Hydration modelling of an ettringite-based binder

Cement and Concrete Research 76 (2015) 51–61

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Hydration modelling of an ettringite-based binder J.F. Georgin ⁎, E. Prud'homme University of Lyon, INSA Lyon, LGCIE, F-69621 Villeurbanne, France

a r t i c l e

i n f o

Article history: Received 21 November 2014 Accepted 7 May 2015 Available online xxxx Keywords: Calcium aluminate cement Ettringite Modelling Particle size distribution Pore solution

a b s t r a c t Aluminous cement, when mixed with calcium sulfate and water, produces a binder called an ettringite binder. A model of the hydration of an ettringite binder that accounts for the reactive surface of the solid phases in a solute was proposed. Mechanisms of precipitation and dissolution were considered. Calibration was performed based on the kinetics of the conductivity of the solute. Experimental pH kinetics are in accordance with the model prediction. Based on this good agreement, the final part of this article discusses the model predictions of the hydration of an ettringite-based binder. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Compositions based on mixtures of calcium aluminate cement and calcium sulfate are used extensively in the field of building chemistry as repair mortars, tile adhesives, grouts and self-levelling floor screeds [1,2]. In these formulations, products dry and harden rapidly. The development of material microstructure is associated with many factors [3–7], such as temperature, humidity, cement content, various additives and the ratios of mixture compositions. Aluminous cement, when mixed with calcium sulfate and water, produces a binder called an ettringite binder because the main hydrate formed is ettringite (C6A$3H32). The formation of an ettringite binder is mainly based on the reaction between calcium aluminate cement (CAC) and calcium sulfate (C$), which leads to the formation of ettringite (C6A$3H32), aluminium hydroxide (AH3) and calcium monosulfoaluminate (C4A$H12). The formation of these hydrates depends on the mineralogy and crystallochemistry of the CAC, which varies greatly. Three different mineral phases are generally present in various amounts as a function of the following CAC components: calcium mono-aluminate (CA), calcium di-aluminate (CA2) and mayenite (C12A7). The hydration reactions are indicated by chemical Eqs. (1), (2) and (3). 3CA þ 3CDH x þ ð38−3xÞH→C 6 AD3 H 32 þ 2AH3

ð1Þ

3CA2 þ 3CDHx þ ð47‐3xÞH→C6 AD3 H32 þ 5AH3

ð2Þ

C12 A7 þ 12CDH x þ ð137‐12xÞH→4C6 AD3 H32 þ 3AH3

ð3Þ

⁎ Corresponding author. Tel.: + 33 472437137. E-mail address: [email protected] (J.F. Georgin).

http://dx.doi.org/10.1016/j.cemconres.2015.05.009 0008-8846/© 2015 Elsevier Ltd. All rights reserved.

When the calcium sulfate in the system is exhausted, ettringite dissolves and calcium monosulfoaluminate precipitates. In practice, mixture compositions can be more complex because they are often based on several other components, such as Portland cement, slag, pozzolan or calcite. Additionally, the aforementioned reaction scheme can be more complex as well, as shown in [8] and [9]. As example, when limestone is introduced into the mixture, carbonates react with aluminate phases promoting the formation of carboaluminate phases. Therefore the extra aluminates do not participate to the formation of monosulfoaluminate. Indeed, it was observed in [10] that calcium monosulfoaluminate does not form in cements containing limestone. Instead, monocarboaluminate (Mc) and hemicarboaluminate (Hc) phases form and more ettringite remains. The mechanical performance and durability of this type of material depend on the features of the matrix microstructure and its kinetic development at an early age. The different characteristics as solubility or chemical stability of the hydrates formed at an early age govern the sensitivity of the mineral matrix sustainability toward carbonation and cracking. Improving knowledge about the early-age behaviour of these cement-matrix-based materials is a major objective in the development of new binders. The binders can show significant volume variations at an early age because of the shrinkage or swelling imposed by the hydration conditions and the hydric and thermic environment. Many studies have already been performed on the subject by researchers examining the calcium aluminate based binders. Particularly in the case of sulphoaluminate cement [11,12], a modelling approach was proposed in the aforementioned studies to describe the chemohydric-mechanical behaviour of the cement-based material at an early age. In this study, the behaviour of CAC cement was investigated. A model of ettringite binder hydration in a solute that accounts for the reactive surface compound of both the anhydrous and hydrate phases

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J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61

