Chapter 18
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Heterogeneity in Cement Hydrates K. Ioannidou* Multiscale Materials Science for Energy and Environment, MIT-CNRS-AMU, Department of Civil and Environmental Engineering, MIT Energy Initiative, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States *E-mail:
[email protected].
Cement hydrates named C-S-H are the main products of the reaction of cement with water. The C-S-H phase is the most important phase of cement paste as it glues all other phases together in a solid rock-like material. C-S-H gels form and densify via out-of-equilibrium precipitation and aggregation of nano-grains during cement hydration. In this chapter, the link between the making and densification of C-S-H gels and amorphous solids is discussed by coarse-grained models based on the evolution of interaction potentials and an out-of-equilibrium simulation approach for particle precipitation. In particular, we characterize and correlate mesoscale structural and mechanical heterogeneities resulting into residual local eigenstresses. This underlying microscopic picture explains recent macroscopic measurements of the volume change of hydrating cement in fully saturated conditions.
Introduction Calcium-Silicate-Hydrate (C–S–H) is the primary hydration product of Portland cement that precipitates, upon mixing it with water, as nano-scale colloids in the pore solution. The nano-colloids form a highly cohesive gel, a soft glue whose properties in the early stages of the hydration affect the strength that cement and concrete attain after setting, when the material eventually hardens. Hence, in spite of the fact that the use of cement for concrete infrastructures and buildings relies upon its properties as a hard solid, those properties are controlled © 2018 American Chemical Society Horkay et al.; Gels and Other Soft Amorphous Solids ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
by its soft matter state, mainly the C–S–H gel precursor of the hardened paste in the early stages of the hydration. The investigations of the gel during its formation are very limited, most of the structural characterization being done on the hardened paste via neutron scattering, atomic force microscopy (AFM) or scanning electron microscopy (1–3). These studies have been mainly performed on tricalcium silicate, the main reactant in Portland cement, that it is also often used as a model system for cement hydration. They indicate amorphous mesoscale organization built upon structural units of typical size 5-15nm (4). The sub-nanoscale structure of C-S-H building blocks is complex, resembling that of Tobermorite minerals at low calcium over silica ratio (Ca/Si) and that of disordered glasses at high Ca/Si (5). Recent advances of atomistic modeling of C-S-H have revealed characteristic of the structure and mechanical properties of the nano-colloids (5, 6) and have provided a first description of the effective interaction potentials between such nano-colloids (7–9). The effective interactions between the C– S–H units depend strongly on the chemical environment that evolves in time, with the dissolution of the cement grains and the precipitation of various hydration products (10). In fact, the effective net attraction between the C-S-H particles progressively increases with the changing physico-chemical environment, mainly due to strong charge heterogeneities and ion correlations growing in the pore solution. Such effective interactions are favored by the presence of multivalent ions (Ca+2) and may feature a combination of short-range cohesion and longer-range electrostatic repulsion resulting in an attractive well and a repulsive shoulder. A series of controlled experiments with fixed calcium ionic concentration provided useful indications on how the effective nano-scale interactions may indeed evolve over time in conditions relevant to cement hydration (11). Atomic force microscopy measurements show that the main change in time consists in the progressive reduction of the repulsive shoulder with time, which most probably completely disappear by the end of setting (11). Building on this experimental information we have investigated how equilibrium properties and aggregation changes in colloidal suspensions in presence of effective interactions that correspond to different physico-chemical conditions of hydration (12). The emerging picture is that the evolving effective interactions provide a thermodynamic driving for the growth of C-S-H gel and densification towards amorphous solid porous structures that are crucial to cement strength (13). In the past, with the idea that the gel can be regarded as a random assembly of sticky units, the gel growth has been prevalently addressed in terms of an irreversible aggregation of colloidal nanoparticles, as in a diffusion-limited cluster aggregation (DLCA) process (14). Nevertheless, this scenario has little or no correspondence to the morphology revealed by the experiments on the hardened cement paste. It does not explain how high packing fractions required for the final strength are reached. Moreover, C–S–H is produced through an exothermic reaction and its growth kinetics, measured in terms of the time dependence of the heat flow, is strongly non-monotonic, being characterized by an acceleration and followed by a deceleration regime. This feature is difficult to reconcile with the monotonic growth kinetics expected in the case of DLCA. 358 Horkay et al.; Gels and Other Soft Amorphous Solids ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
A recent simulation approach for C–S–H gelation and densification based on colloidal particles precipitation was able to reconcile several experimental results of the early and late stages of cement hydration (12, 15). In particular, the non-monotonic growth kinetics and the densification of the structure locally up to volume fractions 75% was rationalized. Moreover, these studies provided a quantitative description of local densities, pore size distributions, effective surface areas, hardness and elastic moduli that characterize C-S-H porous solids in hardened cement paste. Here, we discuss the mesoscale modeling scheme with emphasis on the effective interactions and how features of local packing connect to out of equilibrium phenomena of reactive densification in cement hydration. This is related to the development and (recently measured) eigenstresses resulting to bulk volume changes (16). We demonstrate this at the level of local packing and mesoscale organization of matter by correlating locally dense and locally stressed regions.
