A numerical algorithm for evaluating the chloride diffusion coefficient of concrete with crushed aggregates

Construction and Building Materials 171 (2018) 977–983

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

A numerical algorithm for evaluating the chloride diffusion coefficient of concrete with crushed aggregates Jian-Jun Zheng, Jian Zhang, Xin-Zhu Zhou ⇑, Wen-Bing Song School of Civil Engineering and Architecture, Zhejiang University of Technology, Hangzhou 310023, PR China

h i g h l i g h t s
The chloride diffusion coefficient of concrete with crushed aggregates is evaluated.
The distribution of polygonal aggregates is simulated within a square element.
The validity of the numerical method is verified with two sets of experimental data.

a r t i c l e

i n f o

Article history: Received 16 November 2017 Received in revised form 4 March 2018 Accepted 21 March 2018

Keywords: Concrete Chloride diffusion coefficient Crushed aggregate Random walk algorithm

a b s t r a c t Since the deterioration of marine reinforced concrete structures is, to a great extent, related to the movement rate of chloride ions in concrete, it is essential to determine the chloride diffusion coefficient of concrete through experiment or theoretical prediction. This paper proposes a numerical algorithm for evaluating the chloride diffusion coefficient of concrete with crushed aggregates. In the numerical algorithm, the mesostructure of three-phase concrete is reconstructed by generating polygonal aggregates of various sizes and placing them within a simulation element with periodic boundary conditions. The random walk algorithm is then applied to the simulated three-phase concrete for computing the chloride diffusion coefficient. With this algorithm, the reasonable values of the random walk radius and the number of simulations are determined. Finally, comparisons are made between the calculation results and the experimental ones obtained from the literature to verify the numerical algorithm. Based on several numerical examples, three primary factors, the aggregate content, and the thickness and chloride diffusion coefficient of interfacial transition zone, influencing the chloride diffusion coefficient of concrete, are evaluated quantitatively. This paper concludes that the proposed numerical algorithm is effective in evaluating the chloride diffusion coefficient of concrete with crushed aggregates. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction It has been recognized from laboratory experiments and field observations that reinforcement corrosion due to the penetration of chloride ions is the primary cause accounting for the cracking, spalling, and even delaminating of reinforced concrete (RC) structures exposed to chloride environments. In the past thirty years, lots of such cases were reported around the world, such as major spalling in bridge decks [1], significant corrosion in jetty substructures [2], and severe surface cracking of concrete buildings [3]. Therefore, it is important to measure or predict the transport properties of concrete and to analyze the primary influential

⇑ Corresponding author. E-mail address: [email protected] (X.-Z. Zhou). https://doi.org/10.1016/j.conbuildmat.2018.03.184 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.

factors for the service life estimation and durability design of RC structures [4]. On the experimental side, Delagrave et al. [4] adopted two methods to test three series of mortars and found that sand modifies the microstructure by forming a porous interfacial transition zone (ITZ) on the surface. This is due to the fact that cement is a particulate material with a mean size around 10 lm, while the aggregate diameter is at least 0.15 mm in normal concrete. When encountering the much larger aggregate, the cement grains are subjected to the ‘‘packing” constraints imposed by the aggregate surface, resulting in a local increase in porosity. The overall effect of the ITZ, tortuosity, and dilution induced by the addition of sand particles leads to a decrease in chloride diffusion coefficient. However, Halamickova et al. [5] and Asbridge et al. [6] also showed that the ITZ becomes percolating once the sand content reaches a

