microstructural development in Portland cement-based materials can be simulated. ... studies, simulation of intricate and compound process of cement hydration, ...
A Numerical Hydration Model of Portland Cement Ippei Maruyama1, Tetsuro Matsushita2 and Takafumi Noguchi3
ABSTRACT : A computer-based numerical model is presented, with which hydration and microstructural development in Portland cement-based materials can be simulated. Proposed model enables the prediction of hydration curves as a function of the particle size distribution, chemical composition of the cement, water to cement ratio and the actual curing temperature. Mutual relationship between developing the microstrucure and its effect on hydration process is modeled explicitly. In this contribution modeling and validation of this model are discussed. KEYWORDS: Hydration model, particle size distribution, microstructure, cement composition 1. INTRODUCTION Over the past few decades a number of studies have been made on modeling of cement hydration in order to grasp the time dependent properties of cement based materials (Kondo 1968). In those recent studies, simulation of intricate and compound process of cement hydration, especially focused on micro-mechanics, with the potential of modern computer is attempted (van Breugel 1995, Bentz 1991). Two points seem to be helpful in attempting to sketch out what makes it complex to simulate process of cement hydration and why it needs computer power. One is physical aspect. Cement paste structure, which is composed of cement particles and water, is determined by particle size distribution (Bezjak, 1980) and water to cement ratio. In cement hydration process, cement particles are interconnected and make structure of cement matrix. This physical aspect affects the rate of cement hydration through diffusion of ions (Knudsen 1984). Second is chemical aspect. Cement is mainly composed of tricalcium silicate, but is poly-mineral material at the same time. Reactions of the components interact with each other. And temperature is much influential on the rate of hydration from the chemical points of view (Tomosawa 1974). These two aspects have mutual dependent relationship through diffusion of ions and material formation. This relationship can not be solved with simple equation with regard to space-time problem in cement-based material. But using computer power with concept of discrete event system makes it possible to find an answer to this problem. 2. HYDRATION MODEL OF CEMENT PARTICLE 2.1 Basic Assumptions Proposed hydration model is based on the fundamental kinetic model for Portland cement that is developed by Tomosawa (Tomosawa 1997). Tomoswa’s model is expressed as a single equation composed of four rate determining coefficients which determine the rate of formation and destruction of initial impermeable layer, the activated chemical reaction process and the following relent diffusion controlled process. This preliminary approach shows high potential of simulating of hydration process. Four coefficients, however, are just fit parameter in Tomosawa’s model and they are not predictable 1
Research Associate, Graduate School of Engineering, Hiroshima University, Japan Graduate student, Graduate School of Engineering, The University of Tokyo, Japan 3 Associate Prof., Graduate School of Engineering, The University of Tokyo, Japan 2
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from any information of cement properties. This can be deduced by the fact that particle size distribution and interparticle contact, which play important role in hydration process, are not considered. In order to take into account these and get consistent relationship between cement properties and coefficients, reconstruction and modification are conducted from the assumptions of Tomosawa’s model with additional change. The assumptions are listed as follows: 1. The cement particle initiates hydration from the moment that it is brought into contact with water. 2. The hydrate formed by hydration adheres to the cement particle. And hydrate will be covering it up spherically until interparticle contact comes up. And unhydrated cement keeps spherical shape as well. The new gel is formed at the surface with no restriction of interparticle contacts. If the surface contacts with the surface of another particle, new gel is no longer produced on it. 3. The hydrate has a vgel times as much as the original cement in volume. 4. The liquid phase, which is assumed to be water, diffuses through the hydrate layer and reaches the surface of the cement particle (reacting front) and chemically reacts with cement. This process continues through hydration process. And part of the hydrate produced at the reacting surface moves out through the layer of hydrate. Hence, equi-molar counter diffusion of water and hydrate (presumably ions) is assumed to be taking place in the hydrate layer. 5. The diffusion coefficient of hydrate layer for water is not different between outer products and inner products. This diffusion coefficient is affected by tortuosity of gel as well as radius of gel pore in hydrate. This phenomenon is expressed as a function of degree of hydration. 6. The particle size distribution of cement can be approximated by Rosin-Rammler function. And each particle with the same diameter has the same rate of hydration. 