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A multi-phase model for predicting the effective diffusion coefficient of chlorides in concrete
Construction and Building Materials 26 (2012) 295–301
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
A multi-phase model for predicting the effective diffusion coefficient of chlorides in concrete Long-Yuan Li a, Jin Xia a,b,⇑, San-Shyan Lin c a
School of Civil Engineering, University of Birmingham, Birmingham B15 2TT, UK Institute of Structural Engineering, Zhejiang University, Hangzhou, China c Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan b
a r t i c l e
i n f o
Article history: Received 21 February 2011 Received in revised form 5 May 2011 Accepted 13 June 2011 Available online 5 July 2011 Keywords: Chloride Diffusion Concrete Multi-phase Composites
a b s t r a c t In this paper, a mesoscopic structure model is proposed and is used to investigate the chloride diffusion behaviour in concrete. The concrete is treated as a heterogeneous material composed of cement paste and aggregate two phases. The chloride diffusion is assumed to take place only in the cement paste phase. By modelling concretes with different aggregate volume fractions the effect of aggregate content on chloride diffusion in concrete is examined. The results show that the effective diffusion coefficients obtained from the two- and three-dimensional models are different. The two-dimensional series and parallel models provide upper and lower bounds of the effective diffusion coefficient, respectively, whereas the threedimensional model provides the accurate effective diffusion coefficient. The three-dimensional simulation also demonstrates that the Maxwell model is able to predict accurate effective diffusion coefficient of chlorides in concrete. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Chloride-induced reinforcing steel corrosion is one of the most important material deterioration problems in reinforced concrete structures. It affects a large number of infrastructures, particularly those exposed to environments where de-icing salts or coastal/ marine conditions are encountered. In order to protect the reinforcing steel from corrosion one has to understand how chloride penetration taking place in concrete and how the individual components of the concrete mixture influencing the penetration of chlorides. Considerable efforts have been made by using analytical, experimental and numerical methods to investigate the microscopic and mesoscopic scale transport behaviour of chlorides in heterogeneous concrete materials of multiple phases. For example, by using different dimensional scales, Bentz et al. [1] accomplished experiments to investigate the effects of water-to-cement ratio, degree of hydration, aggregate volume fraction, coarse and fine aggregate particle size distributions, interfacial transition zone thickness and air content on chloride diffusion in concrete. They found that among these variables the water-to-cement ratio, degree of hydration and aggregate volume fraction were the three ⇑ Corresponding author at: Institute of Structural Engineering, Zhejiang University, Room 610, Block (A), Anzhong Building, 338 Yuhangtang Road, Hangzhou 310058, PR China. Tel.: +86 (0)571 8820 8733x610; fax: +86 (0)571 8820 8733x601. E-mail addresses: [email protected] (L.-Y. Li), [email protected] (J. Xia), [email protected] (S.-S. Lin). 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.06.024
major variables influencing concrete diffusivity. Meijers et al. [2] developed a coupled transport model of heat, moisture and chloride ions in concrete in which the concrete was treated as a heterogeneous material consisting of three components, namely, mortar, aggregates and interfacial transition zones. The transport equations were solved numerically using finite element methods, in which the three components were meshed separately and continuity in fluxes at interfaces between them was applied. Yang and Su [3] investigated the effect of aggregate content on the chloride migration coefficient in concrete by using electrochemical migration tests. Based on the experimentally obtained results they developed an empirical formula in which the chloride migration coefficient was expressed as a function of the volume fraction of aggregate. Similar experiments, but using diffusion instead of electro-migration, were also conducted by Caré [4] to quantify the influence of aggregate and interfacial transition zone contents on chloride ingress in concrete. Caré and Herve [5] developed an analytical model for calculating the effective diffusion coefficient of ions in concrete, in which the concrete was also treated as a three-phase composite consisting of a cement continuous phase, an aggregates dispersed phase and an interface transition zone. Moon et al. [6] investigated the characteristics of capillary pores using mercury intrusion porosimetry techniques on 12 concrete specimens composed of six types of Portland and blended cements with water–binder ratios of 40% and 50% and examined the effect of capillary pores on chloride diffusion in concrete. It was found that apart from the capillary pores the
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average pore diameter also has a significant influence on chloride diffusion in concrete. Experiments using ponding tests [7] and accelerated chloride migration tests [8] were also reported in literature to quantify the influence of capillary porosity diameter and continuous pore diameter on the diffusion and migration coefficients of chloride ions in cement-based materials. Recently, Zeng [9] developed a two-dimensional structure model to simulate the chloride diffusion behaviour in concrete, in which the concrete was treated as a heterogeneous material composed of cement paste and aggregate phases with different diffusivities. The modelling results showed that the chloride diffusion in hetero-structured concretes appears to lag behind that for the homogeneous ones predicted using the effective diffusion coefficient, indicating an increasingly notable discrepancy between the two chloride concentration profiles as the diffusion proceeds. Jin et al. [10] developed a similar numerical model in which the concrete was treated as a three-phase composite consisting of a cement continuous phase, an aggregates dispersed phase and an interface transition zone. Zheng and Zhou [11] proposed a three-phase composite sphere model to represent the heterogeneous nature of concrete and derived a closed form expression for chloride diffusion in concrete. Later, by utilising the threephase composite sphere model Zheng et al. [12] further investigated the influence of the aggregate-cement paste interfacial transition zone on the steady-state chloride diffusion in mortars and concretes. In this paper, a mesoscopic structure model is proposed to simulate the chloride diffusion in concrete. The concrete is treated as a heterogeneous material composed of cement paste and aggregate two phases. The chloride diffusion is assumed to take place only in the cement paste phase. By modelling concretes with different aggregate volume fractions the effect of aggregate content on the chloride diffusion in concrete is examined. Finally, by using the concept of representative elementary volume, chloride concentration distribution profiles obtained from the mesoscopic and macroscopic structure models are also compared, which illustrates some important features appeared in the mesoscopic model but cannot be seen in the macroscopic structure models.
