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Non-linear modeling of yield stress increase due to SCC structural build-up at rest
Cement and Concrete Research 92 (2017) 92–97
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Cement and Concrete Research journal homepage: www.elsevier.com/locate/cemconres
Non-linear modeling of yield stress increase due to SCC structural build-up at rest Thibaut Lecompte, Arnaud Perrot ⁎ Univ. Bretagne Sud, FRE CNRS 3744, IRDL, F-56100 Lorient, France
a r t i c l e
i n f o
Article history: Received 9 May 2016 Received in revised form 11 October 2016 Accepted 29 November 2016 Available online 3 December 2016 Keywords: Yield stress Structural build-up Thixotropy
a b s t r a c t It is commonly considered that the yield stress of SCC increases linearly at rest. This increase is due to the SCC structural build-up. This behavior can be explained by the cement particle nucleation at the contact points. During the first dozens of minutes, the cement particle nucleation induces a linear increase of the yield stress in time. However, recent studies have shown that the trend of the yield stress increase becomes non-linear when considering a longer time (before the setting time). This study aims to provide a theoretical frame to describe this nonlinear increase. The proposed model tends toward the linear model during the first dozens of minutes and follows the observed exponential law for a longer resting time. The discrepancy with the linear model is explained by two phenomena: an increase in the solid volume fraction and a decrease in the packing fraction due to cement particle nucleation. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The rheology of SCC evolves in time due to different physical phenomena acting on a different timescale. This evolution is induced by both reversible structural build-up (also called thixotropy) and irreversible hydration of the cement grain that is responsible for the loss of concrete workability. Among the rheological properties of cement pastes, yield stress is involved in numerous forming processes or concrete characteristics such as aggregate stability [1], the pressure exerted on formwork by SCC, multi-layer casting [2–5], or 3D printing [6]. It is also directly linked to the slump or spread flow values and then to the consistency of the concrete [7]. It is therefore important to understand and predict the evolution of yield stress over time, especially when the concrete is left at rest. The yield stress of cement-based materials increases over time at rest [8–12]; this phenomenon is due to the nucleation of cement grains at their contact point by CSH formation during the dormant period before the setting time [12]. This yield stress increase is commonly modeled using a linear relationship with the resting time during the first hour of rest [9,10]. Recently, Perrot et al. have proposed an exponential relationship that describes the yield stress increase up to the setting time [5]. This function describes a smooth transition from the initial linear increase to the exponential evolution and asymptotically tends to the Roussel model for time of rest of a few dozens of minutes. In this model, Perrot
⁎ Corresponding author. E-mail address: [email protected] (A. Perrot).
http://dx.doi.org/10.1016/j.cemconres.2016.11.020 0008-8846/© 2016 Elsevier Ltd. All rights reserved.
et al. have used a critical time that is not based on physical arguments but chosen to obtain the best fit with experimental values. This exponential trend, observed in several studies [5,6,11,13,14], has been explained by the increase of the solid volume content within the cement paste due to the creation of the CSH gel [14]. This paper aims to provide a physically based model that is able to predict the evolution of yield stress with time of rest. In the first part, we present a state of the art of the modeling of yield stress increase with time of rest. Then, we propose a new model that uses both the structural build-up rate defined by Roussel and the hydration degree. Finally, our model is compared with experimental values and other models provided in the literature. 2. State of the art regarding structural build-up modeling The increase in yield stress is commonly considered to be linear during the dormant period prior to setting. Roussel [9,10] has defined the structuration rate Athix as the constant rate of increase in yield stress τ0 over the time at rest trest: τ0 ðt rest Þ ¼ τ0;0 þ Athix t rest
ð1Þ
where τ0,0 is the yield stress of the material with no time at rest. The associated dimensionless yield stress is: τ0 ðt rest Þ ¼
τ 0 ðt rest Þ A ¼ 1 þ thix t rest τ0;0 τ0;0
ð2Þ
The concrete structural build-up is due to complex and coupled phenomena: flocculation due to colloidal interactions and CSH nucleation at
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Nomenclature Greek symbols α hydration degree (−) ϕ solid volume fraction (−) ϕm maximum solid volume fraction (−) percolation solid volume fraction (−) ϕperc τ0 yield stress of the mortar (Pa) dimensionless yield stress of the mortar (−) τ0⁎ τ0,0 initial yield stress of the mortar (no resting time) (Pa) Latin symbols a* radius of curvature of cement particles (m) A0 non-retarded Hamaker constant (J) Athix rate of increase of yield stress with time at rest (Pa·s−1) d cement particles average diameter (m) d0 initial average diameter of the cement particles (m) H surface-to-surface separation distance between cement particles (m) kn, n Constants of the Avrami reaction modeling for hydration degree (−) p Hydrates solid volume fraction around the cement particles (−) r cement particles average radius (m) r0 initial average radius of the cement particles (m) trest time at rest (s) t age of the cement paste (s) tc critical time used in the exponential model (s)V0: Initial volume of anhydrous cement (m3) Vanh Volume of anhydrous cement particle (m3) νc specific volume of cement in hydration products (m3/ kg) Vc Volume of cement in the hydrates (m3) Vh Volume of hydrates (m3) νn specific volume of water in hydration products (m3/kg) Vw Volume of linked water in the hydrates (m3) c mass of reacted cement (kg) wn mass of reacted water (kg)