Table 1 Chemical composition of the anhydrous materials. Raw materials

Main oxides/%wt Al2O3

CaO

SiO2

Fe2O3

MgO

TiO2

K2O

Na2O

SO3

MnO

CAC α-Plaster

69.68 –

29.78 38.70

0.26 0.27

0.16 0.03

0,15 0.1

0.04 0.003

– –

0.23 –

0.27 52.40

0.01 –

was also developed. The hydration process is based on the dissolution– precipitation mechanism that occurs in solution. The model was calibrated by experimental conductimetry data, and its efficiency was confirmed by comparing calculated and measured pH kinetics. The results improve the understanding of the hydration of an ettringite binder at an early age. 2. Anhydrous materials The anhydrous materials used in this study were a CAC and a plaster (α-plaster). The CAC was supplied by Kerneos Aluminates Technologies1 (France) and was mainly composed of mayenite (12CaO·7Al2O3, denoted as C12A7) and ferrite (Al2O3·Fe2O3·4CaO, denoted as C4AF) and small amounts of calcium aluminate (CaO·Al2O3, denoted as CA) and calcium di-aluminate (CaO·2Al2O3, denoted as CA2). The second anhydrous material was an α-plaster with the trade name Prestia supplied by Lafarge.2 The plaster contained a small amount of calcium carbonate (approximately 3%wt). The fact that this impurity is not taken into account in modelling of plaster is justified by the work of Martias et al. [13], which shows that a small amount of calcium carbonate has no effect on the hydration of the plaster. In the mixture with CAC, the presence of calcium carbonate can lead to Mc and Hc formation and to the stabilization of ettringite [10]. However, due to the CAC/α-plaster ratio, calcium carbonate represents only 0.75%wt, which is then negligible. The chemical composition in terms of oxide content for each anhydrous material is presented in Table 1. 3. Experimental techniques Particle size distribution curves and mean diameters were obtained by laser granulometry using a dry process (Mastersizer 2000, Malvern Ltd). The air pressure was 0.5 bar, and the vibration rate of the hopper used to introduce the powder was 20%. Particle diameters were determined to range from 0.3 to 300 μm. Conductivity and pH value measurements were performed using a suspension containing demineralized water and CAC and/or α-plaster on a SevenExcellence device (Mettler Toledo). The ratio between the demineralized water and solid (w/s) in solution was 20. This ratio allowed for the formation of a homogeneous suspension during the experiment without precipitation around the probe or on the wall of the beaker. Data acquisition began with the demineralized water (200 g). After the stabilization of the pH value and conductivity, the anhydrous material powder was added (10 g) using magnetic stirring. The pH value and conductivity were recorded every 5 s for 24 h using the software LabX direct pH (Mettler Toledo). The plaster was investigated first to account for its rate of dissolution and precipitation. A mixture composed of 75%wt CAC and 25%wt α-plaster, which led to the formation of ettringite, was then investigated. Measurements by in situ infrared spectroscopy were performed using a ThermoFisher Scientific IS50 device in the attenuated total reflectance (ATR) mode [14]. To study the structuralization of the cement paste, a mixture of powder and water with a w/s ratio of 0.3 was studied by simply placing the mixture into a cylinder which is fixed around the diamond crystal with a seal at the bottom (Fig. 1). To conduct representative measurements of the material's behaviour, the cylinder was filled 1 2

Kerneos Aluminate Technologies, 8 Rue des Graviers, 92521 Neuilly sur Seine, France. Lafarge, 61 rue des Belles Feuilles, 75116 Paris, France.

to the top (12 cm3). A cover was placed on the surface to minimise water evaporation. The material was then studied under endogenous conditions. Spectra were recorded every minute for 6 h. Measurements were performed from 4000 cm−1 to 450 cm−1 (number of scans: 32, resolution 4.0). The software Omnic Series (Nicolet instrument) was used to record and process the data. To eliminate the contribution of carbon dioxide to the spectra, the spectra were corrected with a straight line between 2400 and 2000 cm−1. The spectra were then corrected with an automatic baseline. The result of these tests was the temporal evolution of the spectra. This follow-up could show, for example, the formation of gypsum, ettringite and aluminium hydroxide due to the absorbance variation of the bands at 1110 cm−1 and 1020 cm−1 [15–18]. By monitoring the band absorbance over time, the competition between species formation and consumption in a paste could be observed. 4. Conductivity results To understand the basic hydration mechanisms of gypsum and the ettringite binder mixture (75%wt CAC and 25%wt α-plaster), the species' behaviour in solution was investigated by measuring the electrical conductivity and pH. Experiments were performed on the suspension of powder and water (w/s = 20) to monitor the dissolution/precipitation process. Although the dissolution kinetics of these anhydrous materials are documented in the literature [19], experiments were required to calibrate the model to the materials actually used. The kinetics are highly dependent on the crystal habit and lattice order, which vary with the origin of the anhydrous materials [20,21]. The results were collected over a period of 24 h after mixing with water but they are presented on shorter period (6–10 h) (Fig. 2). This short interval was chosen because of the rapid setting of the ettringite binder, which under normal conditions can occur in less than 1 h [22]. 4.1. α-Plaster The conversion of hemihydrate to dihydrate appears to be a simple process but has nevertheless fuelled much research. The hydration of hemihydrate occurs through a solution mechanism leading to the formation of calcium sulfate dihydrate [22–24]. The evolution of the conductivity and that of the pH value of the α-plaster suspension (w/s = 20) over time are reported in Fig. 2(A). First, as soon as the hemihydrate comes into contact with water, the hemihydrate partially dissolves, which makes the solution saturated with Ca2 + and SO24 − ions, and the suspension reaches its maximum conductivity (5940 μS.cm−1). This step is very quick and takes only 4 min. The conductivity then remains quite stable for 10 min, which corresponds to the induction period. According to Singh and Middendorf [22,13], during the inoccurs, leading duction period, clustering of hydrated Ca2+ and SO2− 4 to the formation of calcium sulfate dihydrate nuclei. Then, the nuclei grow, and the gypsum begins to crystallize. During this stage, the conductivity suddenly decreases after 14 min of reaction due to the germination and crystallization of CaSO·2H2O. As the dihydrate crystallizes in the solution, the conductivity decreases. Finally, the suspension conductivity stabilizes at a value of approximately 2090 μS.cm−1 after 80 min, which corresponds to the solubility of gypsum [25]. The variation of the pH value is similar, with a rapid increase followed by a decrease due to the dissolution of the plaster and its precipitation as gypsum. After 6 h, the pH value stabilized near a value of 7.9, which is similar to that of gypsum, which naturally occurs at a pH value level of 7. Despite the