Dcem Dagg ð1 /ÞDcem þ /Dagg
ð2Þ
min where Dmax eff and Deff are the upper and lower bounds of the effective diffusion coefficient of chlorides in concrete, / is the volume fraction of the cement paste in the concrete, Dagg and Dcem are the diffusion coefficients of chlorides in the aggregate and cement paste phases, respectively. Considering the aggregate is relatively impermeable, it is reasonable to assume that Dagg = 0. In this case, the upper and lower bounds of the effective diffusion coefficient in Eqs. (1) and (2) min become Dmax eff ¼ /Dcem and Deff ¼ 0. For two-dimensional problems as illustrated in Fig. 1, the upper bound of the effective diffusion coefficient represented by the parallel model (see Fig. 1a) is still given by Eq. (1), but the lower bound of the effective diffusion coefficient represented by the series model (see Fig. 1b) is no longer zero. The reason for this is because the diffusion of chlorides in the two-dimensional parallel model can be still assumed to be only in the direction parallel to the longitudinal axis of the aggregate cylinders (see Fig. 1a), while the diffusion of chlorides in the two-dimensional series model can take place in any directions perpendicular to the longitudinal axis of the aggregate cylinders (see Fig. 1b). It is obvious that the parallel and series models in the two-dimensional case are more accurate than those in the one-dimensional case. Hence, if the lower bound of the effective diffusion coefficient represented by the two-dimensional series model can be predicted then one can estimate the effective diffusion coefficient of chlorides in real concrete by using the obtained upper and lower bound values. It is worth mentioning herein that, although the effective diffusion coefficient of chlorides in concrete can be predicted directly by using three-dimensional simulations, this kind of simulations is not only very time consuming but also extremely expensive in computing. Most of existing numerical studies used the twodimensional model to investigate the influence of individual components of concrete mixture on chloride diffusion [5,9–12]. However, the implication of the results obtained from the twodimensional simulation is not discussed.
3. FE model for determining the lower bound of the effective diffusion coefficient
2. Two-phase prediction models Concrete is a multi-phased heterogeneous material, which consists of aggregates, cement paste, and voids. The aggregates constitute a dense phase randomly distributed in a continuous cement paste matrix that contains numerous, almost uniformly distributed nano-sized gel pores and capillary pores. Since both gel pores and capillary pores are much small compared to the sizes of aggregates, they can be approximately considered as a homogeneous phase. When the concrete is fully saturated, ionic transport mainly takes place in the cement paste matrix through the connective pores. In contrast, the aggregates within the cement paste are usually of very poor diffusivity. Existing data have shown that the diffusion coefficient of chlorides in cement paste is well correlated with the water-to-cement ratio [13] and the relationship between them can be approximately expressed by an exponential function. However, the diffusion coefficient of chlorides in concrete depends on not only the water-to-cement ratio, but also the aggregates [14]. In the two-phase model, the concrete is assumed to be a two-phase composite. One is the homogeneous cement paste and the other is the aggregates. The upper and lower bounds of the effective diffusion coefficient of chlorides in concrete can be obtained by using the simple, one-dimensional parallel and series configuration assumptions of the two constituents, as follows [14],
Dmax eff ¼ ð1 /ÞDagg þ /Dcem
Dmin eff ¼
ð1Þ
In order to determine the lower bound of the effective diffusion coefficient of chlorides in concrete, a simple two-dimensional mesoscopic structure model of two phases is constructed here. Fig. 2 shows a schematic illustration of the model, in which the size of the plain concrete is 100 100 mm, all circular areas represent the aggregates and the rest part is the cement paste matrix. The distribution of aggregates of different sizes employed in the present plain concrete is determined by using a sieve analysis for a given aggregate volume fraction [15]. Several different types of ideal curves were determined on the basis of practical experiments and theoretical calculations. The most known and acceptable curve is the Fuller curve [16,17]. Therefore, Fuller mix [18] was exploited here to determine the size of an aggregate to be generated. Note that aggregate particles in real concrete may not be perfectly circular. However, as it will be demonstrated lately, the effect of aggregate shapes on the chloride diffusion in concrete seems very small. Chloride diffusion taking place in the cement paste matrix with a given water-to-cement ratio can be described using the wellknown diffusion equation as follows,
@C ¼ Dcem r2 C @t
ð3Þ
where C is the concentration of chlorides in the cement paste matrix (moles per unit volume of cement paste), t is the time, and r is the
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Fig. 2. The two-dimensional numerical model for calculating the lower bound of chloride diffusion coefficient in concrete (/ = 0.5). (a) Geometry (l = h = 100 mm) and (b) finite element mesh.
where C1 is the chloride concentration at the line x = 0, l = 100 mm and h = 100 mm are the length and height of the plain concrete used in the numerical model, respectively. For given values of Dcem and C1 one can solve Eq. (3) to obtain the chloride concentration distribution profile at any time. Of particular interest is the x-component of the chloride diffusion flux, Jx, at the steady state. The total chloride flux, q, along the boundary x = l at the steady state is given by,
q¼
Z
h o
J x ð1; l; yÞ dy ¼ Dcem
Z
h
o
@Cð1; l; yÞ dy @x
ð7Þ
The average flux, Jcon, in the plain concrete in x-direction thus is given by,
J con ¼ Fig. 1. Chloride diffusion in two-dimensional models (arrows represent the chloride diffusion directions). (a) The parallel model in which chloride diffusion takes place on in the direction parallel to the axis of cylinders and (b) the series model in which chloride diffusion takes place in the cross-section normal to the axis of cylinders.
