a simulation model for microstructure of portland cement …

2nd International RILEM Workshop on Concrete Spalling due to Fire Exposure 5-7 October 2011, Delft, The Netherlands

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A SIMULATION MODEL FOR MICROSTRUCTURE OF PORTLAND CEMENT PASTE AT HIGH TEMPERATURE

Qi Zhang1, G. Ye1, E.A.B. Koenders1,2 1) Microlab, Section of Materials and Geosciences, Faculty of Civil Engineering and Geoscience, TU Delft, The Netherlands ([email protected]) 2) COPPE-UFRJ, Rio de Janeiro, Brasil Abstract At high temperature, the hydration products in Portland cement paste dehydrate gradually. As a result of this degradation process, the material properties change, initiating micro-cracks and affecting the tensile strength of the material. The rearrangement of the material, induced by dehydration, affects the transport properties of the material, which, on its turn, influences the boiling process in the internal pore structure. In this study, a simulation model for microstructure change of heated Portland cement paste was proposed. The chemical reaction, volume change of calcium-silicate-hydrate (CSH) and microstructure response were discussed respectively.

1. INTRODUCTION At elevated temperature, the cement paste will start to dehydrate, causing irreversible

changes in microstructure. The hydrated Portland cement consists of unhydrated cement, liquid water, air and hydration product (mainly including calcium-silicate-hydrate (CSH) and calcium hydroxide (CH)). It is reported that the dehydration of CSH starts at about 105°C, and that the dehydration of CH occurs at around 420°C. The CSH gel with long silicate chains could be broken during its crystallographic rearrangement and converted into wollastonite. These reactions will change the porous microstructure of cement paste, which determines the meso mechanical properties and transport properties of cement paste.

In the past decades, many microstructure models were proposed to simulate the hydration process of cement, such as HYMOSTRUC3D [1] and CEMHYD3D [2]. However, the dehydration induced microstructure change is hardly studied and simulated. In this contribution, a simulation model of microstructure of Portland cement paste at high temperature was proposed. The theoretical model of microstructure change was introduced first, which consisted of dehydration kinetics, volume change of CSH and microstructure response. Secondly, the numerical method of simulation was proposed and discussed.

2. THEORETICAL MODEL

2.1 General During the dehydration process at high temperature, the volume changes of each phase in

cement paste are different. The hydration product (CSH and CH) lose the chemical water and shrink. However, the unhydrated clinker, which is surrounded by hydration product, expands along with temperature. This strain mismatch can results in microcracking and microstructure change. Therefore, the simulation procedure consists of 3 parts.

� Simulating the dehydration degree under elevated temperature.


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� Simulating the volume change of CSH due to dehydration. � Simulation the microstructure response to the volume change of CSH. The details of each part were discussed in the following.

2.2 The dehydration of Portland cement paste The dehydration process of Portland cement paste is considered as the reactions as

following: 2 2 2 2 2( ) ( ) ( ) ( ) vapor

a b a b cCaO SiO H O CaO SiO H O c H O�AAB �CAA (1)

2 2( ) vaporCa OH CaO H OAAB CAA (2)

Factors that have been identified as controlling rates of the most solid state reactions include chemical reaction, microstructure and diffusion. Many different kinetics models are proposed to explain other isothermal curve [3]. The selection of kinetics model usually depends on the experimental data. The Friedman model [4] was used for kinetics of nth-order reaction in this study.

( ) (1 )nf � � � (3)

where n is the reaction order, � is the reaction degree. Arrhenius equation is used to describe

the influence of temperature.

0( ) exp( / )aK T A E RT � (4)

where 0A , the pre-exponential factor and Ea, the activation energy are referred to as the Arrhenius parameters. Given attention to the reaction order and temperature, the kinetics of dehydration of each phase can generally be expressed as following equation:

( ) ( )d K T fdt� � (5)

Both dehydration kinetics of CSH and CH can be characterized by equation (5). But their parameters are different. Because the dehydration of CH is a one-step reaction from calcium hydroxide to calcium oxide, its kinetics parameters are constants in the whole process. Unlike CH, the dehydration of CSH is multi-step reaction. There are several equilibrium states of CSH during the whole dehydration process. Therefore, the Arrhenius parameters of CSH dehydration, including the activation energy and the pre-exponential factor A0, vary with its dehydration degree. By integrating of equation (5), the dehydration degree of CSH and CH under arbitrary temperature program can be characterized as equation (6) (7).

0, ,0 0(1 ) exp( / )

t t nCSHCSH CSH CSH a CSH

d dt A E RT dtdt�� " "

(6)

0, ,0 0(1 ) exp( / )

t t nCHCH CH CH a CH

d dt A E RT dtdt�� " "

(7)

All above kinetics parameters (A0,CH, Ea,CH, A0,CSH and Ea,CSH), together with WCSH and WCH, were determined by experiment. The measured A0,CSH and Ea,CSH were showed in Fig 1.