J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61

Fig. 1. In situ infrared spectroscopy experimental set-up.

high w/s ratio, these results are consistent with observations (electrical resistivity [25] and strength development [26]) made for plaster with a w/s ratio of 0.6, which is commonly used in various applications. 4.2. Mixture of 75%wt of CAC and 25%wt of α-plaster Investigating the mixture of the two materials was complex because of the competition that can occur between the components. As previously described, the hydration of a mixture of calcium aluminate oxide and calcium sulfate leads to the formation of ettringite (Ca 6 Al 2 (SO 4 ) 3 (OH) 12 ·26H 2 O), monosulfoaluminate (Ca 4 Al 2 O 6 (SO) 4 ·14H 2 O) and aluminium hydroxide (Al(OH) 3 ). Even if the global reaction is well known, there is a lack of information about the interactions that occur between the components at an early age. The behaviour of this mixture in solution is presented in Fig. 2(B). Two phases can be easily distinguished. The pH value immediately increases because of the addition of powder to water and then decreases and stabilizes to a value of approximately 10.5 at 100 min. Stabilization occurs again at approximately 220 min; after that, the pH value increases and stabilizes after 6 h to a value of 12.2. With respect to conductivity measurements, during the first step, the conductivity rapidly increases within 7 min to a conductivity value of 5014 μS.cm− 1 because of the fast dissolution of the surface species. Then, the conductivity stabilizes after 30 min, gradually increasing to 5320 μS.cm−1. The conductivity then suddenly decreases to a value of 2340 μS.cm− 1 after 110 min of reaction. This first step is quite similar to the precipitation of gypsum from plaster.

14

7000

13

6000

12

4000

11 10

3000 (a)

2000

(b)

1000 0 0

50

100

150

200

Time / min

250

300

350

9

(B)

14 13 (b)

5000

12

4000

11 10

3000 (a)

2000

8

1000

7

0

pH value

5000

pH value

Conductivity / µS.cm-1

6000

The system exhibits a period of stability of 110 min. Then, the second step begins. This step is more complex and does not appear in the evolution of the plaster in solution, which indicates an interaction between the ions released in solution. This interaction suggests an important decrease in the conductivity after 220 min of reaction, where for 40 min, the conductivity reaches its lowest value of 1230 μS.cm−1. This precipitation reaction is followed by an increase in conductivity for 10 min (2541 μS.cm−1); then, a weak decrease and stabilization of the solution conductivity at 2025 μS.cm−1 occurs. From these results, assumptions about the mechanisms controlling the reaction during the three first hours are difficult to establish without using microstructural characterization techniques. XRD is commonly used to characterize the material microstructure after the hardening phase. However, for such a short time, species were not well detectable with this technique due to their low crystallinity or their small amount. This is why in this study, the conductivity measurements were completed by in situ infrared spectroscopy to propose an experimental explanation for the observed phenomena. Fig. 3 presents the evolution of the infrared spectra of a 75/25 paste over time to highlight the abovedescribed steps. Initially, only water could be easily detected by the presence of vibrations bands at 3305 cm− 1 (νOH of water) and at 1635 cm− 1 (δH2O) [27]. After 10 min of reaction, the first bands of the infrared spectra relative to gypsum began to appear. Gypsum was characterized by the development of bands at 1131, 1102, 1005 and 668 cm−1 associbonds, bands at 1683 and 1620 cm−1 associated with ated with SO2− 4 the HOH deformation of the water of crystallization and bands at 3490, 3392 and 3250 cm−1 associated with the −OH stretching of the water of crystallization [15]. The bands related to the HOH deformation of water of crystallization reached their maximum after 110 min, corresponding to the end of precipitation, which can be partly attributed to the precipitation of Ca2 + and SO24 − ions of the plaster. This finding suggests that the first step is very similar to the plaster dissolution/ precipitation process. However, infrared spectra do not completely correspond to gypsum, with the presence of a very weak shoulder at 3630 and 1083 cm − 1 (Fig. 3(B)), which could be attributed to the presence of ettringite [28]. After 110 min of reaction, the signal associated with the plaster tended to decrease, and the weak shoulder previously observed at approximately 3630 and 1083 cm− 1 grew significantly and induced the displacement of the maximum peak intensity to a low wavenumber, which is attributed to the development of ettringite [29]. After 6 h of reaction, two new well-defined bands were observed at 990 and 845 cm− 1, which are attributed to the formation of AH3. However, these experimental techniques only provide information about the detectable phases, which can lead to mistakes because information about the reaction is lost. Moreover, these results do not indicate

Conductivity / µS.cm-1

(A)

7000

53

9 8 7

0

50

100

150

200

250

300

350

Time / min

Fig. 2. (a) Conductivity measurements and (b) pH measurements of the suspension based on (A) α-plaster and (B) a mixture of 75%wt CAC and 25%wt α-plaster (w/s = 20).