Laplace differential operator. For simplicity, the initial condition and boundary conditions are assumed as follows,
Cð0; x; yÞ ¼ 0 Cðt; 0; yÞ ¼ C 1 ; @Cðt; x; 0Þ ¼ 0; @y
ð4Þ Cðt; l; yÞ ¼ 0 @Cðt; x; hÞ ¼0 @y
ð5Þ ð6Þ
q h
ð8Þ
By taking the plain concrete as a representative elementary volume in a macroscopic structure model, the average flux, Jcon, thus can be also expressed as,
J con ¼ Dmin eff
min @C Deff C 1 ¼ @x l
ð9Þ
Substituting Eqs. (7) and (9) into (8), it yields,
Dmin eff ¼
Dcem l C1h
Z
h o
@Cð1; l; yÞ dy @x
ð10Þ
Eq. (10) indicates that if the chloride flux in the cement paste matrix is computed from the present two-dimensional mesoscopic
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structure model then the lower bound of the effective diffusion coefficient of chlorides in concrete can be evaluated. 4. Results and discussion The partial differential equation defined by Eq. (3) with initial and boundary conditions defined by Eqs. (4)–(6) can be solved using finite element methods. Fig. 2b shows a typical finite element mesh used in the analysis for the plain concrete of aggregate volume fraction being 0.5. The analyses are performed for the plain concrete that has various different aggregate volume fractions and the corresponding lower bounds of the effective diffusion coefficient of chlorides in concrete calculated using Eq. (10) are plotted in Fig. 3. For the purpose of comparison, the upper bounds of the effective diffusion coefficient calculated using Eq. (1) and the theoretical predictions of the effective diffusion coefficient using Maxwell’s model [19] and Bruggeman equation [20] are also superimposed in the figure. It can be seen from the figure that both the lower and upper bounds of the effective diffusion coefficient decrease with the increase of the aggregate volume fraction. The gap between the lower and upper bounds also increases with the aggregate volume fraction. As is to be expected, the effective diffusion coefficients predicted by using Maxwell’s model and Bruggeman equation are between the lower and upper bounds of the effective diffusion coefficient. It is also observed from the figure that the prediction of Bruggeman equation is more close to the lower bound, whereas the prediction given by Maxwell’s model is almost in the middle of the lower and upper bounds. In order to examine the influence of aggregate size on the lower bound of the effective diffusion coefficient, numerical analyses for the plain concrete with different maximum diameters of circular aggregates are performed. Fig. 4 shows the influence of the maximum diameter of circular aggregates used in the plain concrete on the lower bound of the effective diffusion coefficient. The figure shows that increasing or reducing the maximum diameter of circular aggregates has almost no influence on the obtained results as long as the random distribution of the aggregates is created using the sieve analysis. This implies that the influence of the size of circular aggregates in the concrete on the lower bound of the effective diffusion coefficient is negligible. To examine the effect of aggregate shape on the lower bound of the effective diffusion coefficient, numerical analyses for the plain concrete with circular, ellipse, triangle, square and mixed aggregates are also performed. Fig. 5 shows the meshes used for the plain concrete with the four different aggregate shapes (mesh for the plain concrete with circular
Fig. 4. Comparison of the lower bounds of the effective diffusion coefficient in concrete with different maximum diameters of circular aggregates.
aggregates can be found in Fig. 2b). The comparisons of the lower bounds of the effective diffusion coefficient obtained from the plain concrete with circular, ellipse, triangle, square and mixed aggregates are provided in Fig. 6. It is observed from the figure that, except for the lower bound of the effective diffusion coefficient in the concrete of triangular aggregates that is marginally small, the lower bounds of the effective diffusion coefficients in all other four concretes are more or less the same. The small value of the lower bound of the effective diffusion coefficient found in the concrete of triangular aggregates is mainly attributed to the effect of tortuosity of the pore system. For the same volume fraction of aggregates the concrete of triangular aggregates is believed to have larger tortuosity than the concretes of other four aggregate shapes. This demonstrates that the chloride diffusion in concrete depends on not only the volume fraction of aggregates but also the tortuosity of the pore system. 5. FE model for determining the effective diffusion coefficient In order to demonstrate that the two-dimensional finite element model described in the preceding sections can indeed provide a good lower bound for the effective diffusion coefficient of chlorides in concrete, a three-dimensional finite element analysis model is developed herein. Because of the limitation in computing, only the concrete cube of 50 30 30 mm with spherical aggregates is considered (see Fig. 7). Analyses are performed for the concrete cube with different volume fractions of aggregates under a steady state case in which the front (x = 0) and rear (x = l) surfaces are assumed to have specified concentration boundary conditions and all other four surfaces are assumed to have zero flux boundary conditions. The effective diffusion coefficient is calculate using the following equation,
Deff ¼
Fig. 3. The lower and upper bounds of the effective diffusion coefficient of chlorides in concrete.
Dcem l C1 h
2
Z o
h
Z o
h
@Cð1; l; y; zÞ dy dz @x
ð11Þ
where h = 30 mm and l = 50 mm are the cross-section size and length of the cube. The effective diffusion coefficients obtained from the three-dimensional finite element analysis for different aggregate volume fractions are plotted in Fig. 8. It can be seen from the figure that the effective diffusion coefficient calculated from the three-dimensional model lies almost in the middle of the lower and upper bounds of the effective diffusion coefficient and agrees very well with the prediction given by Maxwell’s model. According to [19,21–23], the difference between the series and Maxwell’s models is the tortuosity, which in the series model is equal to
L.-Y. Li et al. / Construction and Building Materials 26 (2012) 295–301
299
Fig. 6. Comparison of the lower bounds of the effective diffusion coefficient in concrete with different aggregate shapes.
Fig. 7. Finite element analysis model for a cubic concrete with spherical aggregates (/ = 0.5).
Fig. 5. Finite element meshes employed in the plain concrete with different aggregate shapes (/ = 0.5). (a) Ellipse, (b) triangle, (c) square and (d) mixed aggregates.
Fig. 8. Comparison of effective diffusion coefficients of chlorides in concrete obtained from different models.
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one, whereas in the Maxwell’s model is equal to 2//(3 /). As is explained in Fig. 1, the difference between the series and parallel models in the two-dimensional case is also the tortuosity. Interestingly, if take the tortuosity as (1 + /)/(3 /) in the two-dimensional parallel model, then the following analytical solution can be used to predict the lower bound of the effective diffusion coefficient of chlorides in concrete,
Dmin eff ¼ Dcem
/ð1 þ /Þ 3/
ð12Þ
The demonstration of Eq. (12) is also shown in Fig. 8. 6. Non-steady state diffusion The most commonly used equations for chloride ingress in concrete are the Fick’s laws of diffusion. Fick developed two relations that are sometimes referred to as Fick’s first and second laws of diffusion. Fick’s first law is a constitutive equation that describes the relationship between the diffusive flux and concentration of the species. Fick’s second law is the conservation of mass of the species, which holds for non-steady state diffusion. When applying Fick’s laws to a porous material such as the concrete these two relations can be expressed as follows:
J con ¼ Deff rC
ð13Þ
@ð/CÞ ¼ rJ con @t
ð14Þ
Eq. (13) represents the flux of chlorides pass through the unit area of concrete in the unit time, whereas Eq. (14) is a mass conservation equation of chlorides in the unit volume of concrete. Substituting Eq. (13) into (14), it yields,
@C Deff 2 ¼ r C @t /
ð15Þ
Note that if the tortuosity could be neglected then we would have Deff = /Dcem. Hence, Eq. (15) can be interpreted as the diffusion of chlorides in the cement paste. This implies that the influence of the aggregates on the chloride diffusion in concrete is only by the tortuosity generated by the presence of aggregates rather than by the volume of the aggregates, if the mass of chlorides in concrete is defined based on the mole number of chlorides in the unit volume of cement paste. In order to demonstrate Eq. (15), Fig. 9 shows the comparisons of chloride concentration distribution profiles at three different times, obtained from the two-dimensional finite element analyses of the mesoscopic and macroscopic structure models. In the
Fig. 9. Comparison of chloride concentration profiles obtained from the mesoscopic and macroscopic models (/ = 0.5).