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represents a quasi-linear evolution during the first hour. This function proposed by Perrot [5] describes a smooth transition from the initial linear increase to the exponential evolution and asymptotically tends to the Roussel model as trest tends to zero: τ0 ðt rest Þ ¼
Athix t rest =t c tc e −1 þ 1 τ 0;0
where tc is a characteristic time, the value of which is adjusted to obtain the best fit with experimental values. 3. Theoretical framework of the proposed model The yield stress of unstructured cement suspensions (with no resting time) can be modeled using the yield stress model “YODEL” defined by Flatt and Bowen [16–18]. This model defines yield stress as a function of both the amplitude of Van Der Waals interaction between two cement particles at the contact point and the solid volume fraction of the paste. This model writes: 2 A0 a ϕ ϕ−ϕperc τ0;0 ≅m 2 2 d H ϕm ðϕm −ϕÞ
ð4Þ
where m is a pre-factor that depends on the particle size distribution, a* is the radius of curvature of the “contact” points, H is the surface-tosurface separation distance at “contact” points, A0 is the non- retarded Hamaker constant, ϕ is the grain solid volume fraction, ϕm is the maximum solid volume fraction, ϕperc is the percolation volume fraction, and
d is the average diameter of the spherical particles. In Eq. (4), m A20 a 2 repd H
resents the magnitude of interaction forces between two particles and the other part
φ2 ðφ−φperc Þ φm ðφm −φÞ
is linked to the spatial congestion within the
particle network. The basic idea of our model is to add the effect of the reversible nucleation of cement grains using the Roussel model [9,10] with the effect of the irreversible solid volume increase due to cement hydration using Eq. (4) with the increasing solid volume fraction (linked to the hydration degree α). In this case, Eq. (1) can be rewritten as follows: τ0 ðt; t rest Þ ¼ τ0;0 ðt Þ þ Athix t rest
the contact points between cement grains [12]. These two factors are also identified by Wallevik [15] as being responsible for reversible and permanent bridges between cement grains. According to Roussel et al. [12], the flocculation process lasts only a few dozens of seconds. After flocculation, at a timescale of several dozens of minutes, the structural build-up occurs due to the formation of CSH bridges between cement grains at the pseudo contact points. The authors assume that the rate of formation of CSH bridges is constant because the evolution rate of heat due to hydration is constant during the so-called “dormant” period. Therefore, they conclude that the increase of yield stress with time (or the elastic modulus) must be linear. This correlates well with the measurements of yield stress increase between 10 and 60 min after mixing. After this first period of linear increase, numerous experimental data have shown that the evolution of the yield stress of cement-based materials is highly non-linear between 1 h after placing and the setting time [5,6,11,14]. It would therefore appear that there is a nonnegligible increase of the solid volume fraction that leads to an exponential yield stress increase (up to 2% at 2 h for a W/C ratio equal to 0.4) [14]. It is worth noting that the Taylor series of the exponential function et at t = 0 is 1 + t relates to the linear form of yield stress time evolution assumed by Roussel [9,10]. Consequently, an exponential-based function introduced in Eq. (2), as a first-order Taylor series at trest = 0,
ð3Þ
ð5Þ
where t is different from trest and is defined as the paste age that starts with the contact between the cement particles and the water. It is important to keep in mind that at very early time (before 30 min), the yield stress increases only with resting time trest, while later τ0,0 increases with hydration time t, corresponding to the time of the first contact between cement particles and liquid water. The key to this model is the definition of the evolution of the solid volume fraction over the very early age of the paste. A literature survey shows that it is possible to link this solid volume fraction increase to the hydration degree α(t) as proposed by many researchers. As seen in Fig. 1, it is well-known that anhydrous cement creates hydrates in contact with water. The hydration kinetics can be expressed with a hydration, or maturation degree α(t): V anh ¼ ð1−α ðt ÞÞV 0
or
V c ¼ α ðt ÞV 0
ð6Þ
In a first approach, we assume that the hydration of the cement particles creates a network of entangled needles like hydrates of the solid volume fraction p that increases the spatial congestion within the cement particles assembly [19–21]. It increases the radius of the initial cement particles. Due to the fibrous structures of the hydrates, we assume that the volume increase of the particles is equal to the volume of created hydrates divided by their solid volume fraction p. This enables to compute the radius of the envelope volume of the formed hydrates. In this study, we assume that at this stage of hydration, the formation of
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Fig. 1. Solid volume growth of a grain during hardening. In the blue area surrounding the cement particles, a porous structure of needle-like structures is formed.