54

J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61

(A)

(B)

Fig. 3. Evolution of the infrared spectra of the CAC/α-plaster paste at (a) the initial time, after (b) 110 min, (c) 220 min and (d) 360 min of reaction ((A) global view and (B) focus on the sulfate bonds region).

what actually occurred in terms of ionic concentration, which is important for the development of materials with controlled structures. The use of modelling to understand this part of the reaction is an inevitable step in determining whether hypotheses are correct.

called the solubility product and is represented by the symbol Ksp. The solubility product is defined by solubility equation (Table 2), which is expressed with a given solid to the left of the equilibrium sign and dissolved products to the right of the equilibrium sign. The Ksp for each reaction is

5. Hydration model n

j ¼ ∏ ½C i  K sp

νi

ð4Þ

In the case of a mineral binder, the mechanisms that are responsible for hydration are based on the dissolution of anhydrous cement or sulfate and the precipitation of the least soluble hydrates in an aqueous solution. Eqs. (1), (2) and (3) simplify the hydration mechanisms. Indeed, phases CA, CA2, C12A7 and C$Hx dissolve in an aqueous solution and yield several ion species. We consider that Ca2 + is the only ion calcium containing species in solution; SO24 − is the only ion sulfate containing species in solution; andAl(OH)− 4 is the only ion aluminium containing species in solution.

where [Ci] is the activity (considered, to first approximation, equal to the molar concentration) of the ith ion containing dissolved species in solution and νi is the stoichiometric factor associated with equation j. The activities of the solid and water are close to unity.

5.1. Dissolution–precipitation mechanism

Congruent dissolution–precipitation kinetics can be estimated from the definition of the reaction rate, Rj, which is defined by the following equation:

Dissolution–precipitation stoichiometric equations are presented in Table 2. A saturated solution is a solution that contains the maximum amount of dissolved products at a given temperature. In dissolution–precipitation reactions, the equilibrium constant is

i¼1

5.2. Hydration kinetics

Rj ¼

d ½C i  IAP j j ¼ k ln j RT 0 ν i dt K sp

Table 2 Chemical equations. Calcium mono-aluminate Calcium di-aluminate Mayenite Plaster Water Ettringite Gibbsite Monosulfoaluminate Gypsum

(CaO)(Al2O3) + 4H2O ⇔ Ca2+ + 2 Al(OH)− 4 (CaO)(Al2O3)2 + 11H2O ⇔ Ca2+ + 4 Al(OH)‐4 + 2H3O+ 2+ (CaO)12(Al2O3)7 + 33 H2O ⇔ 12 Ca + 14 Al(OH)‐4 + 10 OH− (CaO) (SO3) (H2O)0.5 ⇔ Ca2+ + SO24 ‐ + 0.5 H2O 2H2O ⇔ H3O+ + OH+ (CaO)6(Al2O3) (SO3)3 (H2O)32 ⇔ 6 Ca2+ + 2 Al(OH)‐4 + 3 SO24 ‐ + 26 H2O + 4 OH− 2AlðOHÞ3 þ 4H2 O⇔2AlðOHÞ4 ‐ þ 2H3 Oþ (CaO)4(Al2O3) (SO3) (H2O)12 ⇔ 4 Ca2+ + 2 Al(OH)‐4 + SO24 ‐ + 6 H2O + 4 OH− (CaO) (SO3) (H2O)2 ⇔ Ca2+ + SO24 ‐ + 2 H2O

ð5Þ

J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61

7

Plaster CAC

6

dV/dlog(r)

5 4 3 2 1 0 0,1

1

10

100

1000

55

approximately several μm or less with a narrow size distribution range. Grinding provides benefits with respect to early-age hydration and compressive strength because the higher fineness of the anhydrous material, measured as Blaine fineness (cm2 /g), indicates that very fine particles with a larger surface area are in equilibrium with an aqueous solution. The particle size distribution of α-plaster and CAC anhydrous powder were evaluated by laser diffraction (Fig. 4). Both materials showed the same range of granulometry, with a volume-median-diameter D50 of 32.9 μm for α-plaster and 25.3 μm for CAC. The repartition in volume for CAC is composed of four families, approximately 0.7 μm, 8 μm, 40 μm and 300 μm, representing, respectively, 4.5%, 36.5%, 58.9% and 4.6% of the volume. For α-plaster, two families can be distinguished, approximately 1 μm (9.4% of the volume) and approximately 45 μm (90.6% of the volume).

Particle size / µm Fig. 4. Particle size distribution of the CAC and α-plaster.

where IAP j is the Ion Activity Product, which is equal to the j at equilibrium; k j is the constant velocity product solubility Ksp of reaction j; R is the universal gas constant; and T0 is the absolute temperature.
 j The Saturation Index (SI ¼ ln IAPj ) controls the sign of the reaction K sp

rate. If SI = 0, the solution is at equilibrium. If SI b 0, the solution is undersaturated and the solid dissolves. If SI N 0, the solution is supersaturated and the solid grows.