Fig. 10. A typical distribution profile of chloride concentration obtained from the mesoscopic model (/ = 0.5).
mesoscopic model, the governing equation is described by Eq. (3) with the initial and boundary conditions defined by Eqs. (4)–(6), whereas in the macroscopic model, the governing equation is described by Eq. (15) with the initial and boundary conditions also defined by Eqs. (4)–(6). Note that the concentration obtained from the macroscopic model is only the function of time and x-coordinate because of the symmetric nature of the problem, while that obtained from the mesoscopic model is the function of not only time and x-coordinate but also y-coordinate. For the purpose of the comparison, the chloride concentrations shown in Fig. 9 at a given time and a given x-value are the average value of the chlorides at the section from y = 0 to y = h. It is evident from the comparisons shown in the figure that, the two models match rather well, although the macroscopic model provides slightly higher concentration in the region near to the surface and lower concentration in the region far away from the surface. The variation of the chloride concentration with y-coordinate in the mesoscopic model depends on the structure and configuration of aggregates. Fig. 10 shows a typical distribution profile of the chloride concentration obtained from the present mesoscopic model. It can be seen from the figure that the concentration variation along y-axis is noticeable but not significant. 7. Conclusions This paper has presented an investigation on the chloride diffusion in concrete by using a two-phase model. From the results obtained the following conclusions can be drawn: (1) The two-dimensional series and parallel models can provide the upper and lower bounds of the effective diffusion coefficient of chlorides in concrete. The analytical expressions for the upper and lower bounds can be approximated as min Dmax eff ¼ Dcem / and Deff ¼ Dcem /ð1 þ /Þ=ð3 /Þ. (2) The size of aggregates has a negligible influence on the diffusion coefficient of chlorides in concrete as long as the random distribution of the aggregates is created using the sieve analysis. (3) The shape of aggregates has small influence on the diffusion coefficient of chlorides in concrete. This is believed to be attributed to the effect of aggregate shape on the tortuosity of the pore system. (4) The effective diffusion coefficient of chlorides in concrete can be approximated as the average of the upper and lower bounds of the effective diffusion coefficient given by the two dimensional series and parallel models.
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(5) The effective diffusion coefficient of chlorides in concrete predicted by using the three-dimensional finite element simulations is very close to that predicted using the Maxwell’s model. (6) The effective diffusion coefficient can be directly used in Fick’s first law. However, when it is applied to Fick’s second law the diffusion coefficient appeared in the finally deduced ‘‘diffusion equation’’ involves only the tortuosity effect. Finally, it should be stressed that the present model does not take into account the effect of the interfacial transition zone between the cement and aggregates. Thus, strictly speaking, the diffusion coefficient used for the cement paste in this paper should be the combined diffusion coefficient of chlorides in the cement paste and interfacial transition zone. Acknowledgement The work is supported by The Royal Society through the two international joint projects under the Grants (JP0867232 and JP090130), which is gratefully appreciated. References [1] Bentz DP, Garboczi EJ, Lagergren ES. Multi-scale microstructural modeling of concrete diffusivity: identification of significant variables. Cem Concr Aggr 1998;20(1):129–39. [2] Meijers SJH, Bijen JMJM, De Borst R, Fraaij ALA. Computational modelling of chloride ion transport in reinforced concrete. Heron 2001;46(3):207–16. [3] Yang CC, Su JK. Approximate migration coefficient of interfacial transition zone and the effect of aggregate content on the migration coefficient of mortar. Cem Concr Res 2002;32(10):1559–65. [4] Caré S. Influence of aggregates on chloride diffusion coefficient into mortar. Cem Concr Res 2003;33(7):1021–8. [5] Caré S, Herve E. Application of a n-phase model to the diffusion coefficient of chloride in mortar. Transp Porous Med 2004;56(2):119–35. [6] Moon HY, Kim HS, Choi DS. Relationship between average pore diameter and chloride diffusivity in various concretes. Constr Build Mater 2006;20(9): 725–32.
