PMF. A similar expression has been used in previous studies to fit intermolecular forces in watertobermorite systems. 14 The corresponding fitting parameters for the temperature and water pressures considered in GCMC simulations are gathered in Tab. 2. Table 2: Parameters of fitted PMF using Eq. 5 (Uf it in kJ/mol). Pw [MPa] 0 0 0 0 0 10 100 1000

T [K] 290 300 310 320 330 300 300 300

a1 2.08601×106 2.18195×106 2.37755×106 1.61680×106 1.61039×106 2.49964×106 1.74458×106 2.4032×106

a2 5.27821 6.08641 7.14630 4.15102 4.17123 5.92788 5.55778 7.53021

a3 451.242 367.602 275.266 551.284 535.034 411.288 359.313 202.4

a4 0.274818 0.273448 0.265897 0.288499 0.28639 0.26497 0.281973 0.246166

a5 -2.92433×1016 -1.37316×1016 -7.63028×1015 -4.16629×1016 -4.4253×1016 -6.6390×1016 -2.65003×1016 -7.48295×1016

a6 6.57752 6.56038 6.53504 6.57443 6.57761 6.61896 6.59766 6.65309

a7 0.450013 0.449921 0.451142 0.451319 0.451095 0.447413 0.447449 0.443308

a8 0.312794 0.319699 0.325298 0.309689 0.309155 0.305592 0.313632 0.304488

Emerging behaviors at the mesoscale: coarse-grained simulation The PMF fitted using Eq. 5 are used as an input in mesoscale simulations of calcium silicate hydrates. Such mesoscale simulations allow upscaling critical physical chemistry information from the molecular scale and represents a huge computational advantage compared to full atomistic simulations. 56 Coarse-grained simulations were performed in a system with 2000 particles interacting via the PMF in Eq. 5 and the parameters in 2. The particles were randomly generated in a simulation box with a packing density of 0.64, which corresponds to low density C-S-H. 57 The system was equilibrated during 1 ns in a NVT simulation. Figure 9 (a) shows a snapshot of the granular system after equilibration. As observed in previous mesoscale studies, 58 the system tends to form a network (Fig. 9 (b)) with denser domains and a mesoscale porosity. To further validate this mesocale simulation, the elastic constants are computed via a finite difference method in which the simulation box is slightly deformed in each one of the three

19


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4 HL-L0 L @kJêHmol.Å2 LD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 33

2 0 -2 -4 -6

300 K

300 K HFitL

-8 -10

15 20 25 30 35 40 45 50 Basal spacing @ÅD

Figure 8: Free energy under pressure control: PMF and fitted potential using Eq. 5 with the parameters in Tab. 2 for a system at 300 K and water pressure of 0 MPa. The other cases are shown in the Supporting Information.

20


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axial directions and the three shear planes. The obtained values (in Voigt notation) are for the terms C11 , C22 and C33 are 19.8±7.7 GPa in average and for the terms C12 , C13 and C23 13.9± 3.93 GPa in average. The out-of-diagonal terms Cij for which i, j > 3 are expected to be zero for isotropic materials; here, these terms were approximately one order of magnitude smaller that the other components. The corresponding bulk and shear moduli are 13.9 and 8.8 GPa, respectively, which are consistent with the elastic moduli experimentally measured for the varieties of C-S-H. 57 Asymmetry tension/compression is observed in the uniaxial stress-strain response of this granular system as shown in Fig. 9 (c) and (d) according to three true strain rates ε˙ = Ln (L(t)/L(0)) /dt. The elastic moduli computed from the slopes of the elastic regimes portions of the curves are similar for all strain rates tested and for both tension and compression. These elastic moduli are also consistent with the values Young modulus that can be obtained from Tab. ??. The yield stress σy (limit of the elastic behavior) of the system is larger and much more dependent on the strain rate under compression than under tension. Hysteresis emerges at the mesoscale for both thermal and mechanical loads as can be seen in Fig. 9 (e) and (f), respectively. The PMFs identified in the last section are temperature dependent. To investigate the effects of this temperature dependency on the mesoscale behavior, simulations using the PMF at the five temperatures considered in Tab. 2 were performed. Fig. 9 (e) shows the volumetric strain resulting of a thermal loading from the temperature associated to the PMF (only the curves corresponding to 290, 300 and 330 K are shown, see the Supporting Information for the curves of 310 and 320 K) up to 400 K then a decrease in the temperature to 270 K. The corresponding coefficients of thermal expansion (CTE) at increasing temperature (αT + (in ×10−5 K −1 )= 53.96 (290 K), 53.61 (300 K), 55.47 (310 K), 56.56 (320 K) and 43.43 (330 K)) and the CTE at decreasing temperature (αT − (in ×10−5 K −1 )= 40.61 (290 K), 48.40 (300 K), 39.34 (310 K), 50.50 (320 K) and 41.28 (330 K)) exhibit differences in the order of tens of 10−5 K −1 . For comparison, Qomi et al. 59 obtained a CTE of 45×10−5 K −1 for high and low density C-S-H at the molecular scale.