Mesoscale Model for Reactive Solidification of Cement Despite the remarkable complexity of colloidal physical chemistry, the equilibrium phases of colloidal suspensions are usually well-described with coarse-grained models assuming short range interactions, reminiscent of simple liquids (17–19). However, in complex fluids the relaxation times are typically much longer than in simple fluids and often it is important to understand the interplay between kinetic and thermodynamic trends (20). For instance, a variety of arrested, out-of-equilibrium states like gels or glasses can occur upon changing the control parameters or environmental conditions (e.g., varying the solid volume fraction or the strength of the interactions or applying a rate-dependent mechanical perturbation). The structural complexity of the metastable states typically emerges from the combination of a local/medium range order, bearing a signature of the underlying thermodynamics, with the slow kinetics eventually leading to an amorphous self-organization of the material over larger length-scales. Several experiments have been reported where complex aggregation pathways and structural arrest have been observed, or where equilibrium phases and metastable states resulted from complex effective interactions: arrested spinodal decomposition (21, 22), fluid cluster phases, unusual microcrystalline gels, colloidal membranes (23–25) or string-like or tetrahedrally packed aggregates (26–28). For investigations of cement hydrates gels and amorphous solids, coarse-grained modeling was employed based on nano-grains/colloids precipitation interacting with evolving attracto-repulsive potentials (12). The potentials are composed of short range attraction with a longer-range repulsion resulting in an attractive well, a shoulder and a repulsive tail. To account for the creation of new particles in the precipitation simulation, the μVT statistical ensemble is assumed, where μ is the chemical potential, V the volume of the simulation box and T the temperature. The interaction potential of the particles is fixed for each simulation and the chemical potential determines the particle 359 Horkay et al.; Gels and Other Soft Amorphous Solids ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
equilibrium density. However, cement hydration occurs in short time. Therefore, non-equilibrium simulations of different kinetic rates were performed. The chemical potential was chosen such that it promotes densification of the system. In this type of μVT precipitation approach, the kinetic path is build up upon the progressive increase of particle density. A metastable state of density ρ1 is the precursor of state ρ2, ρ2> ρ1. This has a significant implication on the structural and mechanical heterogeneities of amorphous solids states of volume fraction 40% or larger, relevant for the hardened cement paste (see Discussion section). Gels and solids with the same potentials were also obtained via variation of traditional control parameters such as the solid volume fraction and temperature (13). In more details, during cement hydration C-S-H colloids precipitate in the pore solution because of the ongoing chemical reactions causing dissolution of anhydrous cement grains and precipitation of the C-S-H phase along with other minor various hydration products. In this scheme, newly precipitated C-S-H colloidal particles in the simulation box interact with existing one, stick together, and form aggregates that eventually create a stress-bearing gel. A free energy gain related to the chemical potential μ drives the precipitation of C-S-H particles. The effective inter-particle forces for cement hydrates have been obtained from sub-nanoscale simulations and AFM experiments (11, 29). The simulations consist of a grand canonical Monte Carlo (GCMC) scheme, where the chemical potential corresponds to the free energy gain of a new particle creation coupled to molecular dynamics (MD) allowing to follow both the densification and the dynamics of the aggregated structures. The formation and growth of the C-S-H mesoscale structures are determined by the interplay between free energy of the system (resulting from the chemical potential and the inter-particle potential) and the imposed out-of-equilibrium conditions (corresponding to the rate at which GCMC events take place with respect to the structural relaxation time of the aggregated structures). A relatively common feature of effective colloidal interactions is a competition between a short-range attraction with a longer-range repulsion resulting in an attractive well, a shoulder and a repulsive tail. The physical origin of attraction at small separations can be very different, e.g. van der Waals forces as in the DLVO theory (30), polymer-mediated depletion interactions (31), polarization effects in case of induced magnetic interactions (25), micro-ion correlation in case of multivalent electrolytes (32), bond formation in DNA-coated colloids (33) or even activity of self-propelled particles (34). The long range repulsion typically