5.3.1. Anhydrous solid The distribution of the particle size shown in Fig. 4 is expressed in versus the radius of the particle. The cumulative distriterms of d dV logðr Þ bution is calculated as follows: Z V ðRÞ ¼

R 0

ð6Þ

where the unit volume V(Rmax) = 1. It is assumed that the granular medium can be represented by a set of spherical grains (ball model). Let nb(r) be the number of balls with radii less than r. The cumulative volume of the granular material is the integral of particle radii less than R. Therefore,

5.3. Reactive surface model

Z

The reaction rate law defined in Eq. (5) is used to account for the equilibrium of the dissolution–precipitation mechanism that depends on the activities of the ion-containing species in the water solution. The surface reactivity of the solid phase is also an important factor that affects the reaction kinetics. In the cement industry, mill grinding, which consumes a large amount of energy, produces irregular cementitious particles with an average particle size of

dV d logðr Þ d log ðr Þ

V ðRÞ ¼

0

R

4 π r 3 dnb : 3

ð7Þ

Additionally, we can equate the experimental distribution of particle size in Eq. (6) and the model distribution in Eq. (7) as follows: Z R Z R 4 dV π r 3 dnb ¼ d logðrÞ: ð8Þ V ðRÞ ¼ 0 3 0 d log ðr Þ

Fig. 5. Principle of the surface reactivity scheme.

56

J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61

Z V ðRÞ ¼

R

0

4 dnb dr ¼ π r3 dr 3

Z 0

R

dV dr d logðr Þ rlnð10Þ

ð9Þ

and we have the following particle size distribution: dnb dV 3 1 1 ¼ : dr d log ðr Þ 4 π r 4 lnð10Þ

ð10Þ

Considering that the amount of dissolved solid mdis is located in the periphery of the beads and corresponds to a thickness e (Fig. 5), we can calculate the amount of undissolved mndis using the following expression: RZmax

mndis ¼ m0 

150

100

50

0

0.5

1

ð11Þ

1.5 Particle size [nm]

2

2.5

3

Fig. 7. Cumulative distribution of the particle number of the precipitated solid versus b.

where m0 represents the initial amount of anhydrous material according to the model of the particle size distribution (Eq. (10)). Additionally, based on the ball model, when the amount mdis is dissolved, the active surface area of the solid that is in equilibrium with the water solution can be evaluated as follows: RZmax

sactive ¼ m0  ω 

dnb 4π ðr−eÞ2 dr dr

ð12Þ

e

where ω represents the shape factor that accounts for the fact that the particles are not perfectly spherical. Because the size distribution of particles number is known, we can define a function g versus the thickness e of the dissolved material:

g ðeÞ ¼ R e Zmax

b=0.5 b=1. b=2.

200

0

dnb 4 π ðr−eÞ3 dr ¼ m0  f ðeÞ dr 3

e

RZ max

Cumulative distribution of the particle number

Thus,

The active surface area of the solid can also be evaluated with the following expression obtained by combining Eqs. (11), (12) and (13): 

 mndis −1 mndis ¼ mndis  ω  h sactive ¼ mndis  ω  g f m0 m0

ð14Þ

The h functions matching the anhydrous phases are plotted in Fig. 6. Let the specific surface sref corresponding to the surface of the undissolved anhydrous material be mndis = m0 = mref. The shape factor is also evaluated using the following expression: sre f ¼ mre f  ω  hð1Þ:

ð15Þ

Combining Eqs. (14) and (15) yields

dnb 4π ðr−eÞ2 dr dr :

ð13Þ

dnb 4 π ðr−eÞ3 dr dr 3

sactive sre f

e


 mndis h m m0 : ¼ ndis  hð1Þ mre f

ð16Þ

1

Distribution of the particle number

Plaster CAC

0.8 0.7 h(mndis /m0)/h(1) [.]

b=0.5 b=1. b=2.

300

0.9

0.6 0.5 0.4 0.3 0.2

250

200

150

100

50 0.1 0

0

0.1

0.2

0.3

0.4 0.5 0.6 mndis/m0 [.]

0.7

Fig. 6. h function of the surface reactivity model.

0.8

0.9

1

0 0

0.5

1

1.5 2 Particle size [nm]

2.5

Fig. 8. Distribution of the particle number of the precipitated solid versus b.

3

J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61

57

10

10

Table 3 [16,31–34] Solubility products obtained from the literature. Solubility products 25 °C

Mayenite Plaster Gypsum Ettringite Monosulfoaluminate Gibbsite

– 5.32e-4 3.72e-5 2.80e-45 3.71e-30 1.59e-31

Objective function

Species

9

10

Thus, the reaction rate becomes Rj ¼

d½C i  s Kj j ¼ active k ln j RT 0 : sre f ν i dt K sp

ð17Þ 8

10

5.3.2. Precipitated solid If we consider the precipitated solid, the foregoing arguments are not valid because the distribution of the particle size is unknown and can vary over time. The problem arises because there are two degrees of freedom, i.e., the number of particles and their radii. In this work, it is assumed that precipitated particles with a cumulative distribution of hydrated particles follow the simple law: nb ¼ a r b :

ð18Þ

The amount of solid that is precipitated, mp can also be evaluated as follows: Z mp ¼ ρ 

0

rp

dnb 4  π r 3 dr dr 3

ð19Þ

where rp is the radius of the grains of maximum size and ρ is the density of the precipitated solid. Thus, for a given amount of precipitated solid, the corresponding radius rp is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3mp ðb þ 3Þ : rp ¼ 4ρabπ bþ3

0

5

10

15

20 25 30 Iteration number

35

40

45

50

Fig. 10. Objective function evolution versus the iteration number (plaster–gypsum).

