J. Phys. Chem. C 2009, 113, 2347–2351
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Hydration Processes in Tricalcium Silicate: Application of the Boundary Nucleation Model to Quasielastic Neutron Scattering Data Vanessa K. Peterson* and Andrew E. Whitten Bragg Institute, Australian Nuclear Science and Technology Organisation, PMB 1, Menai, NSW, Australia 2234 ReceiVed: August 12, 2008; ReVised Manuscript ReceiVed: NoVember 20, 2008
Tricalcium silicate hydration is followed using quasielastic neutron scattering (QENS) and the kinetics of product nucleation and growth modeled using a boundary nucleation (BN) model. The BN model, previously applied once to calorimetry data of hydrating tricalcium silicate, is applied to QENS data that includes components that do not follow the evolution of heat (measured using calorimetry). Variations in modeling conditions are explored, including combination with a second model describing later, diffusion-limited, behavior. The BN model produces comparable quality fits to those from the commonly used Avrami-derived model and is found to support the theory that nucleation and growth begins at the time of mixing and proceeds with a single kinetic process. Introduction Tricalcium silicate, the main constituent of ordinary Portland cement, reacts quickly with water and is responsible for the early strength development of cement paste. Hydration begins with the irreversible dissolution of the outer tricalcium silicate grains, followed by a seemingly dormant period in the hydration known as the induction period, which ends with accelerated nucleation and growth of products:1
Ca3SiO5 + (3 + y - x)H2O f (CaO)x(SiO2) · (H2O)y + (3 - x)Ca(OH)2 (1) The first product on the right-hand side of eq 1 is calcium silicate hydrate (C-S-H) with a Ca:Si ratio, x, and a degree of protonation, y. The C-S-H is the product that determines the mechanical properties of the hydrated cement and these properties are dependent on the C-S-H stoichiometry and morphology, which vary during the course of the hydration. Calcium hydroxide is also formed, which unlike C-S-H is a crystalline product with well-defined stoichiometry. As nucleation and growth progresses, the hydrating tricalcium silicate grains become coated in hydration product and the evolving cementitious matrix creates a series of (sometimes isolated) pockets and pores. The development of the hydration products make it increasingly difficult for the water to reach the hydrating grains. The hydration rate eventually becomes controlled by the rate at which water can diffuse through the matrix, which is somewhat dependent on the morphology and density of the C-S-H that forms. Two kinetic processes of tricalcium silicate hydration are recognized after initial hydrolysis: Nucleation and growth followed by diffusion-limited hydration. Studies of these kinetic processes are important as an understanding of the mechanisms by which C-S-H develops may enable the control and design of C-S-H with favorable properties, and this is particularly relevant for systems more complex than for tricalcium silicate, such as whole cement. Following the disappearance of tricalcium * To whom correspondence should be addressed. E-mail: vanessa.peterson@ ansto.gov.au.
silicate, or the appearance of hydration products, is equivalent to following the progress of the overall hydration. Due predominantly to easy accessibility, calorimetry is commonly used to follow the progress of the hydration by tracking the evolution of heat with time, which is correlated with the dissolution of tricalcium silicate. Calorimetry has good timeresolution as many data points can be collected. A more detailed insight into the formation of hydration products during these kinetic processes can be gained by following the conversion of water into the reaction products, which can be monitored through the quantitative analysis of the state of hydrogen as a function of time. During the hydration process the motions of hydrogen, initially present only in the freely diffusing water, change as the water becomes trapped, adsorbed on surfaces, and transferred into the hydration products. Quasielastic neutron scattering (QENS) can quantitatively measure these changes and has been successfully applied through time-resolved experiments to measure the hydration of cementitious systems.2-22 Unlike calorimetry, which indirectly follows the consumption of tricalcium silicate, QENS follows directly the evolution of the hydrogen containing components. In the QENS method used here for hydrating tricalcium silicate, the time-evolution of the fraction of hydrogen considered immobile in a QENS experiment is scalable with the evolution of heat measured using calorimetry; both have similar kinetics.21 QENS measures hydrogen containing components that are involved in