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critical value, leading to an increase in chloride diffusion coefficient. Yang and Su [7] applied a migration test to estimate the transport properties of mortar and evaluated the effect of the aggregate content on the chloride diffusion coefficient. Based on experimental results, Caré [8] quantified the competing effects of the ITZ and the tortuosity and concluded that addition of aggregates results in a denser cement paste and hence a smaller chloride diffusion coefficient. On the theoretical side, various effective medium approximations and numerical techniques are used to analyze the chloride regress into cementitious materials. By introducing the concept of ITZ percolation and modeling mortar as an ordered periodic aggregate geometry, Garboczi et al. [9] expressed the chloride diffusion coefficient as a function of those of ITZ and cement paste. For low aggregate content concrete, the chloride diffusion coefficient and other properties were derived analytically [10]. By considering the characteristics of three phases in concrete, Caré and Herve [11] formulated the chloride diffusion coefficient. Zheng et al. [12] developed a semi-empirical, three-phase composite sphere model to evaluate the effect of the ITZ on the chloride diffusion coefficient of concrete. Aggregate shape was also shown to exert a certain influence on the transport coefficient of concrete [13]. Sun et al. [14] presented a multi-scale method to evaluate the chloride diffusion coefficient of concrete and analyzed the main influential factors. Šavija et al. [15] developed a lattice model to simulate the transport of chloride ions in sound and cracked concrete. With the ionic binding effect, Liu et al. [16] presented a two-dimensional, multi-component model for the ionic transport in concrete. Besides the ITZ and aggregate, they also evaluated the effect of ionic binding on the chloride diffusion coefficient. Li et al. [17] introduced the concept of double porosity to describe the ingress of chloride ions into concrete. Du et al. [18] evaluated the effects of various mesotructural parameters on the chloride diffusion coefficient of concrete. However, in the above-mentioned analytical methods, aggregates are all assumed to be spherical (circular) or ellipsoidal (elliptical) and hence they are not appropriate for concrete with crushed aggregates. In these numerical methods, heterogeneous concrete is reconstructed and the governing differential equations have to be solved for different random configurations, which proves to be time-consuming [19]. It is therefore essential to present an efficient numerical algorithm for estimating the chloride diffusion coefficient of concrete with crushed aggregates. This paper presents a numerical algorithm for simulating the mesostructure of concrete with crushed aggregates and for evaluating the chloride diffusion coefficient. In placing polygonal aggregates into a square element, periodic boundary conditions are introduced to avoid artificial wall effects. Based on the simulated mesostructure, the random walk method is applied to the computation of the chloride diffusion coefficient. Comparisons with experimental results verify the validity of the numerical algorithm and numerical examples are given to quantify the main influential factors.

2. Mesostructural simulation of concrete with polygonal aggregates At present, it is still difficult to simulate the mesostructure of concrete with polyhedral aggregates. For this reason, the random walk algorithm is limited to two-dimensional concrete with polygonal aggregates in this paper. Thus, aggregates are modeled as convex polygons and randomly distributed in the cement paste. For a convex polygonal aggregate, the diameter d is defined as the smallest width of all excribed rectangules which are just big enough to contain the particle wholly inside. From the sieve analysis, the

aggregate size distribution can be expressed, with respect to the number of aggregates, by a cumulative distribution function P(d) [20]. For the purpose of computer simulations, an element with sizes of a  a is chosen. When the aggregate area fraction fa, which is defined as the area ratio of all the aggregates to the simulation element, is given, the computer simulation procedure consists of the generation and distribution of aggregates. For convenience, W is defined as a random variable uniformly distributed on the interval (0, 1). The aggregates conformed to P (d) are generated as follows: 1. In general, the number of sides of a crushed aggregate is between 3 and 10 [21]. If there is no further information on this parameter, the number of sides mi of the i-th aggregate may be taken as a random integer variable uniformly distributed between 3 and 10, i.e.,

mi ¼ ½3 þ 8wi1 

ð1Þ

where wi1 is sampled from W and [x] denotes the largest integer smaller than or equal to x. 2. In the polar coordinate system, the j-th polar angle hj (j ¼ 1; 2; . . . ; mi ) of the aggregate can be obtained as

hj ¼ 2pwj2

ð2Þ

where wj2 is sampled from W. The polar angles are rearranged from the smallest to the largest. If the elongation ratio is controlled to vary randomly between A1 and A2, the corresponding polar radius rj is given by

r j ¼ A1 þ ðA2  A1 Þwj3

ð3Þ

where wj3 is sampled from W. 3. If the aggregate is concave, the coordinates are all invalid and need to be regenerated by returning to step 2 until it becomes convex. 4. The diameter di of the aggregate is given by solving the following equation

Pðdi Þ ¼ wi4

ð4Þ

where wi4 is sampled from W. The aggregate is then enlarged or shrunk in all directions so that the diameter is equal to di exactly. The area si is calculated and added up to the total area Si of i aggregates that have been generated so far, using

Si ¼ Sði1Þ þ si

ð5Þ

If Si P f a a2 , stop the aggregate generation, and the number of aggregates Na is equal to i. Otherwise, repeat steps 1 to 4 until Si P f a a2 . When these generated aggregates are placed within the simulation element, a periodic boundary condition is introduced to avoid any wall effects. The details are as follows: 1. The aggregates are rearranged from the largest to the smallest. 2. The center coordinates (xi, yi) of the i-th aggregate (i = 1, . . ., Na) are determined by

xi ¼ awi5 ;

yi ¼ awi6

ð6Þ

where wi5 and wi6 are both sampled from W. The aggregate (in black) will be situated completely within the element Fig. 1(a) or extend beyond the element Fig. 1(b) and (c). 3. If the aggregate extends beyond the element, additional aggregates (in grey) are generated at a distance of a from the aggregate in the two directions, as shown in Fig. 1(b) and (c). 4. If the newly placed aggregates do not overlap with those already placed, repeat steps 2 and 3 for the next aggregate.