To illustrate the meaning given in the expression of Eq. (18), the cumulative distribution and the distribution of the particle number of the precipitated solid versus the value of b, are respectively plotted in Figs. 7 and 8 in the case of a volume of precipitated solid which is mp =ρ ¼ 1000 nm3 . When b = 1, that means we assume a homogeneous distribution of the particle size on the range of radius [0, rp] as depicted in Fig. 8. But, this proposal which is not supported by experimental evidence needs obviously further experimental investigations to determine the real distribution of the particle size of the precipitated solid. The active surface area of the solid can also be evaluated by the following expression: Z sactive ¼ ω 

0

rp

dnb  4π r 2 dr: dr

ð21Þ

Therefore, we have

ð20Þ

sactive ¼ ω 

4π ab bþ2 rp : bþ2

ð22Þ

-6

3

7000

x 10

calculus experience

6000

2.5

Velocity constant kj

Conductivity [ µS cm-1]

5000

4000

3000

plaster gypsum

2

1.5

1

2000

0.5

1000

0

0

1

2

3 Time [hours]

4

5

Fig. 9. Iterative calibration of the model (plaster–gypsum).

6

0 0

5

10

15

20 25 30 Iteration number

35

40

45

Fig. 11. Iterative calibration of the velocity constant (plaster–gypsum).

50

58

J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61 -4

6

6000

x 10

calculus experience 5000

Conductivity [ µS cm-1]

Product of solubility Kjsp

5

4

plaster gypsum

3

4000

3000

2000

2 1000

1 0

0

0

5

10

15

20 25 30 Iteration number

35

40

45

Combining Eqs. (20) and (22) yields 4πab  bþ2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi!bþ2 3mp : 4ρabπ

bþ3

ð23Þ

Let the specific surface sref correspond to the surface of the precipitated solid for a given mref. The shape factor is evaluated by the following expression: ω¼ 4πab  bþ2

sre f sffiffiffiffiffiffiffiffiffiffiffiffiffiffi!bþ2 bþ3 3mre f : 4ρabπ

ð24Þ

sactive ¼ sre f

4πab bþ2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi!bþ2 3mp 4ρabπ sffiffiffiffiffiffiffiffiffiffiffiffiffiffi!bþ2 ¼ bþ3 3mre f 4ρabπ bþ3

1

1.5 Time [hours]

2

2.5

3

gypsum and calcium monosulfoaluminate. In this work, we considered b to be equal to 1. The values of certain solubility products may be found in the literature [30], which is reported in Table 3. Nevertheless, both the state values of the solubility products and the velocity constants must be calibrated. In this work, we used conductimetry measurements to calibrate all of these parameters. Concerning the term sactive sre f dealing with the reactive surface, the reference quantity mref of the anhydrous material in Eq. (16) or the precipitated solid in Eq. (25) is respectively equal to the molar mass of the anhydrous material or the hydrate.

6.1. Plaster–gypsum

Therefore, we finally have 4πab bþ2

0.5

Fig. 13. Best response model in terms of the conductivity (plaster–gypsum) of the surface reactivity model.

Fig. 12. Iterative calibration of the solubility product (plaster–gypsum).

sactive ¼ ω 

0

50

sffiffiffiffiffiffiffiffiffiffi!bþ2 mp : mre f

bþ3

ð25Þ

An inverse method based on Kalman filtering [31] was used based on the experimental measurements previously presented in this article of the conductivity of a diluted solution of α-plaster. To evaluate the consistency of the proposed reactive surface model (RSM), the inverse method was performed for the configurations with and without the RSM. The final estimate of the model parameter vector x corresponds

7000 Anhydrous phase RSM Without RSM Hydrate phase RSM Experience

6. Model calibration 6000

5000 Conductivity [ µS cm-1]

Before using the above-described modelling procedure to simulate the chemo-hydro-mechanical problem at an early age by accounting for the coupling between hydration kinetics and drying kinetics, modelling parameters must be evaluated. In the case of anhydrous j must be determined for each dissolution, the parameters kj and Ksp mineral phase of the CAC (calcium mono-aluminate, calcium dialuminate and mayenite) and for the sulfate source (plaster). In the j and b must be case of precipitated hydrates, the parameters kj, Ksp determined for each potential hydrate, such as ettringite, gibbsite,

4000

3000

2000

1000 Table 4 Parameter values for plaster and gypsum with RSM.

0 Species

k

Ksp

S (g/l)

Plaster Gypsum

2.7e-06 2.0e-07

4.0e-04 5.6e-05

3. 1.28

0

0.5

1

1.5 Time [hours]

2

2.5

Fig. 14. Best response model in terms of conductivity (plaster–gypsum).

3

J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61 Table 5 Parameter values for plaster and gypsum in different configurations of RSM. Species

Plaster alpha Gypsum

Table 6 Parameter values for the anhydrous and hydrate phase of the ettringite binder with RSM.