7. Dormant period in the initial process of hydration is assumed that there is a process in which the reaction resistance increases with the increase of degree of hydration in each particle (film formation) followed by a period in which the reaction resistance decreases with increasing thickness of outer products. 2.2 Paricle size distribution in space Cement particle distribution will make a big difference in cement hydration process. Defining degree of cement hydration as ratio of reacted cement volume to initial cement volume, in this sense, each particle shows different degree of hydration and degree of hydration of total cement paste should be accounted for this different degree of hydration of each particle. In the proposed model, it is assumed that the cement particle distribution can be expressed with Rosin-Ramler function:
(
V ( d p ) = 1 − exp −bd p
n
)
(1)
On the other hand, cement particles develop an interparticle contacts as hydration proceeds. This phenomenon is determined by the reaction rate, initial position and size of each cement particle in paste matrix. After formation of interparticle contact, cement hydration will be inhibited by decrease of area to suck available water to hydrate. In regard to this aspect, these exist several models of location of cement particles. To cite instances, random distribution model (Kunders 1997) and flocculation model (Bentz 2000) are proposed so far. In this proposed model, it is assumed that each cement particle has the same ratio of available water volume to its cement volume and unit cubic cell is determined as centered spherical cement particle surrounded water in cube. The size of unite cubic cell is according to entire water to cement ratio of cement paste. (see Figure 1) This is one of the methods to arrange Figure 1. Schematic representation of particle size distribution model in cement paste
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the cement particles in space homogeneously. In this case, the cement particles are located as pseudo6-neighborhood. The 6-neighborhood arrangement is not efficient. If the closest packing is assumed, the arrangement should be 12-neighborhood among the same size of particles. But when flocculation is considered (Taylor 1997), the arrangement of spherical cement particles, which assumes homogenous arrangement with regard to volumetric ratio, must be less efficient and the number of neighboring particles will be reduced. 2.3 Hydration model of each cement particle According to the Tomosawa’s model hydration process is expressed as a single equation that is dominated by three different rate-determining phenomena, i.e. producing initial protective layer, inward diffusion of external water equivalent to the outward diffusion of reaction products and chemical reaction on the surface of unhydrated surface (Tomosawa 1997):
− ρ c (γ + Wa , g ) rt
2
drt Cw∞ 2 = dRt ( vgel − 1) rt drt 1 r − 1 Rt 1 1 , = ⋅ dt 2 + t + dt dt Rt 2 2 kd rt De kr rt
(2)
where rt is radius of unhydrated cement particle, Rt is total radius including the gel layer, De is effective diffusion coefficient of water in the cement gel, kr is coefficient of reaction rate per unit area of reaction front, γ is the stoichiometric ratio by mass of water to cement, Wa , g is the ratio of water entrapped in the gel pore to cement, ρc is density of unhydrated cement and vgel is volumetric ratio of hydrated cement paste gel to unhydrated cement. With Eq.(2), degree of hydration of each particle, α d is calculated: 4
4
r
α d = 1.0 − π rt 3 π r03 = 1.0 − t 3 3 r0
3
(3)
where α d is degree of hydration of the cement particle whose diameter is d = 2r0 . The total degree of hydration α is defined as accumulation of each degree of hydration over the cement particles whose distribution is according to Eq.(1). The first term of the denominator on the right side of former Eq.(2) relates to the initial reaction, indicating its reaction resistance. This affects only a very early stage when total degree of hydration almost equals to 0, and in this case, the reaction rate is determined by kd . Here kd is assumed to be expressed as the sum of the term of increase of mass transfer resistance as increase of total degree of hydration and the term of degrease of mass transfer resistance as increase of thickness between original boundary and reaction front ( r0 − rt ) of each hydrating cement particle: kd =
B
α 1.5
+ C ⋅ ( r0 − rt )
4.0
(4)
As the hydration process progresses the gel density has been found to increase (Relis 1977). The effective diffusion coefficient De is assumed to be decrease with increase of the gel density: 1.5
1 De = DE ln αd
(5)
where DE is initial diffusion coefficient that is dependent on the composite of cement particle. 2.4 Structural limitation by interparticle contact Cement particles are expanding with hydration process by a factor vgel . Available space is occupied on a first come and each particles are interconnected with hardening of cement paste matrix. This formed cement paste matrix structure has effects on physical aspect of matrix as well as hydration process of cement with feedback from it. In this regard, the structural limitation is modeled explicitly.