hydrates increases only the spatial congestion within the cement particle network. Our study focuses on the very early age when the hydration degree is lower than a few percent. Therefore, we think it is reasonable to assume that the hydrate volume increase can be modeled by a simple radius increase of the initial “spherical” particle. Several functions α(t) have been expressed and confronted to experiments in the literature [22–24]. Khawam and Flanagan [25] have reviewed the mathematical models for solid-state reaction kinetics. Taylor [26] has investigated the kinetics of C3S consumption at different time scales. At an early age, the kinetics corresponds to a germination process, divided into different periods: the initial reaction period, the induction period, and the acceleratory period [26]. This early-age behavior can be expressed by a sigmoid, corresponding to the Brown [22], or Avrami-Erofeyev reaction modeling [27,28]: α ðt Þ ¼ 1− expð−kn t n Þ
ð7Þ
where kn is a constant, and n can take values ranging from 1 to 3 depending on the studies. We compute the hydrate volume by using the terms of Powers & Brownyard and Brouwers [29,30]: V h ¼ vc c þ vn wn
ð8Þ
where vc and vn are the specific volumes of cement and water into the hydration product, c is the mass of reacted cement, and wn is the mass of reacted water. The ratio (wn/c) is constant for a given type of cement. For a CEMI cement type, Brouwers [30] provides the values of 0.32 cm3/ g for vc, 0.62 cm3/g for vn, and a value of 0.199 for (wn/c). The supplementary volume due to hydration corresponds to the volume of linked water in the hydrates produced: V w ¼ vn wn
ð9Þ
It is then possible to express the radius evolution of the hydrates envelope by taking into account the average hydrate volume fraction p: r ðt Þ ¼ r 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 vn wn 3 α ðt Þ 1þ p vc c
ð13Þ
The evolution of the radius with time enables to write the evolution of the cement particles diameter and the solid volume fraction of particles. dðt Þ ¼ d0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 vn wn 3 α ðt Þ 1þ p vc c
ϕðt Þ ¼ ϕ0 ð1 þ χα ðt ÞÞ with χ ¼
ð14Þ 1 vn wn p vc c
ð15Þ
Eqs. (14) and (15) can be used in Eq. (5) to obtain the evolution of yield stress over time and time of rest: 4=3 2 ϕ0 ð1 þ χα ðt ÞÞ−ϕperc A0 a ϕ0 ð1 þ χα ðt ÞÞ τ0 ¼ m 2 2 þ Athix t rest ð16Þ ϕm ðϕm −ϕ0 ð1 þ χα ðt ÞÞÞ d0 H with α(t) defined in Eq. (7) and depending on constant parameters kn and n. It is worth noting that the developed model assumes that ϕm, m, A0, a* and H remain constant and are not affected (or in a negligible way) by both the early hydration and the nucleation of the cement grains. In this study, we consider that this is acceptable for a cement paste age lower than the setting time. It is worth noting that when t tends to 0, Eq. (16) tends toward the Roussel linear model. For trest = 0, the model is able to predict the workability loss brought by cement hydration as the reversible bonds created by nucleation of cement grains are broken. However, our model works until the volume fraction of anhydrous cement particles plus needlelike hydrates reaches the maximum packing value ϕm. 4. Experimental validation
Furthermore, the reacted cement mass c can be deduced from Eq. (6): c ¼ V c =vc ¼ α ðt ÞV 0 =vc
ð10Þ
wn can then be calculated from Eq. (8), (9), and (10): wn ¼ α ðt Þ
vn wn vc c
ð11Þ
and the supplementary “solid” volume due to the reaction Vw writes: 3
V w ¼ αðtÞð4πr30 Þ vvnc ðwcn Þ (12)
The experimental validation was carried out on three cement pastes with different water-to-cement ratios (0.3, 0.32, and 0.35). The CEM I 52.5 cement and the polycarboxylate-based high range waterreducing admixture (HRWRA) were the same as those used in a previous study of Perrot et al. [18]. The HRWRA dosage was 0.4% of the cement mass. Quasi-adiabatic calorimetric tests were carried out on fresh samples 10 min after the first contact of water with cement on cylindrical samples of 8 cm in diameter and height. The temperature was recorded during 72 h. In order to compute the evolution of the heat of hydration and then of the hydration degree, the data was analyzed using the method proposed by [31]. Then, Eq. (7) is fitted on experimental curve of the increase of the hydration degree during the first 6 h of tests in order to
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obtain the kn and n values for the tested mortars. Experimental and modeled hydration degree curves are plotted on Fig. 2a to 2c. Table 1 shows the estimated values of kn and n. It shows that the hydration rate increases with the W/C ratio. This is in agreement with the literature [32,33]. The yield stress of the three different pastes was measured using a stress-growth procedure (low constant shear rate) as described in Mahaut et al. and Perrot et al. [18,34]. Yield stress measurements were carried out for different resting times ranging from 0 to 8000 s. In this period, the setting time has not been reached but the yield stress increase cannot be considered as linear. A rheometer Anton Paar Rheolab QC equipped with a vane geometry well adapted for cement paste and mortar was used. The vane geometry
Fig. 2. Comparison of measured hydration degree with modeled hydration degree computed with Eq. (7). a) W/C = 0.3, b) W/C = 0.32, c) W/C = 0.35.