Anhydrous phase RSM

Without RSM

Hydrate phase RSM

k

Ksp

k

Ksp

k

Ksp

5.1e-07 1.0e-08

5.3e-04 3.7e-05

5.3e-08 2.6e-08

5.8e-04 6.0e-05

6.1e-08 2.2e-07

4.7e-04 5.6e-05

to the minimization of an objective function, which may be generally written as follows:  T   f ob j ðxÞ ¼ yexp −ycomp ðxÞ C −1 exp yexp −ycomp ðxÞ þ ðx−x0 ÞT C −1 0 ðx−x0 Þ

59

ð26Þ

where yexp is the measurement result of conductivity versus time; ycomp(x) is the computed result of the conductivity versus time for a set of model parameters x; Cexp is computed using the scatter of the repeated experiments; and C0 is the covariance, i.e., the uncertainty associated with the initial guess x0. 6.1.1. With RSM Fig. 9 shows the successive model responses in terms of conductivity j ) according to the inversus different sets of model parameters (kj, Ksp verse method, in which the objective function was minimised in each iteration. The decreasing objective function is clearly shown in Fig. 10, and the associated variations of the parameters versus the iteration number are also depicted in Figs. 11 and 12. The end values are reported in Table 4. We note that only the velocity constants were greatly modified by the inverse process. The solubility products were near the values reported in the literature (Table 3). The best response of the model considering a reactive surface is depicted in Fig. 13. Differences can be observed, but they are mainly in the amplitude values. The kinetics predicted by the model are in good agreement with the measured conductivity. The first phase of increasing conductivity corresponds to the dissolution of the plaster, when the molar concentrations of calcium and sulfate species increased in the solute. The solution was undersaturated with respect to the plaster. The second phase of decreasing conductivity coincided with the supersaturation of the solution with respect to gypsum. This phase lasted until the plateau, during which the solution was at equilibrium with gypsum.

Species

k

Ke

S (g/l)

Mayenite Ettringite Gibbsite Monosulfoaluminate

5.2e-08 1.2e-08 1.1e-07 9.8e-08

8.9e-07 4.1e-45 1.8e-31 2.9e-30

77.7 0.33 2.3e-6 0.20

6.1.2. Without RSM To assess the relevance of the RSM proposed in this study, the inverse method was also performed for several configurations of the model without reactive surface modelling, referred to as “without RSM”, with only the reactive surface modelling of the anhydrous phase, referred to as the “anhydrous phase RSM,” and with only the reactive surface of the hydrate phase, referred to as the “hydrate phase RSM”. The best response of the model in each configuration is presented in Fig. 14. The end values are reported in Table 5. Large differences can be observed. All of the kinetics predicted by the model are in poor agreement with the measured conductivity. If we first consider the case “without RSM”, the first phase of the conductivity increases, which again corresponds to the dissolution of the plaster. The solution is undersaturated for the plaster until the solution becomes supersaturated for gypsum. The first phase of constant conductivity is also associated with the perfect balance between the dissolution of the plaster and precipitation of gypsum. This constant phase persists until the plaster is completely consumed. The third phase is the steep slope of the conductivity that coincides with the decreasing concentration of calcium and sulfate species in the solution. The last plateau corresponds to the situation in which the solution is at equilibrium with gypsum. In the two other cases, “anhydrous phase RSM” and “hydrate phase RSM”, the effect of each reactive surface model is clearly apparent based on the results depicted in Fig. 14. Globally, the RSM brings the kinetics of the dissolution–precipitation mechanism into continuity, which is observed in the experimental measurements. The values of the solubility products obtained in these configurations are not different from those obtained from modelling the reactive surface, but this result demonstrates the necessity of considering the reactive surface in hydration modelling to describe the dissolution– precipitation mechanism over time, especially when several chemical equations are coupled.

6000

13

12

Calculus Experience

5000

10 pH

Conductivity [ µS cm-1]

11

4000

3000

Calculus Experience

9

2000 8

1000

0

7

0

1

2

3 Time [hours]

4

5

6

Fig. 15. Best response model in terms of conductivity for the ettringite mixture with RSM.

6

0

1

2

3 Time [hours]

4

5

Fig. 16. Response model in terms of pH for the ettringite mixture with RSM.

6

60

J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61

0.025

0.09

Ca2+ 0.08

Al(OH)-4 0.02

Mayenite Plaster Gypse

0.06

Molar concentration [mol/l]

Molar concentration [mol/l]

0.07

0.05 0.04 0.03 0.02

S0-4

0.015

0.01

0.005

0.01 0

0

1

2

3

4

5

6

7

8

9

0

10

0

1

2

3

4

5 6 Time [hours]

Time [hours]

7

8

9

10

Fig. 19. Model predictions of the concentration of ions in the solute.

Fig. 17. Model predictions of the consumption of anhydrous phases.

6.2. CAC–plaster

7. Model predictions

To produce an ettringite binder, a mixture of CAC and plaster was mixed with water. Understanding the process by which the binder is formed requires that all of the chemical equations presented in Table 2 be considered. The number of model parameters j ) for that must be determined is 8, corresponding to the sets (k j, Ksp mayenite, ettringite, gibbsite and monosulfoaluminate. The same inverse method process was performed, but convergence was difficult to obtain. The shape of the experimental conductivity curve was too rugged to lead to a smooth calibration process. The choice of parameter values was also manually performed based on our expertise in response surface modelling. Fig. 15 shows the best calibration result obtained. The associated parameters are reported in Table 6. The next section of this paper discusses the analyses of dissolution and precipitation of species in the solute given by the model corresponding to each distinct phase of the conductivity kinetics. First, however, the kinetics of the pH given by the model are compared to the pH measurement results shown in Fig. 16. Good agreement between the model prediction and experimental results confirms the validity of the proposed approach.