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3. Determination of the chloride diffusion coefficient of concrete

y

y

x

(a)

x

(b)

In this paper, only saturated and uncracked concrete is considered. On the macroscopic level, chloride ions drived by a concentration gradient tend to diffuse from high to low concentration zones. On the microscopic level, the phenomenon is a statistical result of the random walks of a great number of chloride ions. According to the Einstein-Smoluchowski equation for twodimensional problems [19,22], the diffusion coefficient D is related 2

to the total mean square displacement hl i within the mean time t by 2

D¼

y

x

(c) Fig. 1. Polygonal aggregate situated: (a) within simulation element; (b) on left side; and (c) at upper left vertex.

hl i 4t

ð7Þ

This shows that the random walk method is an effective means for computing the chloride diffusion coefficient of concrete as follows. On one hand, concrete is viewed as an effective homogeneous medium of chloride diffusion coefficient Dcon. A Brownian particle starts a random walk from the center o until it hits the circumference C of a circle with radius R0 for the first time, as shown in Fig. 3. From Eq. (7), the mean hitting time can be expressed as 2 tðR0 Þ ¼ R0 4Dcon

ð8Þ

On the other hand, it is well known that, on the mesoscopic level, concrete is actually composed of aggregates, ITZs, and a cement paste, as shown in Fig. 4. Since the aggregate is considerably denser than the ITZ and the cement paste, it is reasonable to assume that the chloride diffusion coefficient of aggregate, Da, is zero, i.e.,

Da ¼ 0

ð9Þ

The Brownian particle still starts a random walk from the center o. Since the center is randomly selected, it is probably located in the aggregate, the ITZ, or the cement paste. If the center is in the aggregate, the Brownian particle will stay there forever, and if the center is in the ITZ or the cement paste, the Brownian can never enter the aggregate. Since the propability for the center to be in the aggregate is fa, the chloride diffusion coefficient of concrete will be reduced by a factor of (1 – fa) [19]. When the Brownian particle is situated within phase k, a maximum circle tangent to the interface between two phases centered at the Brownian particle is constructed, as shown in Fig. 5. On the circumference of the circle, a random point is selected as

hi ¼ 2pwi7

Fig. 2. Distribution of convex polygonal aggregates in cement paste at fa = 0.5.

5. Otherwise, the newly placed aggregates are all invalid and need to be regenerated by returning to step 2 until all the aggregates are placed within the element. Since the ITZ thickness h appears to be independent of the aggregate size [13], the mesostructure of concrete is reconstructed by putting an ITZ layer around each aggregate. As a computer simulation example, it is assumed that the aggregate size follows the Fuller curve, a = 60 mm, fa = 0.5, and the smallest and largest aggregate diameters are 0.3 and 9.5 mm, respectively [20]. With these inputs, the simulated distribution of polygonal aggregates is shown in Fig. 2, which shows that it looks like real concrete.

ð10Þ

where wi7 is sampled from W. Thus, the Brownian particle jumps directly to the random point and the mean hitting time tðri Þ is obtained from Eq. (7) as 2 tðr i Þ ¼ r i 4Dk

ð11Þ Concrete

Γ R0 o

Fig. 3. Effective homogeneous medium of concrete and constructed circle of radius R0.

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Cement paste

tðr j Þ ¼ R0

o

(


2 Dð2Þ r s 2ðDð1Þ þ Dð2Þ Þ Dð1Þ r j "

2
#) ð2Þ ð1Þ D D rs rs rj rs rj arctan þ þ  þ rj rj rs rj rs pDð1Þ r 2j

1

ð14Þ

where D(1) and D(2) denote the chloride diffusion coefficients of phases 1 and 2, respectively. Thus, the total mean time that the Brownian particle hits the circumference C of the circle with radius R0for the first time is equal to

ITZ Aggregate Fig. 4. Heterogeneous medium of concrete encompassing polygonal aggregates, ITZs, and cement paste.

tðR0 Þ ¼

X X tðr i Þ þ tðr j Þ i

ð15Þ

j

By equating Eqs. (8) and (15) and considering the reduction coefficient (1 – fa), we can formulate the chloride diffusion coefficient of concrete as ri

Dcon ¼

o

ð1  f a ÞR20 X X tðr j ÞÞ 4ð tðr i Þ þ i

ð16Þ

j

4. Determination of radius R0 and number of simulations M Fig. 5. Constructed largest circle tangent to interface centered at Brownian particle.