originates from colloids being charged (35, 36) or from other system-specific interactions like magnetic, steric etc. Often, such systems are inherently many-body (37), which adds further complexity to the problem. However, in some cases the macroscopic properties are well-described by pairwise additive effective interactions (38). In case of charged suspensions, this might be true under high salt conditions when the physical charges on the particle surfaces are efficiently screened. For highly charged particles, i.e. in charged stabilized suspensions, the repulsive part of the interaction potential dominates the behavior (the height of the shoulder is much larger than the thermal energy), while more complex behavior can be observed in weakly charged colloids where 360 Horkay et al.; Gels and Other Soft Amorphous Solids ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
both the attractive well and the repulsive shoulder are of the order of kBT (39–45). Such a scenario can arise in systems like heterogeneously charged surfaces and membranes (46) and in materials like clays (47) or C-S-H (10). The electric double layer theory of Gouy and Chapman (GC), based on the solution of the Poisson–Bolzmann (PB) equation, describes the formation of a diffuse electric double layer when a charged surface is in contact with an electrolyte solution. It allows to calculate the interaction between two colloids. Combination of the GC or PB theory with van der Waals interactions led some 70 years ago to the so-called DLVO theory which is still the reference frame for many colloidal chemistry applications and is extensively developed in textbooks (35). The interaction between two identical particles immersed in an electrolyte, as predicted by the DLVO theory, is composed of two independent and additive parts: the double layer interaction, always repulsive, and the van der Waals interaction, always attractive. It is now clear that this was not correct and that a system composed of two charged particles with like-charges may generate attractive configurations due to correlations in the local concentration of ions due to thermal fluctuations. This has been known for more than twenty years (47, 48) and the suggestion that this may contribute to the cohesion of set cement was made more than two decades ago (49). The attraction appears when the electrostatic coupling in the electrolyte is sufficiently large. The electrostatic coupling can be evaluated from the colloidal surface density of charge, the ionic charge and radius, the solvent dielectric constant (in an implicit solvent description) and the temperature (50). In aqueous media, this happens with divalent or multivalent ions at short separation, say in the nm range. The reason why the PB treatment and DLVO theory miss the possibility of attractive double layer interactions is due to the neglect of thermally-driven ionic concentration fluctuations. The PB equation is therefore an example of the so-called mean field approximation. Taking into account electrostatic interactions only, the overall shape of the force (or pressure) versus inter-colloid distance can therefore vary from a continuously decreasing function (as in the DLVO theory or at weak electrostatic coupling) to a single well curve. For large enough and charged enough ions in interaction with highly charge colloids, this pressure-distance curve exhibits a short-distance negative well continued with a positive shoulder that eventually dies out at large distance. Note that in this case, the van der Waals contribution will be a small correction. The systems of C-S-H gels and solid investigated composed of spherical particles with diameter σ interacting via effective potentials, which are a combination of a short-range generalized Lennard-Jones (LJ) attraction and a long-range Yukawa repulsion (35):
where r is the inter-particle distance, γ the exponent in the generalized LJ interaction and κ the inverse screening length. The interaction potentials are truncated and shifted to zero at 4σ. Prefactors ε and Α measure the relative strength of the attractive and repulsive interaction terms. The functions V(r,γ,κ,ε,A) of Equation 1 with all possible parameter values define a family of interaction potentials with an attractive minimum at r=rmin and a repulsive barrier at r=rmax. 361 Horkay et al.; Gels and Other Soft Amorphous Solids ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
A sub-family of potentials was considered with fixed γ=12 and κ-1=0.5σ and combinations of A and ε values such that the depth of the attraction well at rmin is fixed to V(rmin)=-1kBT and the height of the repulsive shoulder is between 0 < V (rmax) ≤ 0.5 kBT.
Simulations Parameters Specifically, we focus on three cases the high shoulder HS (A=4, ε=1.5), the low shoulder LS (A=12, ε=2.4) and no repulsive shoulder LJ (A=0, ε=1.5). The three interaction potentials are displayed in Figure 1. In all cases under consideration the attraction is short-ranged, i.e. the width of the attraction well is in the range between 0.1σ and 0.3σ.