tricalcium silicate hydration and are not detected by calorimetry. The time-evolution of one such component has been correlated with the evolution of surface area as measured using small-angle neutron scattering21 and the time-evolution of this component is not scalable with the heat evolution as measured using calorimetry. Therefore, the additional components detected by QENS compared to calorimetry provide detailed information on the progress of the reaction of the free water and the appearance of hydrogen containing products, making it a more sensitive probe of the formation of hydration products. QENS data treatment is important and in a study of hydrating tricalcium silicate the number of components in the QENS model needs to effectively represent the populations of hydrogen
10.1021/jp807209w CCC: $40.75 Published 2009 by the American Chemical Society Published on Web 01/16/2009
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in the sample. All of the contributions from the hydrogen are assigned to the components of the model, regardless of the model chosen, and contributions that are not well represented will be assigned to a component in the model where they do not belong, causing problems with data interpretation. The QENS model used here is well defined for hydrating tricalcium silicate and has been described elsewhere.13-20 The nucleation and growth processes occurring during tricalcium silicate hydration, as measured using QENS, have been characterized previously using kinetic models derived from work of Avrami.2,6,14-20,22 An assumption that nucleation and growth occurs randomly is made in the Avrami-derived model; however, this is known not to be true for cementitious matrices where nucleation and growth occurs at surfaces. Recent work applied an alternative model that considered boundary nucleation (BN) and demonstrated that this model could be used as a more effective descriptor of the nucleation and growth processes occurring in hydrating tricalcium silicate than the Avramiderived model, as measured using calorimetry.23 One of the key findings of this paper was that nucleation and growth occurs at the time of mixing, and the so-called “induction period” is a consequence of the same nucleation and growth process occurring in the acceleratory period. The main aim of this work is to test the effectiveness of the BN model to describe the nucleation and growth processes in hydrating tricalcium silicate, as measured using QENS, relative to the commonly used Avrami-derived model. It is preferential to examine the BN model using a variety of data for the hydration process, including the components measured by QENS that give more information about the formation of hydration product than measured using calorimetry. Experimental Methods 1. QENS Data Collection and Treatment. The tricalcium silicate used here was obtained from Construction Technology Laboratories (CTL, Skokie, IL) and hydrated using a water to tricalcium silicate mass ratio of 0.4. QENS measurements were made using the time-of-flight Fermi Chopper Spectrometer at the NIST Center for Neutron Research24 with an incident neutron wavelength (λ) of 4.8 Å, at 30 °C. Data were collected continuously for up to 64 h and the results time-averaged over 33 min intervals. The QENS model used here has been described elsewhere.13-20 The scattering function, S(Q,ω), (where Q ) (4π/λ) sin(θ/2), θ is the scattering angle, the energy transfer ) p ω, p ) h/2π, and h is Planck’s constant) is dominated by the incoherent scattering from hydrogen. Data were summed from several detectors covering the Q -range (2.0-2.3 Å-1) and modeled using one Gaussian and three Lorentzian functions and attributed to various populations of hydrogen
S(Q,ω) )
B
√2π(WB/2.355)2
( (
exp -
1 x - x0 2 WB ⁄ 2.355
)) 2
F2
2π (x - x ) + W /2 2 0 ( F2 ) 2
IH )
B B + C + F1 + F2
(3)
The time evolution of the IH follows the evolution of heat (measured using calorimetry), when they are compared as fractions of the same final value.21 The constrained water component in the QENS model is associated with a C-S-H type in which the water is less tightly held than for other C-S-H types and does not scale with the evolution of heat.21 Using QENS, the time-evolution of the fraction of water not considered as part of the bulk water phase, known as the bound water index (BWI) and including both the constrained hydrogen and IH components, can be used to follow the hydration kinetics
BWI )
B+C B + C + F1 + F2
(4)