where ri is the radius of the circle and Dk is the chloride diffusion coefficient of phase k. When the Brownian particle is situated a point x near the interface between the ITZ and the cement paste within a prescribed distance, rs denotes the distance between x and its projection x0on to the interface. A circle with radius rj centered at x0 is constructed, as shown in Fig. 6. Since rs is much smaller than rj, the curved interface can be approximated as a straight one. If the interface divides the circumference into C1 and C2, the probabilities p1 and p2 that the Brownian particle hits C1 and C2, 

respectively, and the mean hitting time t ðr j Þ are given by [22]

p1 ¼

Dð1Þ Dð1Þ þ Dð2Þ

"

1þ

4Dð2Þ

pðDð1Þ þ Dð2Þ Þ

arctan


# rs rj

p2 ¼ 1  p1

ð12Þ ð13Þ

Phase 1

Γ1

x

rj

rs

It can be seen from the above algorithm that the walk path of a Brownian particle is a zigzag line composed of straight line segments. Owing to the finite element size, the Brownian particle possibly moves beyond the element. To overcome this difficulty, a periodic boundary condition is introduced, i.e., the extended part of a straight line segment is shifted to the opposite side, as shown in Fig. 7. To compute the chloride diffusion coefficient of concrete, it is important to determine the reasonable value of R0. As expected, the larger R0 is, the more sufficiently the Brownian particle explores the mesostructure of concrete and hence the more accurate Dcon is, but at the cost of computational time. It is also found that, for a given R0, a different initial random seed will result in a slightly different value of Dcon. This slight difference is caused by the finite element size. To eliminate the difference, the ergodic hypothesis is introduced [23]. With the hypothesis, the computer simulation is performed for a sufficiently large number M until the average value of Dcon tends to be smooth and stable. To determine the reasonable values of R0 and M, the Fuller aggregate gradation is considered. a = 28.5 mm, fa = 0.2, h = 0.03 mm, Dcp = 8.21  1012 m2/s, Di = 43.0  1012 m2/s, and the smallest and largest aggregate diameters are 0.30 and 9.5 mm, respectively. With these inputs, the mesostructure of concrete is simulated and the walk path of a Brownian particle is shown in Fig. 8 for R0 = 5, 10, 15, and 20 mm, respectively. Fig. 8(a) shows that, when R0 < 5 mm, the Brownian particle wonders in the vicinity of the upper right corner. When R0 increases from 5 to 10 mm, the Brownian particle passes through the upper side and walks near the lower side. After a while, it goes back to the upper right

z

x0

y x

x1

y1 z1

y1

Interface z1 y1

Phase 2

Γ2

Fig. 6. Constructed circle centered at Brownian particle situated near interface.

w x y z

(a)

(b)

Fig. 7. Reflection of random walk path of Brownian particle situated near (a) upper side; and (b) lower right corner.

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start

start

end

(a)

(b)

end start

(c)

start

end

(d)

Fig. 8. Random walk path of a Brownian particle at: (a) R0 = 5 mm; (b) R0 = 10 mm; (c) R0 = 15 mm; and (d) R0 = 20 mm.

Fig. 9. Variation of chloride diffusion coefficient Dcon with radius R0.

corner and moves toward the central region, as shown in Fig. 8(b). When R0 increases from 10 to 20 mm, the Brownian particle first moves toward the upper left corner as shown in Fig. 8(c), then wonders there, and finally crosses the left side and stops near the right side, as shown in Fig. 8(d). For M = 300, Dcon is plotted against R0 as shown in Fig. 9, which shows that, when R0 is smaller than 6 mm, Dcon increases with increasing R0. When R0 exceeds 12 mm, Dcon tends to be constant. For R0 = 12 mm, Dcon is plotted against M as shown in Fig. 10. Fig. 10 shows that, when M is smaller than 150, Dcon fluctuates. When M is larger than 150, Dcon tends to be a fixed value. Therefore, R0 and M are respectively taken as 12 mm and 150 in all the following computations.

Fig. 10. Variation of chloride diffusion coefficient Dcon with number of simulations M.

5. Experimental verification and discussions To verify the proposed numerical algorithm, the experimental data of Yang and Su [7] and Zheng et al. [13] are considered. In the first test, ASTM Type I Portland cement was adopted to cast cement paste and concrete specimens with a water/cement ratio of 0.4 and aggregate contents of 0.1, 0.2, 0.3, and 0.4, respectively. The aggregate diameters ranged from 0.15 to 9.50 mm. After the specimens were cast for one day, they were demoulded. Then, the specimens were cured in water with a temperature of 23 °C. After one year of curing, the chloride diffusion coefficients of these specimens were measured with a migration test, as shown in

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Fig. 11. Comparison between calculated values and experimental data of Yang and Su [7].