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Table 1 Model parameters. W/C
0.3
0.32
0.35
Roussel linear model τ0 i n i (Pa) Athix (Pa/s)
50.2 0.016
30 0.0095
16.5 0.011
Perrot et al. Exponental model τ0 i n i (Pa) 50.2 0.012 Athix (Pa/s) tc (s) 3100
30 0.0052 2200
16.5 0.009 2900
Proposed model Փ0 kn n p τ0 i n i (Pa) Athix (Pa/s)
0.49 8.10E−11 2 0.01 28.9 0.0095
0.47 1.00E−10 2 0.01 16.7 0.011
0.51 7.00E−11 2 0.01 47.6 0.016
used in this study consisted of four blades around a cylindrical shaft. The vane height and diameter are respectively 60 mm and 40 mm. These dimensions allow for an accurate measurement of the yield stress from 10 to 700 Pa. After mixing, the material homogeneity was checked. Then, the material was slowly poured in fifteen different cylindrical containers of 10 cm in diameter and 10 cm in height. The containers walls were covered with sandpaper to avoid material slippage during tests. The first vane test is then performed, few seconds after the end of the pouring step and close to 1 min after the end of the mixing step. A measurement stage was performed during 180 s on the Anton Paar rheometer to obtain the yield stress for given resting times on undisturbed samples contained in the containers. Stress growth is used to determine the yield stress with an apparent shear rate of 0.005 s−1. At such a shear rate, viscosity effects are negligible. As a consequence, the yield stress is computed from the maximum torque value which is required for the onset of the flow, i.e. when the apparent yield stress (static yield stress) is reached on the cylindrical shearing surface. About every 10 min, an undisturbed sample was tested. Four replicate tests were also carried out for each cement paste in order to evaluate the dispersion of the results. Evolutions of yield stress with resting times are plotted in Fig. 3a to 3c for cement pastes of W/C ratio of 0.3, 0.32, and 0.35. The best fit of the Roussel linear model [9,10], of the exponential model proposed by Perrot et al. [5], and the proposed model are plotted in these figures. In this case, the time and time of rest start together a) W/C = 0.3, b) W/C = 0.32, c) W/C = 0.35. Perrot et al. [18] provide the values of parameters A0 = 1.6 × 10−20 J, H = 3.5 × 10−9 m, d0 = 10 × 10−6 m, a⁎ = 300 × 10−9 m, ϕm = 0.6 and ϕperc = 0.415 for the studied cement pastes with an admixture dosage of 2% [18]. For a CEMI cement type, Brouwers [30] provides the values of 0.32 cm3/g for vc, 0.62 cm3/g for vn, and a value of 0.199 for the ratio wn/c. The estimated values of hydration heat evolution measured using the calorimetric measurements are used in the curve fitting procedure. All others parameters are summarized in Table 1. We can see in Fig. 3 that the Roussel linear model provides an accurate description of the increase in yield stress in a resting time in the range of 0–3000 s. After that period, the linear model rapidly diverges from the experimental measurements. The Perrot et al. exponential model closely fits the experimental data for all tested W/C in the range of plotted resting time. This model provides a good description of the yield stress evolution and is easy to use. However, fitting parameters are not linked to the microstructure and physical properties of the hydrating cement pastes. The proposed model aims to link the very early age hydration kinetics and cement paste microstructure to the evolution of the yield stress.
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It is important to note that during this study, the hydration degree remains under 1%. Therefore, the volume increase is small, and the assumption of the sphericity of the particle appears valid. It has been recently shown that after this period (more than 2 h), the mechanical behavior of the cement-based material becomes frictional and brittle [35]. The material solid volume fraction tends toward the packing fraction and our approach is no longer valid. 5. Conclusions In this study, we propose a model that predicts the evolution of the yield stress of cement-based materials in terms of both the time of rest and the age of the paste. This model takes into account the nucleation rate of cement grains using the Athix parameter and the growth of the cement grains due to the beginning of the hydration reaction using the hydration degree α. Our proposed model provides a good description of the yield stress increase toward the material setting as shown by the performed experimental validation. This physically based model provides an improvement of the Roussel linear model as it can be used for a longer time and to predict the workability loss and structural build-up of concrete and cementbased materials. References
Fig. 3. Comparison of models with experimental measurements of yield stress evolution with time.
The parameter values of the hydration and coagulation kinetics and the hydrate volume fraction within the spherical envelope (i.e. Athix and p) are adjusted to obtain the best fit between our model and experimental measurements. The solid volume fraction of the hydrate envelope p is equal to 0.01. The Athix parameter is the same as the one obtained for the Roussel model [9,10]. In addition, the kn and n parameters are taken from data analysis of calorimetric tests (Fig. 2a to 2c). This model seems to be able to predict the evolution of the yield stress of the concrete before the setting time. This model extends the Roussel linear model for a longer time and is able to take into account both the effect of rest and the effect of hydration. Fig. 2 shows that when the hydration degree tends toward 0 (i.e. for t b 3000 s), the proposed model is equivalent to the Roussel linear model. After this period, the predicted increase rate of the yield stress rises following the experimental trend.