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Cement particle is assumed be spherical and each particle has cubic space in accordance with water to cement ratio. This assumption means that the each particle is even in terms of available space for expanding. There are three type of mode Mode 1 Mode 2 Mode 3 with degree of expanding (Figure 2). Figure 2. 3-dimensional expression of structural “Mode 1” is the state that the cement limitation of expanding hydrate gel particle does not make contact with surfaces of cell, “mode 2” is the state that the cement particle starts to make contact with surfaces of cell and the contacted parts have circular shape and “mode 3” is the state that the cement particle makes contact with surfaces of cell widely and the contacted parts are connected each other. (i) Initial state of cement particle and cell Volume of cement particle in the cell whose size is 1 in a side is determined with the cement density ρc , the water density ρ w and water to cement ratio W / C : Vc =
1
(6)
ρ W /C ⋅ c +1 ρw
Ratio of radius of cement particle to the length of cubic side r0 holds: 1/ 3
3V r0 = c 4π
(7)
(ii) Surface and volume In “mode 1”, “mode 2” and “mode 3”, the ratio of radius of hydrated cement particle to the length of cubic side r satisfies r0 ≤ r < 0.5 , 0.5 ≤ r < 2 / 2 and 2 / 2 ≤ r < 3 / 2 respectively. The surface that is in contiguity with water is represented by a function of r : 0.5 S (r ) = 4π r 2 , S (r ) = 4π r 2 − 12π 1 − , S (r ) = 8 r
1/ 2
2
∫
1/ 2
2
∫
r −1/ 2 r −1/ 4 − x
r 2
r − x2 − y 2 2
dxdy (8)
And the volume of hydrated cement is represented by a function of r : 4 1 1 4 2 V ( r ) = π r 3 , V (r ) = π r 3 − 6π r 3 − r 2 + 3 3 2 24 3 1 + 2 1 ⋅ 0.5 ⋅ r 2 − x 2 − 1/ 4 0.5 2 16 ∫ 2 r − x2 π 0.5 r 2 −1/ 2 + ⋅ − Arc cos 2 2 2 4 r −x
V (r ) = 2 ⋅ r 2 −
dx
(9)
Each three equation is shown according to an order of “mode 1”, “mode 2” and “mode 3”. The structural limitation of hydration process is introduced to Eq.(2) by using the reduction factor Cst :
− ρ c (γ + Wa , g )rt
2
drt Cw∞ = Cst ⋅ 1 r − 1 Rt 1 1 dt + t + 2 kd rt De kr rt 2
4
dRt 4π ( vgel − 1) rt drt = ⋅ dt S ( Rt ) dt 2
(10)
where Cst is defined as: Cst =
S ( Rt ) 4π Rt
(11)
2
where Cst means the effect of reduction of water through gel surface that is contacted with water and S ( Rt ) is the same function of Eq.(8). 2.5 Temperature effect on hydration process Curing temperature effect on the rate of chemical reaction, i.e. hydration process. Temperature effect is introduced to each mass transfer coefficient and reaction factor in this proposed model (Tomosawa 1997). It is assumed that B in Eq.(4), kr and De in Eq.(2) follow Arrheniu’s law. Hence, with the values of B293 , kr 293 and De 293 that are given for 293 K, the coefficients at T K are expressed as follows: B = B293 exp − β1 (1/ T − 1/ 293) De = De 293 exp − β 2 (1/ T − 1/ 293)
(12)
kr = kr , 293 exp − E / R ⋅ (1/ T − 1/ 293)
3. MODEL PARAMETERS For the determination of the model parameters kr 293 , DE 293 (in Eq.(5)), B293 , C293 , β1 , β 2 and E / R an evaluation was conducted. Degree of hydration is determined by the ratio of amount of heat liberation to maximum heat liberation predicted by Woods equation (Woods 1932). In this evaluation more than 20 hydration tests (i.e. Tomosawa 1997-2) were involved, comprising 9 different types of Portland cement with C3S contents ranging from 20% to 70%. Curing temperatures varied from 10 °C to 60 °C, particle size distribution with blaine value ranging from 200 m2/kg to 550 m2/kg. The kr 293 is expressed as a function of the C3S and C3A contents:
(
kr 293 = 8.05 ⋅ 10−10 × rwC3 S + rwC3 A
)
0.975
(13)
where rwC3 S and rwC3 A are mass contents of C3S and C3A respectively. This parameter indicates the reaction rate per unit area of reaction front and this value dominates the initial rate of hydration process. It seems reasonable to suppose that the kr 293 is associated with C3S and C3A contents showing high reaction speed before dormant period. In this evaluation the value of kr 293 is ranging from 1.8 ·10-8mm/h to 5.6·10-8mm/h. The DE 293 is expressed as a function of the C3S contents: DE 293 = 3.2 ⋅ 10−14 × rwC3 S