2. Modeling the Kinetics. The kinetic processes of a hydrating cementitious component are evident in the features of the time-dependent evolution of the BWI or the IH. QENS data measure scattering from hydrogen, so unlike calorimetry which measures the rate of heat evolution, the BWI and the IH represent evolved product. While the differential form of the nucleation and growth models are more sensitive, the smoothing of the QENS data required for its transformation to this form may introduce errors and was therefore not done. The nucleation and growth processes, occurring from time t ) ti to the start of the diffusion-limited period, at t ) td, observed in the timeevolution of such hydrogen fractions determined using QENS, have been fitted using a model derived from the original work of Avrami2,6,14-20,22 as per previous work14-20,22
+ Y(t) ) Y(ti) + A[1 - exp[-[kav(t - ti)]n]]
WF1 WP F1 C + + 2π (x - x )2 + (W /2)2 2π (x - x )2 + (W /2)2 0 C 0 F1 WF2
width at half-maximum of this function (WB) is fixed to that of the instrument resolution in this configuration, 0.146 meV. The integrated area B of the Gaussian function represents the number density of immobile hydrogen, as found in the Ca(OH)2 and some of the C-S-H. The remainder of the hydrogen is considered mobile and results in quasielastic scattering, modeled using the three Lorentzian functions (last three terms on the right in eq 2). The integrated area C represents the number density of “constrained” hydrogen associated with a C-S-H phase. The width of this function (WC) is determined from the final seven time-averaged spectra and fixed throughout the fitting procedure. Two wider Lorentzian functions of integrated areas F1 and F2 are used to describe the quasielastically broadened components arising from mobile hydrogen in the “free” water. The width of these profiles, WF1 and WF2 are determined from the first seven time-averaged spectra and fixed throughout the fitting procedure. Hence, the five free variables used in the fitting procedure were B, C, F1, F2, and x0. The fraction of hydrogen considered immobile on the dynamic time scale accessible by QENS (IH), can be expressed as
(2)
The energy transfer is x, and x0 is the energy center of each function. The Gaussian function (first term on the right in eq 2) is used to model contributions to from hydrogen atoms that are considered immobile on the time scale measurable. The full-
(5)
Y(t) and Y(ti) are the BWI or the IH at time t and ti, respectively; kav is the nucleation and growth reaction rate constant; A is the asymptotic volume fraction of either the BWI or IH at infinite time in the absence of a change to diffusionlimited hydration, and can be correlated with the amount of product that could ultimately form. The exponent n is traditionally ascribed to the dimensionality of product growth occurring in a three-dimensional pore space. It should be noted that this exponent is not related to the dimensionality of product growth for cementitious systems in any meaningful way.23 This exponent is well-characterized for the tricalcium silicate used
Tricalcium Silicate Hydration: Boundary Nucleation
J. Phys. Chem. C, Vol. 113, No. 6, 2009 2349
here and is fixed to a value of 2.27. The parameters that vary in the model as used here were A, kav, and ti. The BN model is implemented here as derived from Cahn25
Y(t) ) Y(ti) + A[1 - exp[-bS-1/3fS(aS)]]
(6)
where
bS ) IS/8S3G,
as ) (ISG2)1/3(t - ti)
(7)
and
fS(aS) ) aS
∫01 1 - exp(-πaS3[(1 - x3) ⁄ 3 - x2(1 - x)]) dx (8)
A is the normalization constant and is comparable with the A from the Avrami-derived model, IS is the specific nucleation rate, S is the boundary area per unit volume (S ) 0.805 × 10-6 m-1 for the sample used here as calculated from the volume of hydration product after full hydration, derived from the reaction stoichiometry to be 0.662 cm3 g-1,23 and the specific surface area of the tricalcium silicate, 0.533m2 g-1); G is the linear growth rate. The parameters varied in the model were A, IS, G, and ti. From these parameters other parameters are derived, including kB, the nucleated grain boundary area transformation rate, where kB ) (ISS)1/4G3/4, kG, the non-nucleated grain boundary transformation rate, where kG ) SG.23 The unitless ratio kB/kG is important as it determines the type of kinetic behavior for the process. When kB/kG is large, the boundary region will be densely populated and transform quickly. If the ratio is small, then internal boundaries will be sparsely populated and the entire system will transform at the same rate. It is under this latter condition that spacially random nucleation occurs, as required by the Avrami-derived model.23 By assuming spatially random nucleation, a rate constant for nucleation processes comparable to that derived from Avrami-derived models can be obtained: kav ) (π/3)1/4 kB. At times greater than td, diffusion-limited processes occur. The time evolution of the BWI or the IH are described at times greater than t ) td through combining either the Avrami-derived or BN models with a diffusion-limited hydration model derived from Fuji and Kondo26 and commonly used to describe QENS data for hydrating cementitious materials7,14-20,22