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[7] Yang CC. On the relationship between pore structure and chloride diffusivity from accelerated chloride migration test in cement-based materials. Cem Concr Res 2006;36(7):1304–11. [8] Yang CC, Cho SW, Wang LC. The relationship between pore structure and chloride diffusivity from ponding test in cement-based materials. Mater Chem Phys 2006;100(2–3):203–10. [9] Zeng YW. Modeling of chloride diffusion in hetero-structured concretes by finite element method. Cem Concr Compos 2007;29(7):559–65. [10] Jin WL, Xu B, Wang HL. Numerical simulation and performance analysis of concrete based on material mesostructure. J Architec Civil Eng 2008;25(4):19–22 [in Chinese]. [11] Zheng J, Zhou X. Three-phase composite sphere model for the prediction of chloride diffusivity of concrete. J Mater Civil Eng 2008;20(3):205–11. [12] Zheng JJ, Wong HS, Buenfeld NR. Assessing the influence of ITZ on the steadystate chloride diffusivity of concrete using a numerical model. Cem Concr Res 2009;39(9):805–13. [13] Atkinson A, Nickerson AK. The diffusion of ions through water-saturated cement. J Mater Sci 1984;19(9):3068–78. [14] Hobbs DW. Aggregate influence on chloride ion diffusion into concrete. Cem Concr Res 1999;29(12):1995–8. [15] Zheng JJ, Li CQ, Jones MR. Aggregate distribution in concrete with wall effect. Mag Concr Res 2003;55(3):257–65. [16] Shakhmenko G, Brish J. Concrete mix design and optimization. In: Proceedings of the 2nd International Ph.D. Symposium in Civil Engineering, Budapest; 1998. p. 160–16. [17] Rahmani H, Ramazanianpour AA. Effect of binary cement replacement materials on sulfuric acid resistance of dense concretes. Mag Concr Res 2008;60(2):145–55. [18] Talbot AN, Richart FE. The strength of concrete and its relation to the cement, aggregate, and water. Bulletin no. 137; 1923. p. 1–116. [19] Dormieux L, Lemarchand E. Modélisation macroscopique du transport diffusif. Apport des méthodes de changement d’échelle d’espace. Oil Gas Sci Technol 2000;55(1):15–34. [20] Bruggeman DAG. Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen: I. DielektrizitaÈtskonstanten und LeitfaÈhigkeiten der MischkoÈrper aus isotropen Substanzen. Ann Phys 1935;416(7): 636–79. [21] Garboczi EJ. Permeability, diffusivity, and microstructural parameters: a critical review. Cem Concr Res 1990;20(4):591–601. [22] Van Brakel J, Heertjes PM. Analysis of diffusion in macroporous media in terms of a porosity, a tortuosity and a constrictivity factor. Int J Heat Mass Transfer 1974;17(9):1093–103. [23] Xu K, Daian JF, Quenard D. Multiscale structures to describe porous media part II: transport properties and application to test materials. Trans Porous Media 1997;26(3):319–38.
Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
A multi-phase model for predicting the effective diffusion coefficient of chlorides in concrete Long-Yuan Li a, Jin Xia a,b,⇑, San-Shyan Lin c a
School of Civil Engineering, University of Birmingham, Birmingham B15 2TT, UK Institute of Structural Engineering, Zhejiang University, Hangzhou, China c Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan b
a r t i c l e
i n f o
Article history: Received 21 February 2011 Received in revised form 5 May 2011 Accepted 13 June 2011 Available online 5 July 2011 Keywords: Chloride Diffusion Concrete Multi-phase Composites
a b s t r a c t In this paper, a mesoscopic structure model is proposed and is used to investigate the chloride diffusion behaviour in concrete. The concrete is treated as a heterogeneous material composed of cement paste and aggregate two phases. The chloride diffusion is assumed to take place only in the cement paste phase. By modelling concretes with different aggregate volume fractions the effect of aggregate content on chloride diffusion in concrete is examined. The results show that the effective diffusion coefficients obtained from the two- and three-dimensional models are different. The two-dimensional series and parallel models provide upper and lower bounds of the effective diffusion coefficient, respectively, whereas the threedimensional model provides the accurate effective diffusion coefficient. The three-dimensional simulation also demonstrates that the Maxwell model is able to predict accurate effective diffusion coefficient of chlorides in concrete. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Chloride-induced reinforcing steel corrosion is one of the most important material deterioration problems in reinforced concrete structures. It affects a large number of infrastructures, particularly those exposed to environments where de-icing salts or coastal/ marine conditions are encountered. In order to protect the reinforcing steel from corrosion one has to understand how chloride penetration taking place in concrete and how the individual components of the concrete mixture influencing the penetration of chlorides. Considerable efforts have been made by using analytical, experimental and numerical methods to investigate the microscopic and mesoscopic scale transport behaviour of chlorides in heterogeneous concrete materials of multiple phases. For example, by using different dimensional scales, Bentz et al. [1] accomplished experiments to investigate the effects of water-to-cement ratio, degree of hydration, aggregate volume fraction, coarse and fine aggregate particle size distributions, interfacial transition zone thickness and air content on chloride diffusion in concrete. They found that among these variables the water-to-cement ratio, degree of hydration and aggregate volume fraction were the three ⇑ Corresponding author at: Institute of Structural Engineering, Zhejiang University, Room 610, Block (A), Anzhong Building, 338 Yuhangtang Road, Hangzhou 310058, PR China. Tel.: +86 (0)571 8820 8733x610; fax: +86 (0)571 8820 8733x601. E-mail addresses: [email protected] (L.-Y. Li), [email protected] (J. Xia), [email protected] (S.-S. Lin). 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.06.024
major variables influencing concrete diffusivity. Meijers et al. [2] developed a coupled transport model of heat, moisture and chloride ions in concrete in which the concrete was treated as a heterogeneous material consisting of three components, namely, mortar, aggregates and interfacial transition zones. The transport equations were solved numerically using finite element methods, in which the three components were meshed separately and continuity in fluxes at interfaces between them was applied. Yang and Su [3] investigated the effect of aggregate content on the chloride migration coefficient in concrete by using electrochemical migration tests. Based on the experimentally obtained results they developed an empirical formula in which the chloride migration coefficient was expressed as a function of the volume fraction of aggregate. Similar experiments, but using diffusion instead of electro-migration, were also conducted by Caré [4] to quantify the influence of aggregate and interfacial transition zone contents on chloride ingress in concrete. Caré and Herve [5] developed an analytical model for calculating the effective diffusion coefficient of ions in concrete, in which the concrete was also treated as a three-phase composite consisting of a cement continuous phase, an aggregates dispersed phase and an interface transition zone. Moon et al. [6] investigated the characteristics of capillary pores using mercury intrusion porosimetry techniques on 12 concrete specimens composed of six types of Portland and blended cements with water–binder ratios of 40% and 50% and examined the effect of capillary pores on chloride diffusion in concrete. It was found that apart from the capillary pores the
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average pore diameter also has a significant influence on chloride diffusion in concrete. Experiments using ponding tests [7] and accelerated chloride migration tests [8] were also reported in literature to quantify the influence of capillary porosity diameter and continuous pore diameter on the diffusion and migration coefficients of chloride ions in cement-based materials. Recently, Zeng [9] developed a two-dimensional structure model to simulate the chloride diffusion behaviour in concrete, in which the concrete was treated as a heterogeneous material composed of cement paste and aggregate phases with different diffusivities. The modelling results showed that the chloride diffusion in hetero-structured concretes appears to lag behind that for the homogeneous ones predicted using the effective diffusion coefficient, indicating an increasingly notable discrepancy between the two chloride concentration profiles as the diffusion proceeds. Jin et al. [10] developed a similar numerical model in which the concrete was treated as a three-phase composite consisting of a cement continuous phase, an aggregates dispersed phase and an interface transition zone. Zheng and Zhou [11] proposed a three-phase composite sphere model to represent the heterogeneous nature of concrete and derived a closed form expression for chloride diffusion in concrete. Later, by utilising the threephase composite sphere model Zheng et al. [12] further investigated the influence of the aggregate-cement paste interfacial transition zone on the steady-state chloride diffusion in mortars and concretes. In this paper, a mesoscopic structure model is proposed to simulate the chloride diffusion in concrete. The concrete is treated as a heterogeneous material composed of cement paste and aggregate two phases. The chloride diffusion is assumed to take place only in the cement paste phase. By modelling concretes with different aggregate volume fractions the effect of aggregate content on the chloride diffusion in concrete is examined. Finally, by using the concept of representative elementary volume, chloride concentration distribution profiles obtained from the mesoscopic and macroscopic structure models are also compared, which illustrates some important features appeared in the mesoscopic model but cannot be seen in the macroscopic structure models.