Figure 1. The inter-particle potentials combine a short-range 12-24 Lennard-Jones attraction and a long-range Yukawa repulsion. The black dotted line corresponds to HS, the red dashed line to LS and the green solid line to 12-24 LJ. The attraction to repulsion ratio ε/Α depends upon the concentration of calcium cations. 362 Horkay et al.; Gels and Other Soft Amorphous Solids ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Regarding the simulation approach, GCMC accounts for the interplay between the chemistry -in specific the free energy gain from the production of one new C-S-H particle, and the interactions between the particles. Each GCMC cycle consists of Np particle insertion or deletion attempts succeeded by M=100MD steps in NVT conditions. R=Np/(M·L3·δt) where L is the length of the simulation box and δt=0.0025(mσ2/ε)1/2, is the rate of hydrate production. Simulations with higher values of R bring the system further out of equilibrium (12). Time was measured by MD unit time (mσ2/ε)1/2, the temperature was T=0.15 and the chemical potential μ=−1 all in reduced units. The chemical potential is chosen to promote densification. The simulations have been performed in two system sizes, one of box size 585.54nm with up to 610,000 particles (large) and of 390.36nm with up to 180,000 particles (medium). The diameters of the particles were ranging between 4-10nm.
Discussion The early formation of open gel C-S-H structures is reached due to interactions of HS-type that provide two body attraction but also favor the formation of elongated clusters due to the strength (height) of the repulsive shoulder (12, 41, 42, 45). Such HS interactions during cement hydration occur at low Ca+2 ions concentration when large capillary pores are filled with water during the partial dissolution of cement grains (tricalcium silicate). As already mentioned, the physical reason for attracto-repulsive interactions during cement hydration is ion-correlation forces between highly charged C-S-H nano-colloids meditated through divalent ions in the pore solution. The attraction to repulsion strength is controlled from the concentration of Ca+2 ions (11). Moreover, the HS potential facilitates the precipitation of C-S-H particles at lower supersaturation. For temperatures relevant for the formation of solid aggregates (T~0.15kBT), the energy gain to form clusters at low volume fractions is larger with the HS potential than with the LS one. The chemical potentials μ for HS and LS potentials were computed in Reference (13). This is also supported from recent experimental findings in controlled in situ hydration of tricalcium silicate, where clusters of nano-particles of C-S-H are observed in the vicinity of but not yet stuck to the reacting cement grain (51). Overall, type HS potentials at the early stages of hydration are beneficial as C-S-H nano-colloids can precipitate more easily and form elongated clusters that eventually impinge into an arrested gel and contribute to the initial gelation of cement hydrates. An extensive study of this potential can be found in References (12, 13). If HS interaction between C-S-H particles was present throughout hydration, cement would not densify to form the compact solid material we know. From precipitation (12) and equilibrium (13) simulations it is shown that in order to attain higher volume fractions the repulsive shoulder shall reduce and disappear. Simulation data such as densification curves, coordination number and free energies support the fact that the continuous densification of the material is intimately related to the underlying equilibrium-phase behavior of the evolving effective interactions between the hydrates (12, 13). 363 Horkay et al.; Gels and Other Soft Amorphous Solids ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Figure 2 shows the distributions of nearest neighbors for amorphous solid structures of volume fraction η=0.37 resulted from precipitation simulations using the hybrid GCMC and MD scheme for HS and LS potentials. The LS potential enhances the number of nearest neighbors allowing for denser local packing compared to HS. The inset shows the average number of nearest neighbors for HS and LS for configurations resulted from equilibrium MD simulations. The nearest neighbors of particles interacting with LS potential are around 10 irrespectively of the volume fraction whereas the one with HS transition from 6 neighbors to 10 as the η increases. The mesoscale structures of particles interacting with HS are elongated clusters that form gels at lower η than LS. This is related to the fact that local particles pack in Bernal spirals (6 neighbors) (13). HS provides the initial gelation and LS favors local densification.
Figure 2. Distributions of nearest neighbors for samples of volume fraction η=0.37 produced from precipitation simulations with potentials HS (circles) and LS (triangles). The inset shows the average number of nearest neighbors for HS and LS from MD simulations. The data plotted here are from Reference (12).