Y(t(t>td)) ) 1 - {[1 - Y(td)]1/3 - R-1(2Di)1/2(t - td)1/2}3 (9) Y(t) and Y(td) is the BWI or the IH at time t and td, respectively. Di is an effective diffusion coefficient that is a variable and represents the average ion-mobility for the sample. Di is not a direct measure of the diffusion-limited reaction rate, although the term 2Di〈R-1〉2 is an effective diffusion-limited reaction rate constant22 where R is the initial mean radius of the hydrating tricalcium silicate grains (R ) 4.9 µm for the tricalcium silicate sample used here). We include the diffusionlimited model in this work to assess the effect of using data that extends into the diffusion-limited region on the determination of the parameters describing the nucleation and growth process using either the BN or Avrami-derived models. Although the nucleation and growth and diffusion-limited hydration processes occur independently, there is a time period where the rate is dependent on both mechanisms; the time at which the hydration becomes dominated by diffusion-limited kinetics occurs at t ) td. The models were fitted to the data using a simulated annealing method, followed by a least-squares minimization procedure to ensure a converged solution and allow the extraction of estimated standard deviations (esds).
Figure 1. Time-evolution of the bound water index (BWI), used as a measure of the degree of hydration, and the immobile hydrogen (IH), which scales with calorimetry data, for hydrating tricalcium silicate measured using QENS. Two models to describe nucleation and growth are fitted to the data, an Avrami-derived model (red) and a boundary nucleation model (blue). Both models are combined with a model derived from Kondo and Fujii for the diffusion-limited hydration.
Results and Discussion The data and model outputs are shown in Figure 1, and the model parameters are shown in Table 1. We find that in general, the time evolution of the BWI and IH are described equally well by the Avrami-derived model as they are by the BN model. The fits of the models to the data is not as good for the IH, where the data are less scattered due to exclusion of the “constrained hydrogen” component of the QENS data (Figure 1). The time-resolution of the QENS data is not as good as that of calorimetry data, as it is dependent on the number of scattering events which are relatively few compared to the evolved heat. The relatively small number of data points between ti and td and the correlation between parameters in the BN model result in relatively large estimated standard deviations (esds) derived from the least-squares covariance-matrix. The effect of the small number of data on the precision of the determination of G and Is is effectively demonstrated by the significant decrease in the esds for these parameters when the number of data points used in the fitting of the BN or Avrami-derived model is increased through fitting without the diffusion-limited portion of the model to data collected up to 16 h. The parameter A obtained from these models can be directly compared and is in agreement within one esd between the two models. Similarly, combination with the diffusion-limited model produces values for Di that are in agreement. When these values are determined from the IH, parameter A is in agreement within two esds while Di remains in agreement below one esd. The value for the parameter A is significantly less when determined from the IH than determined using the BWI. Similarly, the value of Di from the IH is smaller than that from the BWI. These results are expected given that the constrained water component is absent from the IH. The values of td from both the Avrami-derived and BN models are approximately 9 h, as determined from both the BWI and the IH. The BWI includes the constrained hydrogen that is associated with some C-S-H, not measured using calorimetry, and is a sensitive probe of the start of hydration. The ti determined using the BN model (from both the BWI and the IH) is significantly smaller that determined using the Avramiderived model and close to the time of mixing, strongly
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TABLE 1: Avrami-Derived and Boundary Nucleation (BN) Model Parameters, and Fujii and Kondo-Derived Model Parameters, for Hydrating Tricalcium Silicate As Determined Using the Bound Water Index (BWI) and Immobile Hydrogen (IH) Fractions from QENS Avrami BWI BN BWI Avrami BWIc BN BWIc Avrami BWIb BN BWIb Avrami IH BN IH a
χ2
Aa
ti, h
td, h
Di × 10-16, m2 h-1
0.367 0.373 0.443 0.372 0.412 0.263 0.449 0.457
0.187(4) 0.20(2) 0.261(2) 0.22(2) 0.206(2) 0.227(4) 0.128(2) 0.15(1)
1.2(1) -0.3(3) 0c 0c 0.8(1) -0.3(2) 1.13(7) -0.3(2)
9.11(4) 9.5(2) 8.7(2) 9.6(2)
4.8(1) 4.6(2) 4.9(2) 4.6(2)
Units reflect those for BWI or the IH. 16 h. c ti fixed to zero.