21


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The differences between αT + and αT − indicate a hysteresis due to thermal loadings. For the CTEs using the PMFs at 290, 300, 310 and 320 K, the values of αT + are relatively closer (average of 52.60 ×10−5 K −1 for all values with standard deviations of 5.26 ×10−5 K −1 ) indicating that the temperature dependence of the PMF does not play a major role in the CTE for system at increasing temperature. Similar reasoning can be applied to the PMFs at 290, 300, 310 and 330 K regarding CTE of system at decreasing temperature (average of 44.02 ×10−5 K −1 for all values with standard deviations of 5.05 ×10−5 K −1 ). Regarding mechanical loads, under isotropic compression, a part of the deformation is irreversible. Similar results were obtained in previous studies on monodisperse systems of C-S-H grains. 60 As can be seen in the fitting of the PMF under different water pressures (see Supporting Information), this pressure dependency of the PMF is expected to be negligible in the range of pressure considered.

Conclusion In this article, the intermolecular interactions between crystalline calcium silicate hydrates were reported according to various temperatures and water pressures. GCMC simulations with configurational bias were performed at the molecular scale in order to identify the potentials of mean force to be used at mesoscale to assess the behavior of granular system representing calcium silicate hydrates. These results leave room to a better understating of the effects of temperature and pressure on the interactions between calcium silicate hydrates and water at the nanoscale. This work leads to the following conclusions: • Under saturated conditions, the disjoining pressure isotherms of calcium silicate hydrates are not very strongly affected by changes in water pressure. This behavior was also observed to other phyllosilicates such as clays. 15,42 Thus, at constant temperature, the approximated response at different fluid pressures can be obtained from a single pressure isotherm. A corollary of this observation is that variations in the water 22


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Page 23 of 33

Stress σ [MPa]

100

. . ε= -0.001/fs .

10

ε= -0.01/fs

1

ε= -0.0001/fs 0.1 -0.25

-0.15 Strain, ε [-]

0.003 0.002 0.001

cr

In

0 -0.001

330 300 290

-0.002 -0.003

-0.05

250

s ea

gT

in

330 300 290

300 350 Temperature [K]

400

5

200

b

e

0.004

as

0.005

re

1000

De c

Volumetric strain, εvol [-]

a

c

in gT

0.006

10000

f

d

180

4

160 140

Pressure [GPa]

Stress, -σ [MPa]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

120 100 80

.

ε= 0.01/fs . ε= 0.001/fs . ε= 0.0001/fs

60 40 20 0.2

0.4 Strain, ε [-]

0.6

Unloading

3 2 1

0 0

Loading

0.8

0 -0.1

-0.08 -0.06 -0.04 -0.02 Volumetric strain, εvol [-]

0

Figure 9: Mesoscale behavior: (a) snapshot of the granular system interacting via the PMF in Eq. 5 and (b) the corresponding network of particles. Uniaxial stress-strain response of the system under uniaxial (c) compression and (d) tension according to three deformation rates ε. ˙ The strain is computed from the matrix of the supercell parameters h = [a, b, c] using ε(t) =

1 2

h−1 0

T

h(t)T h(t)h−1 0 − I , where h0 is the reference super cell parameters

and I is the 3 × 3 identify matrix. The volumetric strain is defined as the trace of the strain tensor: εvol = T r(ε) (e) hysteresis under temperature variation, (f) hysteresis under isotropic compression.

23


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pressure results in the same variations in the cohesive pressure for a fixed temperature. • The surface energy decreases with the temperature, the chemical potential of water and the water pressure. Variation in the cohesive pressure on the order of a MPa were observed with changes in the temperature. The energy barrier needed to be overcome to disjoining the layers is on the order of few hundreds of kT/nm2 , which indicates that the system is prone to metastability. • Under confining pressure control, crystalline calcium silicate hydrates respond differently under tension and compression. For (tensile) confining pressure exceeding 0.6 GPa, the 1W state becomes stable, but with energy barrier necessary to disjoin the layers of only few kT/nm2 . Under compression, the depth of the free energy profile increases with the confining pressure. These energy profiles can be used to upscale C-S-H properties via coarse-grained simulation at the mesoscale. • The anharmonicity of the interactions identified at the molecular scale translates in an asymmetry tension/compression and thermal expansion that are also observed at the mesoscale. Regarding thermal deformations and thermal expansion, the pressure and temperature dependence of the PMF seems negligible within the range of temperature considered in this study (290-330 K). Hysteresis due to thermal and mechanical loads were observed at the mesoscale. Perspectives includes testing pressure and temperature effects on creep on both molecular and mesoscale using for example the simulation protocol recently proposed in the literature. 61

Supporting Information Available The following files are available free of charge. • Supporting Information: Details on pressure isotherms, PMF and coarse-grained simulations. 24


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g

d

unloadin

T

Pressure

Graphical TOC Entry Swelling energy

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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loa d

ing

Strain

Emerging behavior at the mesoscale : - Hysteresis under thermomechanical loads - Asymmetry tension/compression

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Monte Carlo Molecular Modeling of Temperature and Pressure Effects

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