Dcem Dagg ð1 /ÞDcem þ /Dagg
ð2Þ
min where Dmax eff and Deff are the upper and lower bounds of the effective diffusion coefficient of chlorides in concrete, / is the volume fraction of the cement paste in the concrete, Dagg and Dcem are the diffusion coefficients of chlorides in the aggregate and cement paste phases, respectively. Considering the aggregate is relatively impermeable, it is reasonable to assume that Dagg = 0. In this case, the upper and lower bounds of the effective diffusion coefficient in Eqs. (1) and (2) min become Dmax eff ¼ /Dcem and Deff ¼ 0. For two-dimensional problems as illustrated in Fig. 1, the upper bound of the effective diffusion coefficient represented by the parallel model (see Fig. 1a) is still given by Eq. (1), but the lower bound of the effective diffusion coefficient represented by the series model (see Fig. 1b) is no longer zero. The reason for this is because the diffusion of chlorides in the two-dimensional parallel model can be still assumed to be only in the direction parallel to the longitudinal axis of the aggregate cylinders (see Fig. 1a), while the diffusion of chlorides in the two-dimensional series model can take place in any directions perpendicular to the longitudinal axis of the aggregate cylinders (see Fig. 1b). It is obvious that the parallel and series models in the two-dimensional case are more accurate than those in the one-dimensional case. Hence, if the lower bound of the effective diffusion coefficient represented by the two-dimensional series model can be predicted then one can estimate the effective diffusion coefficient of chlorides in real concrete by using the obtained upper and lower bound values. It is worth mentioning herein that, although the effective diffusion coefficient of chlorides in concrete can be predicted directly by using three-dimensional simulations, this kind of simulations is not only very time consuming but also extremely expensive in computing. Most of existing numerical studies used the twodimensional model to investigate the influence of individual components of concrete mixture on chloride diffusion [5,9–12]. However, the implication of the results obtained from the twodimensional simulation is not discussed.
3. FE model for determining the lower bound of the effective diffusion coefficient
2. Two-phase prediction models Concrete is a multi-phased heterogeneous material, which consists of aggregates, cement paste, and voids. The aggregates constitute a dense phase randomly distributed in a continuous cement paste matrix that contains numerous, almost uniformly distributed nano-sized gel pores and capillary pores. Since both gel pores and capillary pores are much small compared to the sizes of aggregates, they can be approximately considered as a homogeneous phase. When the concrete is fully saturated, ionic transport mainly takes place in the cement paste matrix through the connective pores. In contrast, the aggregates within the cement paste are usually of very poor diffusivity. Existing data have shown that the diffusion coefficient of chlorides in cement paste is well correlated with the water-to-cement ratio [13] and the relationship between them can be approximately expressed by an exponential function. However, the diffusion coefficient of chlorides in concrete depends on not only the water-to-cement ratio, but also the aggregates [14]. In the two-phase model, the concrete is assumed to be a two-phase composite. One is the homogeneous cement paste and the other is the aggregates. The upper and lower bounds of the effective diffusion coefficient of chlorides in concrete can be obtained by using the simple, one-dimensional parallel and series configuration assumptions of the two constituents, as follows [14],
Dmax eff ¼ ð1 /ÞDagg þ /Dcem
Dmin eff ¼
ð1Þ
In order to determine the lower bound of the effective diffusion coefficient of chlorides in concrete, a simple two-dimensional mesoscopic structure model of two phases is constructed here. Fig. 2 shows a schematic illustration of the model, in which the size of the plain concrete is 100 100 mm, all circular areas represent the aggregates and the rest part is the cement paste matrix. The distribution of aggregates of different sizes employed in the present plain concrete is determined by using a sieve analysis for a given aggregate volume fraction [15]. Several different types of ideal curves were determined on the basis of practical experiments and theoretical calculations. The most known and acceptable curve is the Fuller curve [16,17]. Therefore, Fuller mix [18] was exploited here to determine the size of an aggregate to be generated. Note that aggregate particles in real concrete may not be perfectly circular. However, as it will be demonstrated lately, the effect of aggregate shapes on the chloride diffusion in concrete seems very small. Chloride diffusion taking place in the cement paste matrix with a given water-to-cement ratio can be described using the wellknown diffusion equation as follows,
@C ¼ Dcem r2 C @t
ð3Þ
where C is the concentration of chlorides in the cement paste matrix (moles per unit volume of cement paste), t is the time, and r is the
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Fig. 2. The two-dimensional numerical model for calculating the lower bound of chloride diffusion coefficient in concrete (/ = 0.5). (a) Geometry (l = h = 100 mm) and (b) finite element mesh.
where C1 is the chloride concentration at the line x = 0, l = 100 mm and h = 100 mm are the length and height of the plain concrete used in the numerical model, respectively. For given values of Dcem and C1 one can solve Eq. (3) to obtain the chloride concentration distribution profile at any time. Of particular interest is the x-component of the chloride diffusion flux, Jx, at the steady state. The total chloride flux, q, along the boundary x = l at the steady state is given by,
q¼
Z
h o
J x ð1; l; yÞ dy ¼ Dcem
Z
h
o
@Cð1; l; yÞ dy @x
ð7Þ
The average flux, Jcon, in the plain concrete in x-direction thus is given by,
J con ¼ Fig. 1. Chloride diffusion in two-dimensional models (arrows represent the chloride diffusion directions). (a) The parallel model in which chloride diffusion takes place on in the direction parallel to the axis of cylinders and (b) the series model in which chloride diffusion takes place in the cross-section normal to the axis of cylinders.