IS, µm2 h-1
G, µm h-1
0.2(2)
0.15(4)
kav, h-1
kB, h-1
kG, h-1
kB/kG
0.153(8)
0.12(3)
1.2(3)
0.163(2)
0.10(2)
1.6(3)
0.150(7)
0.096(6)
1.6(1)
0.145(5)
0.09(2)
1.6(4)
0.179(8) 0.114(6) 0.4(3)
0.13(3) 0.154(4)
0.4(1) 9.3(1) 9.4(1) b
1.92(4) 1.90(4)
0.119(7) 0.170(4)
0.4(3)
0.11(3)
Diffusion-limited (DL) portion of the data and model are excluded, determination using data to
supporting the theory that the induction period is not a separate chemical process to the acceleratory nucleation and growth period. Constraint of the value of ti to zero in the BN model (table 1) results in no change to the fit of the model or to the parameters determined. The effect of this same constraint on the fit of the Avrami-derived model is marked. The Di value obtained from the diffusion-limited model, as combined with the Avrami-derived model obtained from the BWI, is in agreement with that obtained by others22 for a similar tricalcium silicate sample (with the same mean average particle size, water to cement ratio, and temperature) using a similar QENS method. As expected, Di determined from the IH is significantly smaller than for that determined using the BWI, due to the exclusion of the more-mobile hydrogen (the constrained-hydrogen) component. The effect of inclusion of the diffusion-limited kinetics to the determination of the kinetic parameters for the nucleation and growth period are observed using both nucleation and growth models through a comparison of the results using all data (up to 68 h), where td was allowed to vary freely through its combination with the diffusion-limited model, with those using data up to 16 h and without the diffusion-limited model. The determination of td through combination with a second model produces different results for the Avrami-derived model than when the Avrami-derived model is used alone. The parameter A as determined using the Avramiderived model is decreased by approximately 9% and kav increased by approximately 19% when the diffusion-limited region is modeled. The parameter ti obtained from the Avramiderived model increases marginally when the diffusion-limited model is included. All parameters determined using the BN model are unchanged within 1 esd when combined with the diffusion-limited model, suggesting that the BN model is less sensitive to the data region being modeled than the Avramiderived model. The values of the parameter A obtained in the calorimetry study differ between the Avrami-derived and BN models,23 while those reported here are in agreement. The differences between the kinetic parameters determined using the Avramiderived and BN models are assumed to arise from deviations from the condition of random volume nucleation that is assumed in the Avrami-derived model and achieved at small kB/kG ratios in the BN model (such as found at lower temperatures). In the calorimetry results,23 the parameter A found using the BN model is approximately 15% higher than that found using the Avramiderived model, at all temperatures, even at 10 °C where the kB/kG ratio is smallest. The exclusion of the diffusion-limited portion of the model and differences in the data time length may be contributing to these disagreements, as may the manual optimization of the parameters performed in the calorimetry