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[28] Avrami M (1939) Kinetics of phase change. I general theory. J. Chem. Phys. 7: 1103–1112. doi: 10.1063/1.1750380. [29] H.J.H. Brouwers, The work of powers and Brownyard revisited: part 2, Cem. Concr. Res. 35 (2005) 1922–1936, http://dx.doi.org/10.1016/j.cemconres.2005.04.009. [30] H.J.H. Brouwers, The work of powers and Brownyard revisited: part 1, H F W Taylor Commem Issue 34 (2004) 1697–1716, http://dx.doi.org/10.1016/j.cemconres.2004. 05.031. [31] C. Boulay, J.M. Torrenti, J.L. André, R. Saintilan, Calorimétrie quasi adiabatique pour bétons: facteurs d'influence, Bull Lab Ponts Chaussées 278 (2010) 18–36. [32] P. Mounanga, A. Khelidj, A. Loukili, V. Baroghel-Bouny, Predicting Ca(OH)2 content and chemical shrinkage of hydrating cement pastes using analytical approach, Cem. Concr. Res. 34 (2004) 255–265, http://dx.doi.org/10.1016/j.cemconres.2003. 07.006. [33] R. Berliner, M. Popovici, K.W. Herwig, et al., Quasielastic neutron scattering study of the effect of water-to-cement ratio on the hydration kinetics of tricalcium silicate, Cem. Concr. Res. 28 (1998) 231–243, http://dx.doi.org/10. 1016/S0008-8846(97)00260-3. [34] F. Mahaut, S. Mokéddem, X. Chateau, et al., Effect of coarse particle volume fraction on the yield stress and thixotropy of cementitious materials, Cem. Concr. Res. 38 (2008) 1276–1285, http://dx.doi.org/10.1016/j.cemconres.2008.06.001. [35] L.K. Mettler, F.K. Wittel, R.J. Flatt, H.J. Herrmann, Evolution of strength and failure of SCC during early hydration, Cem. Concr. Res. 89 (2016) 288–296, http://dx.doi.org/ 10.1016/j.cemconres.2016.09.004.
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Non-linear modeling of yield stress increase due to SCC structural build-up at rest Thibaut Lecompte, Arnaud Perrot ⁎ Univ. Bretagne Sud, FRE CNRS 3744, IRDL, F-56100 Lorient, France
a r t i c l e
i n f o
Article history: Received 9 May 2016 Received in revised form 11 October 2016 Accepted 29 November 2016 Available online 3 December 2016 Keywords: Yield stress Structural build-up Thixotropy
a b s t r a c t It is commonly considered that the yield stress of SCC increases linearly at rest. This increase is due to the SCC structural build-up. This behavior can be explained by the cement particle nucleation at the contact points. During the first dozens of minutes, the cement particle nucleation induces a linear increase of the yield stress in time. However, recent studies have shown that the trend of the yield stress increase becomes non-linear when considering a longer time (before the setting time). This study aims to provide a theoretical frame to describe this nonlinear increase. The proposed model tends toward the linear model during the first dozens of minutes and follows the observed exponential law for a longer resting time. The discrepancy with the linear model is explained by two phenomena: an increase in the solid volume fraction and a decrease in the packing fraction due to cement particle nucleation. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction The rheology of SCC evolves in time due to different physical phenomena acting on a different timescale. This evolution is induced by both reversible structural build-up (also called thixotropy) and irreversible hydration of the cement grain that is responsible for the loss of concrete workability. Among the rheological properties of cement pastes, yield stress is involved in numerous forming processes or concrete characteristics such as aggregate stability [1], the pressure exerted on formwork by SCC, multi-layer casting [2–5], or 3D printing [6]. It is also directly linked to the slump or spread flow values and then to the consistency of the concrete [7]. It is therefore important to understand and predict the evolution of yield stress over time, especially when the concrete is left at rest. The yield stress of cement-based materials increases over time at rest [8–12]; this phenomenon is due to the nucleation of cement grains at their contact point by CSH formation during the dormant period before the setting time [12]. This yield stress increase is commonly modeled using a linear relationship with the resting time during the first hour of rest [9,10]. Recently, Perrot et al. have proposed an exponential relationship that describes the yield stress increase up to the setting time [5]. This function describes a smooth transition from the initial linear increase to the exponential evolution and asymptotically tends to the Roussel model for time of rest of a few dozens of minutes. In this model, Perrot
⁎ Corresponding author. E-mail address: [email protected] (A. Perrot).
http://dx.doi.org/10.1016/j.cemconres.2016.11.020 0008-8846/© 2016 Elsevier Ltd. All rights reserved.
et al. have used a critical time that is not based on physical arguments but chosen to obtain the best fit with experimental values. This exponential trend, observed in several studies [5,6,11,13,14], has been explained by the increase of the solid volume content within the cement paste due to the creation of the CSH gel [14]. This paper aims to provide a physically based model that is able to predict the evolution of yield stress with time of rest. In the first part, we present a state of the art of the modeling of yield stress increase with time of rest. Then, we propose a new model that uses both the structural build-up rate defined by Roussel and the hydration degree. Finally, our model is compared with experimental values and other models provided in the literature. 2. State of the art regarding structural build-up modeling The increase in yield stress is commonly considered to be linear during the dormant period prior to setting. Roussel [9,10] has defined the structuration rate Athix as the constant rate of increase in yield stress τ0 over the time at rest trest: τ0 ðt rest Þ ¼ τ0;0 þ Athix t rest
ð1Þ
where τ0,0 is the yield stress of the material with no time at rest. The associated dimensionless yield stress is: τ0 ðt rest Þ ¼
τ 0 ðt rest Þ A ¼ 1 þ thix t rest τ0;0 τ0;0
ð2Þ
The concrete structural build-up is due to complex and coupled phenomena: flocculation due to colloidal interactions and CSH nucleation at