Laplace differential operator. For simplicity, the initial condition and boundary conditions are assumed as follows,
Cð0; x; yÞ ¼ 0 Cðt; 0; yÞ ¼ C 1 ; @Cðt; x; 0Þ ¼ 0; @y
ð4Þ Cðt; l; yÞ ¼ 0 @Cðt; x; hÞ ¼0 @y
ð5Þ ð6Þ
q h
ð8Þ
By taking the plain concrete as a representative elementary volume in a macroscopic structure model, the average flux, Jcon, thus can be also expressed as,
J con ¼ Dmin eff
min @C Deff C 1 ¼ @x l
ð9Þ
Substituting Eqs. (7) and (9) into (8), it yields,
Dmin eff ¼
Dcem l C1h
Z
h o
@Cð1; l; yÞ dy @x
ð10Þ
Eq. (10) indicates that if the chloride flux in the cement paste matrix is computed from the present two-dimensional mesoscopic
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structure model then the lower bound of the effective diffusion coefficient of chlorides in concrete can be evaluated. 4. Results and discussion The partial differential equation defined by Eq. (3) with initial and boundary conditions defined by Eqs. (4)–(6) can be solved using finite element methods. Fig. 2b shows a typical finite element mesh used in the analysis for the plain concrete of aggregate volume fraction being 0.5. The analyses are performed for the plain concrete that has various different aggregate volume fractions and the corresponding lower bounds of the effective diffusion coefficient of chlorides in concrete calculated using Eq. (10) are plotted in Fig. 3. For the purpose of comparison, the upper bounds of the effective diffusion coefficient calculated using Eq. (1) and the theoretical predictions of the effective diffusion coefficient using Maxwell’s model [19] and Bruggeman equation [20] are also superimposed in the figure. It can be seen from the figure that both the lower and upper bounds of the effective diffusion coefficient decrease with the increase of the aggregate volume fraction. The gap between the lower and upper bounds also increases with the aggregate volume fraction. As is to be expected, the effective diffusion coefficients predicted by using Maxwell’s model and Bruggeman equation are between the lower and upper bounds of the effective diffusion coefficient. It is also observed from the figure that the prediction of Bruggeman equation is more close to the lower bound, whereas the prediction given by Maxwell’s model is almost in the middle of the lower and upper bounds. In order to examine the influence of aggregate size on the lower bound of the effective diffusion coefficient, numerical analyses for the plain concrete with different maximum diameters of circular aggregates are performed. Fig. 4 shows the influence of the maximum diameter of circular aggregates used in the plain concrete on the lower bound of the effective diffusion coefficient. The figure shows that increasing or reducing the maximum diameter of circular aggregates has almost no influence on the obtained results as long as the random distribution of the aggregates is created using the sieve analysis. This implies that the influence of the size of circular aggregates in the concrete on the lower bound of the effective diffusion coefficient is negligible. To examine the effect of aggregate shape on the lower bound of the effective diffusion coefficient, numerical analyses for the plain concrete with circular, ellipse, triangle, square and mixed aggregates are also performed. Fig. 5 shows the meshes used for the plain concrete with the four different aggregate shapes (mesh for the plain concrete with circular
Fig. 4. Comparison of the lower bounds of the effective diffusion coefficient in concrete with different maximum diameters of circular aggregates.
aggregates can be found in Fig. 2b). The comparisons of the lower bounds of the effective diffusion coefficient obtained from the plain concrete with circular, ellipse, triangle, square and mixed aggregates are provided in Fig. 6. It is observed from the figure that, except for the lower bound of the effective diffusion coefficient in the concrete of triangular aggregates that is marginally small, the lower bounds of the effective diffusion coefficients in all other four concretes are more or less the same. The small value of the lower bound of the effective diffusion coefficient found in the concrete of triangular aggregates is mainly attributed to the effect of tortuosity of the pore system. For the same volume fraction of aggregates the concrete of triangular aggregates is believed to have larger tortuosity than the concretes of other four aggregate shapes. This demonstrates that the chloride diffusion in concrete depends on not only the volume fraction of aggregates but also the tortuosity of the pore system. 5. FE model for determining the effective diffusion coefficient In order to demonstrate that the two-dimensional finite element model described in the preceding sections can indeed provide a good lower bound for the effective diffusion coefficient of chlorides in concrete, a three-dimensional finite element analysis model is developed herein. Because of the limitation in computing, only the concrete cube of 50 30 30 mm with spherical aggregates is considered (see Fig. 7). Analyses are performed for the concrete cube with different volume fractions of aggregates under a steady state case in which the front (x = 0) and rear (x = l) surfaces are assumed to have specified concentration boundary conditions and all other four surfaces are assumed to have zero flux boundary conditions. The effective diffusion coefficient is calculate using the following equation,
Deff ¼
Fig. 3. The lower and upper bounds of the effective diffusion coefficient of chlorides in concrete.
Dcem l C1 h
2
Z o
h
Z o
h
@Cð1; l; y; zÞ dy dz @x
ð11Þ
where h = 30 mm and l = 50 mm are the cross-section size and length of the cube. The effective diffusion coefficients obtained from the three-dimensional finite element analysis for different aggregate volume fractions are plotted in Fig. 8. It can be seen from the figure that the effective diffusion coefficient calculated from the three-dimensional model lies almost in the middle of the lower and upper bounds of the effective diffusion coefficient and agrees very well with the prediction given by Maxwell’s model. According to [19,21–23], the difference between the series and Maxwell’s models is the tortuosity, which in the series model is equal to
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299
Fig. 6. Comparison of the lower bounds of the effective diffusion coefficient in concrete with different aggregate shapes.
Fig. 7. Finite element analysis model for a cubic concrete with spherical aggregates (/ = 0.5).
Fig. 5. Finite element meshes employed in the plain concrete with different aggregate shapes (/ = 0.5). (a) Ellipse, (b) triangle, (c) square and (d) mixed aggregates.