study. The kB/kG ratio is an important parameter obtained from the BN model as it is an indicator of the type of process that is occurring in the paste. The kB/kG obtained here agrees with that obtained at 30 °C from the calorimetry study.23 This is noteworthy as it indicates that the differences between the data and processing used here and in the calorimetry study that cause the parameter A to differ do not affect kB/kG. Conclusions In summary, the kinetics of tricalcium silicate hydration product nucleation and growth, as followed using QENS, are characterized using an alternative to the commonly used Avramiderived model, the boundary nucleation (BN) model, and combined with a second model to describe later, diffusionlimited, behavior. The BN model, applied previously to calorimetry data for hydrating tricalcium silicate, successfully describes the time-evolution of the components measured using QENS, including the immobile hydrogen (IH) component, expected to scale with calorimetry data, and the Bound Water Index (BWI), containing components not detected by calorimetry. The results support the theory that there is a single kinetic process responsible for nucleation and growth that begins at the time of mixing. Acknowledgment. We acknowledge the support of the National Institute of Standards and Technology, U.S. Department of Commerce, in providing the neutron research facilities used. References and Notes (1) Taylor, H. F. W. Cement Chemistry, 2nd ed.; Thomas Telford: London, 1997. (2) Berliner, R.; Popovici, M.; Herwig, K. W.; Berliner, M.; Jennings, H. M.; Thomas, J. J. Cem. Concr. Res. 1998, 28, 231–243. (3) Bordallo, H. N.; Aldridge, L. P.; Desmedt, A. J. Phys. Chem. B 2006, 110, 17966–17976. (4) Faraone, A.; Chen, S. H.; Fratini, E.; Baglioni, P.; Liu, L.; Brown, C. Phys. ReV. E 2002, 65, 040501-1–040501-4. (5) Faraone, A.; Fratini, E.; Baglioni, P.; Chen, S.-H. J. Chem. Phys. 2004, 121, 3212–3220. (6) FitzGerald, S. A.; Neumann, D. A.; Rush, J. J.; Bentz, D. P.; Livingston, R. A. Chem. Mat. 1998, 10, 397–402. (7) FitzGerald, S. A.; Thomas, J. J.; Neumann, D. A.; Livingston, R. A. Cem. Concr. Res. 2002, 32, 409–413. (8) Fratini, E.; Chen, S. H.; Baglioni, P. J. Phys. Chem. B 2003, 107, 10057–10062. (9) Fratini, E.; Chen, S.-H.; Baglioni, P.; Bellissent-Funel, M.-C. Phys. ReV. E 2001, 64, 020201-1–020201-4. (10) Fratini, E.; Chen, S.-H.; Baglioni, P.; Bellissent-Funel, M.-C. J. Phys. Chem. B 2002, 106, 158–166.
Tricalcium Silicate Hydration: Boundary Nucleation (11) Fratini, E.; Ridi, F.; Chen, S. H.; Baglioni, P. J. Phys.-Condes. Matter 2006, 18, S2467–S2483. (12) Harris, D. H. C.; Windsor, C. G.; Lawrence, C. D. Mag. Concr. Res. 1974, 26, 65–72. (13) Nemes, N. M.; Neumann, D. A.; Livingston, R. A. J. Mater. Res. 2006, 21, 2516–2523. (14) Peterson, V. K.; Brown, C. M.; Livingston, R. A. Chem. Phys. 2006, 326, 381–389. (15) Peterson, V. K.; Juenger, M. C. G. Physica B 2006, 385, 222–224. (16) Peterson, V. K.; Juenger, M. C. G. Chem. Mat. 2006, 18, 5798– 5804. (17) Peterson, V. K.; Neumann, D. A.; Livingston, R. A. J. Phys. Chem. B 2005, 109, 14449–14453. (18) Peterson, V. K.; Neumann, D. A.; Livingston, R. A. Mater. Res. Soc. Symp. 2005, 840, Q2.2.1-2.2.6.
J. Phys. Chem. C, Vol. 113, No. 6, 2009 2351 (19) Peterson, V. K.; Neumann, D. A.; Livingston, R. A. Physica B 2006, 385, 481–486. (20) Peterson, V. K.; Neumann, D. A.; Livingston, R. A. Chem. Phys. Lett. 2006, 419, 16–20. (21) Thomas, J. J.; FitzGerald, S. A.; Neumann, D. A.; Livingston, R. A. J. Am. Ceram. Soc. 2001, 84, 1811–1816. (22) Allen, A. J.; McLaughlin, J. C.; Neumann, D. A.; Livingston, R. A. J. Mater. Res. 2004, 19, 3242–3254. (23) Thomas, J. J. J. Am. Ceram. Soc. 2007, 90, 3282–3288. (24) Copley, J. R. D.; Udovic, T. J. J. Res. Natl. Inst. Stan. 1993, 98, 71–87. (25) Cahn, J. W. Acta Metall. 1956, 4, 449–459. (26) Fujii, K.; Kondo, W. J. Am. Ceram. Soc. 1974, 57, 492–497.
JP807209W