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Nomenclature Greek symbols α hydration degree (−) ϕ solid volume fraction (−) ϕm maximum solid volume fraction (−) percolation solid volume fraction (−) ϕperc τ0 yield stress of the mortar (Pa) dimensionless yield stress of the mortar (−) τ0⁎ τ0,0 initial yield stress of the mortar (no resting time) (Pa) Latin symbols a* radius of curvature of cement particles (m) A0 non-retarded Hamaker constant (J) Athix rate of increase of yield stress with time at rest (Pa·s−1) d cement particles average diameter (m) d0 initial average diameter of the cement particles (m) H surface-to-surface separation distance between cement particles (m) kn, n Constants of the Avrami reaction modeling for hydration degree (−) p Hydrates solid volume fraction around the cement particles (−) r cement particles average radius (m) r0 initial average radius of the cement particles (m) trest time at rest (s) t age of the cement paste (s) tc critical time used in the exponential model (s)V0: Initial volume of anhydrous cement (m3) Vanh Volume of anhydrous cement particle (m3) νc specific volume of cement in hydration products (m3/ kg) Vc Volume of cement in the hydrates (m3) Vh Volume of hydrates (m3) νn specific volume of water in hydration products (m3/kg) Vw Volume of linked water in the hydrates (m3) c mass of reacted cement (kg) wn mass of reacted water (kg)
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represents a quasi-linear evolution during the first hour. This function proposed by Perrot [5] describes a smooth transition from the initial linear increase to the exponential evolution and asymptotically tends to the Roussel model as trest tends to zero: τ0 ðt rest Þ ¼
Athix t rest =t c tc e −1 þ 1 τ 0;0
where tc is a characteristic time, the value of which is adjusted to obtain the best fit with experimental values. 3. Theoretical framework of the proposed model The yield stress of unstructured cement suspensions (with no resting time) can be modeled using the yield stress model “YODEL” defined by Flatt and Bowen [16–18]. This model defines yield stress as a function of both the amplitude of Van Der Waals interaction between two cement particles at the contact point and the solid volume fraction of the paste. This model writes: 2 A0 a ϕ ϕ−ϕperc τ0;0 ≅m 2 2 d H ϕm ðϕm −ϕÞ
ð4Þ
where m is a pre-factor that depends on the particle size distribution, a* is the radius of curvature of the “contact” points, H is the surface-tosurface separation distance at “contact” points, A0 is the non- retarded Hamaker constant, ϕ is the grain solid volume fraction, ϕm is the maximum solid volume fraction, ϕperc is the percolation volume fraction, and
d is the average diameter of the spherical particles. In Eq. (4), m A20 a 2 repd H
resents the magnitude of interaction forces between two particles and the other part
φ2 ðφ−φperc Þ φm ðφm −φÞ
is linked to the spatial congestion within the
particle network. The basic idea of our model is to add the effect of the reversible nucleation of cement grains using the Roussel model [9,10] with the effect of the irreversible solid volume increase due to cement hydration using Eq. (4) with the increasing solid volume fraction (linked to the hydration degree α). In this case, Eq. (1) can be rewritten as follows: τ0 ðt; t rest Þ ¼ τ0;0 ðt Þ þ Athix t rest
the contact points between cement grains [12]. These two factors are also identified by Wallevik [15] as being responsible for reversible and permanent bridges between cement grains. According to Roussel et al. [12], the flocculation process lasts only a few dozens of seconds. After flocculation, at a timescale of several dozens of minutes, the structural build-up occurs due to the formation of CSH bridges between cement grains at the pseudo contact points. The authors assume that the rate of formation of CSH bridges is constant because the evolution rate of heat due to hydration is constant during the so-called “dormant” period. Therefore, they conclude that the increase of yield stress with time (or the elastic modulus) must be linear. This correlates well with the measurements of yield stress increase between 10 and 60 min after mixing. After this first period of linear increase, numerous experimental data have shown that the evolution of the yield stress of cement-based materials is highly non-linear between 1 h after placing and the setting time [5,6,11,14]. It would therefore appear that there is a nonnegligible increase of the solid volume fraction that leads to an exponential yield stress increase (up to 2% at 2 h for a W/C ratio equal to 0.4) [14]. It is worth noting that the Taylor series of the exponential function et at t = 0 is 1 + t relates to the linear form of yield stress time evolution assumed by Roussel [9,10]. Consequently, an exponential-based function introduced in Eq. (2), as a first-order Taylor series at trest = 0,
ð3Þ
ð5Þ
where t is different from trest and is defined as the paste age that starts with the contact between the cement particles and the water. It is important to keep in mind that at very early time (before 30 min), the yield stress increases only with resting time trest, while later τ0,0 increases with hydration time t, corresponding to the time of the first contact between cement particles and liquid water. The key to this model is the definition of the evolution of the solid volume fraction over the very early age of the paste. A literature survey shows that it is possible to link this solid volume fraction increase to the hydration degree α(t) as proposed by many researchers. As seen in Fig. 1, it is well-known that anhydrous cement creates hydrates in contact with water. The hydration kinetics can be expressed with a hydration, or maturation degree α(t): V anh ¼ ð1−α ðt ÞÞV 0
or
V c ¼ α ðt ÞV 0
ð6Þ
In a first approach, we assume that the hydration of the cement particles creates a network of entangled needles like hydrates of the solid volume fraction p that increases the spatial congestion within the cement particles assembly [19–21]. It increases the radius of the initial cement particles. Due to the fibrous structures of the hydrates, we assume that the volume increase of the particles is equal to the volume of created hydrates divided by their solid volume fraction p. This enables to compute the radius of the envelope volume of the formed hydrates. In this study, we assume that at this stage of hydration, the formation of
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Fig. 1. Solid volume growth of a grain during hardening. In the blue area surrounding the cement particles, a porous structure of needle-like structures is formed.