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Combining the dehydrations of CSH and CH, the weight loss in the total dehydration

process of cement paste can be characterized as following: cem CSH CSH CH CHW W W� � � � (8)

where Wcem is the total weight loss of cement paste, WCSH is the maximum weight loss caused by dehydration of CSH, WCH is the maximum weight loss caused by dehydration of CH. The total dehydration degree and kinetics of cement paste can be expressed as:

cemcem

CSH CH

WW W

�

(9)

cem CSH CSH CH CH

CSH CH CSH CH

d W d W ddt W W dt W W dt� � �

(10)

2.3 The volume change of CSH at elevated temperature The relationship between the tobermorite volume and dehydration degree was separated

into two stages in term of the following two reactions [5]:

(11)

(12)

CSH is an amorphous phase, which is a mixture of imperfect tobermorite and jennite. Therefore, the relationship between the CSH volume and dehydration degree was simplified into bilinear relationship. Each segment could be determined in the following methods.

2.3.1 Stage I: dehydration degree from 0% to 80% Based on the globules model of CSH [6], the volume of CSH is comprised by the dense

globules particles and inters particles space. The packing density of globules particle was 75% in this study [6]. It was assumed that volume change at this stage is caused by the shrinkage of globules particles, and that inter particles space keeps the same.

In the globules model, the atomic structure of the dense globules particles is similar with tobermorite. 11 tobermorite is transformed into 9 tobermorite at high temperature [7]. Merlino etc. [5] studied the atomic structure of both two tobermorite. The detail atomic structure of tobermorite is listed in Table 1. It can be calculated that the unit cell volume of 9

tobermorite is about 82% of t of 11 tobermorite. Therefore, it is assumed that the

(a) (b)

Figure 1. (a) Cubic spline interpolation of activation energy of CSH. (b) Cubic spline interpolation of lnA of CSH.


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globules particle reduces by 18% in volume when dehydration degree is 80%. The total volume change of CSH was characterized by the following formula:

(13)

Table 1: The atomic structure of tobermorite at different hydration degree Compound a ( ) b ( ) c ( ) Cell Volume ( 3) Tobermorite 11 11.274 7.3439 11.468 949.4945

Tobermorite 9 11.156 7.303 9.566 779.3637

2.3.1 Stage II: dehydration degree from 80% to 100% In this stage, the volume change of CSH is mainly caused by crystallization. The XRD/Reitveld measurement [8] showed that around 27% amorphous CSH was transformed into crystalline . The molar volume of CSH and was list in Table 2. It was assumed that the recrystallization degree is linear with the dehydration degree in this stage. The volume change of CSH in this stage was expressed by the following:

(14)

If one substitute 80% with in Equation 13, the equals 86.5%. Then the formula 14 was simplified as:

(15)

Table 2: The atomic structure of tobermorite at different hydration degree Compound Density (mg/m3) Molar volume (cm3/mole) Reference CSH (Dry) 2.12 108 [5]

3.28 52 [5]

2.4 The response of the microstructure to the volume changes of different phases during dehydration

If one takes the micro-heterogeneity into account, the local strain can be split into uniform stain and fluctuating stain . The displacement and strain in the RVE admit the following decomposition:

(16)

(17)

Where and is the local displacement and strain, respectively, denotes the position of a point. Introduce an angle brackets operator <.> as the average of a field on RVE:

(18)

The periodicity of means that the average of fluctuating strain on the RVE vanishes. The average of local strain equals uniform strain.

(19)

(20)

For the microstructure of heated cement paste, the fluctuating stain results in the change of pore structure, which can effects the gas permeability of cement paste. The uniform stain of microstructure is the macro strain, which can lead to the thermal stress and macro


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cracking. The shrinkage of CSH can not only enlarge the pore structure, but also change the bulk volume of cement paste.

The strains caused by thermal dilation and chemical reaction were analyzed in this micromechanics study. Taking different phenomena into account, the total strain in cement paste can be divided into 3 parts:

(21)

(22)

(23)

where x is the location, is the stress field, E is the stiffness sensor of different phases, u is the displacement field, is the elastic strain, is the thermal dilation, is the dehydration induced strain. This typical mechanics equations together with boundary condition can be solved with by numerical method.