Densification of the initial gel structure occurs due to the reduction of the repulsive shoulder (from HS to LS and to LJ). As more hydrates are produced the pore volume reduces and, in many cases, leads to localization of calcium ions among C-S-H hydrates that are at close distances. The initial open, thin-branched structure obtained with the HS potential, densifies resulting to fewer and thicker strands with larger pores in-between similar to coarsening. A proof of concept of this coarsening process when switching interaction from HS to LS can be found in Reference (13). 364 Horkay et al.; Gels and Other Soft Amorphous Solids ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
An important point to note is that HS interactions switch to LS and eventually LJ due to confinement. The system under investigation takes place into a capillary pore of hundreds of nm up to few μm. If densification continued with HS, the pressure would have increased as HS favors expansive open structures (building up of positive pressure). The fact that the precipitation of hydrates is confined, eventually causes the confinement of calcium ions that localize in-between C-SH particles and modify the effective interaction from HS to LS and eventually to LJ. This signals the transition from attracto-repulsive ion correlation forces to electrostatic interactions (the ions being localized in or at the surface of solid C-SH particles) with strongly attractive wells that can be modelled with a LJ potential, ultimately providing the strength of the hardened cement paste. Recent measurements of bulk volume changes in saturated cement pastes at constant pressure and temperature conditions showed the development of non-monotonic colloidal eigenstresses suggesting a competition of different attracto-repulsive mechanisms occurring in different large capillary pores for different ionic concentrations (16). Hydrating cement initially expands and after hundreds of hours begins to shrink (16). This behavior is a macroscopic average of capillary pores filled with newly formed cement hydrates interacting with HS type potentials, (hence expanding) and of capillary pores with denser solid aggregates of C-S-H interacting with LJ type potentials, hence shrinking. As hydration time proceeds the balance shifts towards the shrinking mechanism due to the tighter confinement of the ions. Such volume changes in hydrating cement play a critical role in many engineering applications that require precise calculation of stress and pressure developments. The mechanical properties of hardened cement paste strongly depend on the evolution of effective interactions and the eventual transition to strongly attractive forces between C-S-H particles. The moduli and hardness measured by nano-indentation experiments are in quantitative agreement with simulated ones by the precipitation scheme (15). Experiments show a distribution of moduli and hardness values indicating that cement is a very inhomogeneous rock-like material (52). The reason for this is that cement forms in very short time (~1 month) compared to geological times taken for minerals or sedimentary rocks. On one hand this fast setting of cement allows for high-rise building. On the other hand, structural and mechanical heterogeneities are inevitable as the C-S-H amorphous solids do not have time to reach equilibrium. This is especially important for concrete buildings and infrastructure’s durability and aging. Figure 3 shows such structural heterogeneities developed during the out-of-equilibrium process of particle precipitation interacting with LJ potential. Hardened cement paste exhibits a broad heterogeneity of local volume fractions ηlocal (15, 53). After thresholding the configuration for increasing local volume fractions, an underlying percolating network of highly packed (>64% RCP) particles is revealed. The percolating network of particles locally packed at higher volume fraction than the random close packing (RCP) covers only ~20% of the simulation box volume. In nano-indentation measurements, these are the regions that are responsible for the remarkable hardness and moduli of cement (15). Here, we move further to understand also the role of particles with local volume fraction below 60%. 365 Horkay et al.; Gels and Other Soft Amorphous Solids ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Figure 3. Snapshots of a configuration of hardened cement paste of volume fraction η=0.52 of polydisperse sized particles. They depict the different local volume fractions ηlocal. a) All local volume fractions are included. Particles with ηlocal ≥0.4, ηlocal≥0.5, ηlocal≥0.64 and ηlocal≥0.66 are shown in b, c, d and e respectively. The color code for all snapshots is shown in f.
Figure 4 shows the correlations between local volume fractions and local pressure Plocal in a configuration of hardened cement paste where the total eigenstress (at the level of the simulation box) has been relaxed (~10kPa) (53). In the population of particles with local volume fractions larger than 60% a tail towards positive local pressure is observed, hence overall particles are under compression. Thresholding the configurations by local volume fraction around 0.6, two types of networks reveal in Figure 5b. On one hand, the one of Figure 5b that contains the particles with ηlocal≥0.6 is very compact and its average pressure is positive. On the other hand, the one of Figure 5a containing particles with ηlocal