hydrates increases only the spatial congestion within the cement particle network. Our study focuses on the very early age when the hydration degree is lower than a few percent. Therefore, we think it is reasonable to assume that the hydrate volume increase can be modeled by a simple radius increase of the initial “spherical” particle. Several functions α(t) have been expressed and confronted to experiments in the literature [22–24]. Khawam and Flanagan [25] have reviewed the mathematical models for solid-state reaction kinetics. Taylor [26] has investigated the kinetics of C3S consumption at different time scales. At an early age, the kinetics corresponds to a germination process, divided into different periods: the initial reaction period, the induction period, and the acceleratory period [26]. This early-age behavior can be expressed by a sigmoid, corresponding to the Brown [22], or Avrami-Erofeyev reaction modeling [27,28]: α ðt Þ ¼ 1− expð−kn t n Þ
ð7Þ
where kn is a constant, and n can take values ranging from 1 to 3 depending on the studies. We compute the hydrate volume by using the terms of Powers & Brownyard and Brouwers [29,30]: V h ¼ vc c þ vn wn
ð8Þ
where vc and vn are the specific volumes of cement and water into the hydration product, c is the mass of reacted cement, and wn is the mass of reacted water. The ratio (wn/c) is constant for a given type of cement. For a CEMI cement type, Brouwers [30] provides the values of 0.32 cm3/ g for vc, 0.62 cm3/g for vn, and a value of 0.199 for (wn/c). The supplementary volume due to hydration corresponds to the volume of linked water in the hydrates produced: V w ¼ vn wn
ð9Þ
It is then possible to express the radius evolution of the hydrates envelope by taking into account the average hydrate volume fraction p: r ðt Þ ¼ r 0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 vn wn 3 α ðt Þ 1þ p vc c
ð13Þ
The evolution of the radius with time enables to write the evolution of the cement particles diameter and the solid volume fraction of particles. dðt Þ ¼ d0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 vn wn 3 α ðt Þ 1þ p vc c
ϕðt Þ ¼ ϕ0 ð1 þ χα ðt ÞÞ with χ ¼
ð14Þ 1 vn wn p vc c
ð15Þ
Eqs. (14) and (15) can be used in Eq. (5) to obtain the evolution of yield stress over time and time of rest: 4=3 2 ϕ0 ð1 þ χα ðt ÞÞ−ϕperc A0 a ϕ0 ð1 þ χα ðt ÞÞ τ0 ¼ m 2 2 þ Athix t rest ð16Þ ϕm ðϕm −ϕ0 ð1 þ χα ðt ÞÞÞ d0 H with α(t) defined in Eq. (7) and depending on constant parameters kn and n. It is worth noting that the developed model assumes that ϕm, m, A0, a* and H remain constant and are not affected (or in a negligible way) by both the early hydration and the nucleation of the cement grains. In this study, we consider that this is acceptable for a cement paste age lower than the setting time. It is worth noting that when t tends to 0, Eq. (16) tends toward the Roussel linear model. For trest = 0, the model is able to predict the workability loss brought by cement hydration as the reversible bonds created by nucleation of cement grains are broken. However, our model works until the volume fraction of anhydrous cement particles plus needlelike hydrates reaches the maximum packing value ϕm. 4. Experimental validation
Furthermore, the reacted cement mass c can be deduced from Eq. (6): c ¼ V c =vc ¼ α ðt ÞV 0 =vc
ð10Þ
wn can then be calculated from Eq. (8), (9), and (10): wn ¼ α ðt Þ
vn wn vc c
ð11Þ
and the supplementary “solid” volume due to the reaction Vw writes: 3
V w ¼ αðtÞð4πr30 Þ vvnc ðwcn Þ (12)
The experimental validation was carried out on three cement pastes with different water-to-cement ratios (0.3, 0.32, and 0.35). The CEM I 52.5 cement and the polycarboxylate-based high range waterreducing admixture (HRWRA) were the same as those used in a previous study of Perrot et al. [18]. The HRWRA dosage was 0.4% of the cement mass. Quasi-adiabatic calorimetric tests were carried out on fresh samples 10 min after the first contact of water with cement on cylindrical samples of 8 cm in diameter and height. The temperature was recorded during 72 h. In order to compute the evolution of the heat of hydration and then of the hydration degree, the data was analyzed using the method proposed by [31]. Then, Eq. (7) is fitted on experimental curve of the increase of the hydration degree during the first 6 h of tests in order to
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obtain the kn and n values for the tested mortars. Experimental and modeled hydration degree curves are plotted on Fig. 2a to 2c. Table 1 shows the estimated values of kn and n. It shows that the hydration rate increases with the W/C ratio. This is in agreement with the literature [32,33]. The yield stress of the three different pastes was measured using a stress-growth procedure (low constant shear rate) as described in Mahaut et al. and Perrot et al. [18,34]. Yield stress measurements were carried out for different resting times ranging from 0 to 8000 s. In this period, the setting time has not been reached but the yield stress increase cannot be considered as linear. A rheometer Anton Paar Rheolab QC equipped with a vane geometry well adapted for cement paste and mortar was used. The vane geometry
Fig. 2. Comparison of measured hydration degree with modeled hydration degree computed with Eq. (7). a) W/C = 0.3, b) W/C = 0.32, c) W/C = 0.35.