Fig. 8. Comparison of effective diffusion coefficients of chlorides in concrete obtained from different models.
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one, whereas in the Maxwell’s model is equal to 2//(3 /). As is explained in Fig. 1, the difference between the series and parallel models in the two-dimensional case is also the tortuosity. Interestingly, if take the tortuosity as (1 + /)/(3 /) in the two-dimensional parallel model, then the following analytical solution can be used to predict the lower bound of the effective diffusion coefficient of chlorides in concrete,
Dmin eff ¼ Dcem
/ð1 þ /Þ 3/
ð12Þ
The demonstration of Eq. (12) is also shown in Fig. 8. 6. Non-steady state diffusion The most commonly used equations for chloride ingress in concrete are the Fick’s laws of diffusion. Fick developed two relations that are sometimes referred to as Fick’s first and second laws of diffusion. Fick’s first law is a constitutive equation that describes the relationship between the diffusive flux and concentration of the species. Fick’s second law is the conservation of mass of the species, which holds for non-steady state diffusion. When applying Fick’s laws to a porous material such as the concrete these two relations can be expressed as follows:
J con ¼ Deff rC
ð13Þ
@ð/CÞ ¼ rJ con @t
ð14Þ
Eq. (13) represents the flux of chlorides pass through the unit area of concrete in the unit time, whereas Eq. (14) is a mass conservation equation of chlorides in the unit volume of concrete. Substituting Eq. (13) into (14), it yields,
@C Deff 2 ¼ r C @t /
ð15Þ
Note that if the tortuosity could be neglected then we would have Deff = /Dcem. Hence, Eq. (15) can be interpreted as the diffusion of chlorides in the cement paste. This implies that the influence of the aggregates on the chloride diffusion in concrete is only by the tortuosity generated by the presence of aggregates rather than by the volume of the aggregates, if the mass of chlorides in concrete is defined based on the mole number of chlorides in the unit volume of cement paste. In order to demonstrate Eq. (15), Fig. 9 shows the comparisons of chloride concentration distribution profiles at three different times, obtained from the two-dimensional finite element analyses of the mesoscopic and macroscopic structure models. In the
Fig. 9. Comparison of chloride concentration profiles obtained from the mesoscopic and macroscopic models (/ = 0.5).
Fig. 10. A typical distribution profile of chloride concentration obtained from the mesoscopic model (/ = 0.5).
mesoscopic model, the governing equation is described by Eq. (3) with the initial and boundary conditions defined by Eqs. (4)–(6), whereas in the macroscopic model, the governing equation is described by Eq. (15) with the initial and boundary conditions also defined by Eqs. (4)–(6). Note that the concentration obtained from the macroscopic model is only the function of time and x-coordinate because of the symmetric nature of the problem, while that obtained from the mesoscopic model is the function of not only time and x-coordinate but also y-coordinate. For the purpose of the comparison, the chloride concentrations shown in Fig. 9 at a given time and a given x-value are the average value of the chlorides at the section from y = 0 to y = h. It is evident from the comparisons shown in the figure that, the two models match rather well, although the macroscopic model provides slightly higher concentration in the region near to the surface and lower concentration in the region far away from the surface. The variation of the chloride concentration with y-coordinate in the mesoscopic model depends on the structure and configuration of aggregates. Fig. 10 shows a typical distribution profile of the chloride concentration obtained from the present mesoscopic model. It can be seen from the figure that the concentration variation along y-axis is noticeable but not significant. 7. Conclusions This paper has presented an investigation on the chloride diffusion in concrete by using a two-phase model. From the results obtained the following conclusions can be drawn: (1) The two-dimensional series and parallel models can provide the upper and lower bounds of the effective diffusion coefficient of chlorides in concrete. The analytical expressions for the upper and lower bounds can be approximated as min Dmax eff ¼ Dcem / and Deff ¼ Dcem /ð1 þ /Þ=ð3 /Þ. (2) The size of aggregates has a negligible influence on the diffusion coefficient of chlorides in concrete as long as the random distribution of the aggregates is created using the sieve analysis. (3) The shape of aggregates has small influence on the diffusion coefficient of chlorides in concrete. This is believed to be attributed to the effect of aggregate shape on the tortuosity of the pore system. (4) The effective diffusion coefficient of chlorides in concrete can be approximated as the average of the upper and lower bounds of the effective diffusion coefficient given by the two dimensional series and parallel models.
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(5) The effective diffusion coefficient of chlorides in concrete predicted by using the three-dimensional finite element simulations is very close to that predicted using the Maxwell’s model. (6) The effective diffusion coefficient can be directly used in Fick’s first law. However, when it is applied to Fick’s second law the diffusion coefficient appeared in the finally deduced ‘‘diffusion equation’’ involves only the tortuosity effect. Finally, it should be stressed that the present model does not take into account the effect of the interfacial transition zone between the cement and aggregates. Thus, strictly speaking, the diffusion coefficient used for the cement paste in this paper should be the combined diffusion coefficient of chlorides in the cement paste and interfacial transition zone. Acknowledgement The work is supported by The Royal Society through the two international joint projects under the Grants (JP0867232 and JP090130), which is gratefully appreciated. References [1] Bentz DP, Garboczi EJ, Lagergren ES. Multi-scale microstructural modeling of concrete diffusivity: identification of significant variables. Cem Concr Aggr 1998;20(1):129–39. [2] Meijers SJH, Bijen JMJM, De Borst R, Fraaij ALA. Computational modelling of chloride ion transport in reinforced concrete. Heron 2001;46(3):207–16. [3] Yang CC, Su JK. Approximate migration coefficient of interfacial transition zone and the effect of aggregate content on the migration coefficient of mortar. Cem Concr Res 2002;32(10):1559–65. [4] Caré S. Influence of aggregates on chloride diffusion coefficient into mortar. Cem Concr Res 2003;33(7):1021–8. [5] Caré S, Herve E. Application of a n-phase model to the diffusion coefficient of chloride in mortar. Transp Porous Med 2004;56(2):119–35. [6] Moon HY, Kim HS, Choi DS. Relationship between average pore diameter and chloride diffusivity in various concretes. Constr Build Mater 2006;20(9): 725–32.
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