The above equation (21-23) presented the derivation on micro elastic response on the RVE under given temperature. The micro-cracking takes place while the local stress reaches the failure criteria. The failed element cannot contribute the global stiffness any more, which leads to the meso nonlinearity of the RVE. In that case, the local damage becomes an evolution problem. Then the linear system equations are showed as the following

(24)

The global stiffness matrix depends on the local stress path. The failure criteria followed maximum principal stress theory:

(25)

Introduce a state variable to characterize the state of each element:

(26)

If the local stress of each element reaches the maximum principal stress theory, the is marked as zero. Therefore the global stiffness matrix reads:

(27)

The damage state vector of the whole RVE depends on micro stress field and strength of each phase, which can be numerical calculated by numerical method.

3. NUMERICAL METHOD The nonlinear response of microstructure to the volume change of CSH should be

analyzed by numerical method. In this study, the fracture analysis was carried out by Lattice model. The microstructure of cement paste was characterized by digital voxel structure (Fig 2). Each voxel represents one homogeneous phase. Therefore, the nanostructure in each voxel was ignored.


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3.1 Lattice modelThe lattice model is a numerical method to simulate the fracture process of continues

matrix. In this model, the continuous solid is schematized as a network of two node beams. After determination of boundary condition and external force, the displacements of nodes are calculated. By removing in each loading step the beam elements with highest stress over strength ratio, fracture is simulated. The basic assumption is that the beam elements have a linear elastic behaviour up to failure, and fail in a purely brittle manner. The fracture criterion of each beam is described as:

(28)

where is the normal force acting in the beam element, and are the bending moments in two nodes. is a factor that regulates the amount of bending taken into account. is a scale factor.

3.2 Initial lattice model generation The initial microstructure of cement paste with side length of 100 micron before heating

used in this study was based on the simulated results of HYMOSTRUC3D. By mapping the sphere particle into digital 3D voxel, the HYMOSTRUC 3D result was transformed into 3D voxel structure. The side length of voxel was 2 micron. Lattice model was utilized to analysis the deformation and cracking in microstructure. The generation of 3D beam in lattice model was followed by below steps (Fig 3):

� Generating a 3D node with 6 free degrees in the centre of each voxel. � Generating a 3D beam by connecting two neighbour nodes. � The length of beam was the side length of voxel. � The cross section of beam is square with the same side length of voxel.

Figure 2: The voxel-based microstructure of heated cement paste

Figure 3: The beam generation from voxel and its cross section properties


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3.3 Nonlinear failure analysis procedure The numerical procedure utilized in this study is outlined:

3.4 Deformed voxel microstructure: from lattice to voxel After nonlinear analysis of microstructure by lattice model, the micro deformation and

micro damage were calculated. The deformed lattice structure was transformed into voxel structure again (Fig 5).

4. CONCLUSIONS � The simulation model of microstructure of heated Portland cement paste was proposed.

The physical theory of this model consisted of dehydration kinetics, volume change of CSH due to dehydration, and the microstructure response to volume change of CSH.

� The kinetics of dehydration was characterized by developed Arrhenius equation.

Figure 4: The schematic flowchart of nonlinear analysis

Figure 5: Determination the voxel structure by deformed lattice model


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� The volume change of CSH was analyzed by using globules model [6] and the atomic structure of nature CSH.

� The nonlinear response of microstructure was derived by elastics mechanics theory. � The lattice method was utilized as the numerical method for nonlinear fracture analysis.

References [1] Breugel, K.v., Simulation of hydration and formation of structure in hardening cement-

based materials. 1991, Delft University of technology: Delft. [2] Bentz, D., Modelling cement microstructure: Pixels, particles, and property prediction.

Materials and Structures, 1999. 32(3): p. 187-195. [3] L'Vov, B., Thermal Decomposition of Solids and Melts. 2007: Springer. [4] Friedman, H.L., Kinetics of thermal degradation of char-forming plastics from

thermogravimetry. Application to a phenolic plastic. Journal of Polymer Science Part C: Polymer Symposia, 1964. 6(1): p. 183-195.

[5] MERLINO, S., E. BONACCORSI, and T. ARMBRUSTER, The real structures of clinotobermorite and tobermorite 9 A: OD character, polytypes, and structural relationships. Eur J Mineral, 2000. 12(2): p. 411-429.

[6] Jennings, H.M., Refinements to colloid model of C-S-H in cement: CM-II. Cement and Concrete Research, 2008. 38(3): p. 275-289.

[7] Taylor, H.F.W., THE DEHYDRATION OF TOBERMORITE. Clays and Clay Minerals, 1957. 6(1): p. 101-109.

[8] Zhang, Q. and G. Ye. QUANTITATIVE ANALYSIS OF HEATED PORTLAND CEMENT PASTE BY X RAY DIFFRACTION. in XIII International Congress on the Chemistry of Cement. 2011. Madrid.

a simulation model for microstructure of portland cement …

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