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Table 1 Model parameters. W/C
0.3
0.32
0.35
Roussel linear model τ0 i n i (Pa) Athix (Pa/s)
50.2 0.016
30 0.0095
16.5 0.011
Perrot et al. Exponental model τ0 i n i (Pa) 50.2 0.012 Athix (Pa/s) tc (s) 3100
30 0.0052 2200
16.5 0.009 2900
Proposed model Փ0 kn n p τ0 i n i (Pa) Athix (Pa/s)
0.49 8.10E−11 2 0.01 28.9 0.0095
0.47 1.00E−10 2 0.01 16.7 0.011
0.51 7.00E−11 2 0.01 47.6 0.016
used in this study consisted of four blades around a cylindrical shaft. The vane height and diameter are respectively 60 mm and 40 mm. These dimensions allow for an accurate measurement of the yield stress from 10 to 700 Pa. After mixing, the material homogeneity was checked. Then, the material was slowly poured in fifteen different cylindrical containers of 10 cm in diameter and 10 cm in height. The containers walls were covered with sandpaper to avoid material slippage during tests. The first vane test is then performed, few seconds after the end of the pouring step and close to 1 min after the end of the mixing step. A measurement stage was performed during 180 s on the Anton Paar rheometer to obtain the yield stress for given resting times on undisturbed samples contained in the containers. Stress growth is used to determine the yield stress with an apparent shear rate of 0.005 s−1. At such a shear rate, viscosity effects are negligible. As a consequence, the yield stress is computed from the maximum torque value which is required for the onset of the flow, i.e. when the apparent yield stress (static yield stress) is reached on the cylindrical shearing surface. About every 10 min, an undisturbed sample was tested. Four replicate tests were also carried out for each cement paste in order to evaluate the dispersion of the results. Evolutions of yield stress with resting times are plotted in Fig. 3a to 3c for cement pastes of W/C ratio of 0.3, 0.32, and 0.35. The best fit of the Roussel linear model [9,10], of the exponential model proposed by Perrot et al. [5], and the proposed model are plotted in these figures. In this case, the time and time of rest start together a) W/C = 0.3, b) W/C = 0.32, c) W/C = 0.35. Perrot et al. [18] provide the values of parameters A0 = 1.6 × 10−20 J, H = 3.5 × 10−9 m, d0 = 10 × 10−6 m, a⁎ = 300 × 10−9 m, ϕm = 0.6 and ϕperc = 0.415 for the studied cement pastes with an admixture dosage of 2% [18]. For a CEMI cement type, Brouwers [30] provides the values of 0.32 cm3/g for vc, 0.62 cm3/g for vn, and a value of 0.199 for the ratio wn/c. The estimated values of hydration heat evolution measured using the calorimetric measurements are used in the curve fitting procedure. All others parameters are summarized in Table 1. We can see in Fig. 3 that the Roussel linear model provides an accurate description of the increase in yield stress in a resting time in the range of 0–3000 s. After that period, the linear model rapidly diverges from the experimental measurements. The Perrot et al. exponential model closely fits the experimental data for all tested W/C in the range of plotted resting time. This model provides a good description of the yield stress evolution and is easy to use. However, fitting parameters are not linked to the microstructure and physical properties of the hydrating cement pastes. The proposed model aims to link the very early age hydration kinetics and cement paste microstructure to the evolution of the yield stress.
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It is important to note that during this study, the hydration degree remains under 1%. Therefore, the volume increase is small, and the assumption of the sphericity of the particle appears valid. It has been recently shown that after this period (more than 2 h), the mechanical behavior of the cement-based material becomes frictional and brittle [35]. The material solid volume fraction tends toward the packing fraction and our approach is no longer valid. 5. Conclusions In this study, we propose a model that predicts the evolution of the yield stress of cement-based materials in terms of both the time of rest and the age of the paste. This model takes into account the nucleation rate of cement grains using the Athix parameter and the growth of the cement grains due to the beginning of the hydration reaction using the hydration degree α. Our proposed model provides a good description of the yield stress increase toward the material setting as shown by the performed experimental validation. This physically based model provides an improvement of the Roussel linear model as it can be used for a longer time and to predict the workability loss and structural build-up of concrete and cementbased materials. References
Fig. 3. Comparison of models with experimental measurements of yield stress evolution with time.
The parameter values of the hydration and coagulation kinetics and the hydrate volume fraction within the spherical envelope (i.e. Athix and p) are adjusted to obtain the best fit between our model and experimental measurements. The solid volume fraction of the hydrate envelope p is equal to 0.01. The Athix parameter is the same as the one obtained for the Roussel model [9,10]. In addition, the kn and n parameters are taken from data analysis of calorimetric tests (Fig. 2a to 2c). This model seems to be able to predict the evolution of the yield stress of the concrete before the setting time. This model extends the Roussel linear model for a longer time and is able to take into account both the effect of rest and the effect of hydration. Fig. 2 shows that when the hydration degree tends toward 0 (i.e. for t b 3000 s), the proposed model is equivalent to the Roussel linear model. After this period, the predicted increase rate of the yield stress rises following the experimental trend.
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