Based on the good agreement between the calculated and measured pH and conductivity, this section focuses on the model predictions. Fig. 17 illustrates the consumption of mayenite and plaster. Fig. 18 shows the formation kinetics of hydrates, i.e., ettringite, gibbsite and monosulfoaluminate. Sulfate is the limiting factor of the binder mixture; plaster dissolves, and gypsum precipitates and then dissolves. After 4 h, the sulfate is completely dissolved. At the same time, mayenite dissolves continuously. For hydrates, ettringite precipitates until the sulfate disappears in the solute. Gibbsite continuously precipitates. This first precipitation is reflected by the first decrease in the conductivity up to 4 h. After 5 h, the ettringite dissolves at the same time as the monosulfoaluminate precipitates, in accord with result reported in the literature. The kinetics of conductivity reveal a second decreasing phase after 5 h. Gibbsite precipitates again at a lower rate than that observed in the first phase. Between 4 and 5 h, only mayenite dissolves because plaster is no longer present in the solute. As shown in Fig. 19, the molar concentrations of the calcium and aluminium ions increase, whereas the sulfate ions completely disappear. This process is associated with the increase in conductivity clearly observed between 4 and 5 h. 7000

0.03

5000 Conductivity [µS cm-1]

Molar concentration [mol/l]

Experience Calc. 1 Calc. 2

6000

0.025

0.02 Gibbsite Ettringite Monosulphoaluminate

0.015

0.01

4000

3000

2000

0.005

0

1000

0

1

2

3

4

5

6

7

8

Time [hours] Fig. 18. Model predictions of the precipitation of hydrate phases.

9

10

0 -4 10

-3

10

-2

10

-1

10 Time [hours]

0

10

Fig. 20. Loop of the calculated and experimental conductivity.

1

10

J.F. Georgin, E. Prud'homme / Cement and Concrete Research 76 (2015) 51–61

Regarding the pH kinetics, an initial increase in pH is linked to the dissolution of the mayenite according to the increase in the OH− molar concentration. Then, the rapid decrease observed at up to approximately 30 min is linked to hydrates that precipitate. A plateau between 1 and 3 h is linked to the precipitation of both ettringite and gibbsite. As reported in Table 2, the formation of ettringite requires the consumption of OH−, and at the same time, the formation of gibbsite requires the consumption of H3O+. When the sulfate is completely consumed, mayenite continues to dissolve. Additionally, the pH increases again at 5 h, when the ettringite is in an undersaturated state. At this point, ettringite begins to dissolve and monosulfoaluminate precipitates. Fig. 18 shows that the model predicts that monosulfoaluminate precipitates and dissolves within a short period over the first 30 min. This finding corresponds to the sharp peak identified in the conductivity profile presented in Fig. 15. The experimental conductivity kinetics appear to lack a sharp peak, but if we create a loop for the first few minutes, a decrease in conductivity is also observed, as shown in Fig. 20. A second calculation was performed and called “calculus 2”, which limits the precipitation of monosulfoaluminate to after one hour. We observe that the local decrease in conductivity is missing, which means that the monosulfoaluminateate precipitates rapidly in the solute and dissolves just as quickly. 8. Conclusions A model for the hydration of an ettringite binder that accounts for the reactive surface model of the solid phases in a solute was proposed. Mechanisms of precipitation and dissolution are considered. For each chemical equation describing the precipitation or dissolution (defined by the sign reaction rate), a reaction velocity constant and solubility product constant have been evaluated. Calibration was performed based on the kinetics of the conductivity of the solute. The experimental pH kinetics are also well consistent with the model prediction. Modelling helps describe the dissolution of mayenite and ettringite and the formation of gibbsite until the sulfate is consumed. Monosulfoaluminate begins to precipitate at the same time that ettringite dissolves. Based on this approach, a different mineralogy of CAC will be calibrated in future work, with further consideration of the effect of temperature. Then, the ettringite binder hydration model will be implemented within a porous media framework. Acknowledgements The authors thank the Kerneos Aluminate Technologies Company for supplying the CAC anhydrous material studied in this work. References [1] Xu. Linglin, Peiming Wang, Guofang Zhang, Formation of ettringite in Portland cement/calcium aluminate cement/calcium sulfate ternary system hydrates at lower temperatures, Constr. Build. Mater. 31 (2012) 347–352. [2] C. Peter, Hewlett, LEA'S Chemistry of Cement and Concrete, Fourth Edition, 2003. [3] C. Evju, S. Hansen, Expansive properties of ettringite in a mixture of calcium aluminate cement, Portland cement and β-calcium sulfate hemihydrate, Cem. Concr. Res. 31 (%12) (2001) 257–261. [4] C. Evju, S. Hansen, The kinetics of ettringite formation and dilatation in a blended cement with β-hemihydrate and anhydrite as calcium sulfate, Cem. Concr. Res. 35 (112) (2005) 2310–2321. [5] G. Le Saout, B. Lothenbach, F. Winnefeld, P. Taquet, H. Fryda, Hydration study of a calcium aluminate cement blended with anhydrite, Calcium Aluminates Proceedings of the International Conference, Avignon 2014, pp. 165–176.

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