2.024
(14)
The DE 293 value represents effective diffusion coefficient of cement gel in initial stage. This ratedetermining value may be affected by the C3S contents. This DE 293 is found from 20·10-12mm2/h to 180 ·10-12mm2/h. For the factors B293 , C293 , β1 , β 2 and E / R with constant value is applicable in Coefficient Value
B293
Table 1. Model parameters in proposed model β1 [K-1] β 2 [K-1] [x10 mm/h] C293 [x10-7/mm3·h] -0
0.3
5.0
1231
7579
E / R [K-1]
5364
5
4. CONCLUSION
500
Simulation result [J/g]
majority cases (See Table 1). The B293 , C293 and E / R are rather sensitive to the behavior before dormant period. 20%-variation of there value may cause 5% of degree of hydration at 72 hours. The β1 and β 2 are insensitive. 20% -variation of these values may leads 2% of difference in degree of hydration at 72 hours. In Figure 3 Comparison of simulation result with experimental data of heat liberation is shown. The experimental heat liberation is value of the end of each experiment. Simulation results are good agreement with experimental results.
400
300
200
100
0
0
100
200
300
400
500
Expeimental Heat liberation [j/g]
Figure 3. Predictive accuracy of heat liberation by proposed model (Comparison is made at the end of experiment.)
Entire hydration process of each cement particle is modeled by a single kinetic equation with assuming that particle size distribution, interparticle contact with cell concept and that each cement particle has the same ratio of available water volume to its cement volume. Entire system of cement paste is presented as accumulation of them. Simulation results shows good agreement with experimental data and its accuracy is better than 10% before 72 hours from mixing. 5. REFERENCES Bentz, D. P. and Garboczi, E. J. (1991). “Percolation of Phases in a Three-Dimensional Cement Paste Microstructural model”, Cem. Concr. Res, Vol.21, pp.325-344 Bentz, D. P. (2000). “CHEMHYD3D: A Three-Dimensional Cement Hydration and Microstructure Development Modelling Package. Version 2.0”, NISTR 6485, National Institute of Standards and Technology Bezjak, A. (1980). “On the Determination of Rate Constants for Hydration Processes in Cement Paste”, Cem. Concr. Res. Vol.10, , pp.553-563 Knudsen, T. (1984). “The Dispersion Model for Hydration of Portland Cement I., General Concepts” Cem. Concr. Res., Vol.14, pp.622-630 Kondo, R. and Ueda, S. (1968). ”Kinetics and Mechanism of the Hydration of Cements”, Fifth International Symposium on the Properties of Cement Paste and Concrete Tokyo, II-4, pp.203-248 Kunders, E. A. B. (1997). ”Simulation of volume changes in hardening cement-based materials”, Ph.D thesis, TU Delft, Relis, M. and Soroka, I. (1977). “Variation in Density of Portland Cement Hydration Products” Cem. Concr. Res, Vol.7, pp.673-680, 1977 Taylor, H.F.W. (1997). “Cement Chamistry 2nd Edition”, Thomas Telford, pp230-231 Tomosawa, F. (1974). “A Hydration model of cement” Proc. of Annual Meeting on Cement Technology, Cement Association of Japan, Vol.28, pp.53-57 Tomosawa, F. (1997). “Development of a kinetic model for hydration of Cement” Proceedings of the 10th International congress on the chemistry of cement, Gothenburg, Sweden, 2ii051, Tomosawa, F., Noguchi, T. and Hyeon, C. (1997-2). “Simulation Model for Temperature Rise and Evolution of Thermal Stress in Concrete Based on Kinetic Hydration Model of Cement” Proceedings of 10th International Congress of Chemistry of Cement, Vol,4, pp.4iv072, van Breugel, K. (1995). “Numerical Simulation of Hydration and Microstructural Development in Hardening Cement-Based Material (I) Theory and (II) Application” Cem. Concr. Res., Vol.25, pp.319-331 and 522-530. Woods, H. (1932). “Effect of Cement Composition on Mortar Strength” Engineering News Record, pp.404-407
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