Fig. 11. To perform the random walk of Brownian particles, three parameters, h, Dcp, and Di need to be defined. Since h ranges from 0.01 to 0.05 mm [24], it is taken as the average, i.e., 0.03 mm. By definition, Dcp was measured as 2.03  1012 m2/s. As it is very difficult to directly determine Di, it can be obtained by the inversion method from measured Dcon = 1.34  1012 m2/s at fa = 0.4 that Di = 5.58  1012 m2/s. Thus, the proposed numerical algorithm is applied for the computation of Dcon. A comparison is made in Fig. 11, which demonstrates that the numerical algorithm is in excellent agreement with the measured values with a relative error of 2.6%, 2.1%, and 1.5% at fa = 0.1, 0.2, and 0.3, respectively. In the second test [13], an equivalent ASTM Type I Portland cement was adopted to cast cement paste and concrete specimens with a water/cement ratio of 0.5 and aggregate contents of 0.3, 0.4, 0.5, 0.6, and 0.7, respectively. The aggregate gradation followed the Fuller curve and the diameters ranged from 0.30 to 9.5 mm. After one day the specimens were cast, they were demoulded. Then,

Fig. 12. Comparison between calculated values and experimental data of Zheng et al. [13].

Fig. 13. Variation of chloride diffusion coefficient of concrete with aggregate content for different chloride diffusion coefficients of ITZ.

the specimens were cured in water with a temperature of 21 °C. After 28 days of curing, an accelerated method was used to measure the chloride diffusion coefficients of these specimens [25], as shown in Fig. 12. As in the last example, h = 0.03 mm. Dcp was measured as 8.21  1012 m2/s and Di is experimentally calibrated as 43  1012 m2/s from the measured Dcon = 4.10  1012 m2/s at fa = 0.7. These parameters are then input to calculate Dcon. A comparison is made in Fig. 12, which again demonstrates that the numerical algorithm agrees well with the measured values with a relative error of 7.95%, 1.49%, 1.76%, and 0.737% at fa = 0.3, 0.4, 0.5, and 0.6, respectively. It is therefore validated that the proposed numerical algorithm is effective in evaluating the chloride diffusion coefficient of concrete. It is seen from the above numerical algorithm that the aggregate content and the thickness and chloride diffusion coefficient of ITZ are three primary factors influencing the chloride diffusion coefficient of concrete. In what follows, the aggregate gradation, the smallest and largest aggregate diameters, and the chloride diffusion coefficient of cement paste are set to be the same as in the first verification example. First, h = 0.03 mm and Di = 3.50  1012, 5.58  1012, and 8.50  1012 m2/s, respectively, to examine the effects of fa and Di on Dcon. The calculated results are shown in Fig. 13. As expected, Dcon increases with increasing Di. Dcon at Di = 8.5  1012 m2/s is larger than that at Di = 3.5  1012 m2/s by 2.45%, 5.62%, 7.62%, and 9.72% at fa = 0.1, 0.2, 0.3, and 0.4, respectively. Fig. 13 also shows that, for a given Di, Dcon decreases with the increase of fa. This can be explained as follows. When impermeable aggregates are added to the cement paste, they cause the effects of dilution and tortuosity, which reduce Dcon. At the same time, an ITZ layer with a higher porosity is formed around each aggregate. The effects of ITZ and percolation between neighboring ITZs increase Dcon. Fig. 13 indicates that the former two effects become dominating and therefore Dcon is a decreasing function of the aggregate content. When fa increases from 0.1 to 0.4, Dcon decreases by 43.8%, 38.7%, and 34.2% at Di = 3.5  1012, 5.58  1012, and 8.5  1012 m2/s, respectively. Second, Di = 5.58  1012 m2/s and h is equal to 0.01, 0.03, and 0.05 mm, respectively, to quantify the effect of h on Dcon, as shown in Fig. 14, which demonstrates that Dcon increases with the increase of h for a given fa. This is attributed to the fact that a larger h results

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Acknowledgements The financial support from the National Natural Science Foundation (Grant No. 51779227) and the Science and Technology Plan Project of Zhejiang Province (Grant No. 2016C33106), of the People’s Republic of China, is gratefully acknowledged. References

Fig. 14. Variation of chloride diffusion coefficient of concrete with aggregate content for different ITZ thicknesses.

in a larger ITZ content [26]. When fa = 0.1, 0.2, 0.3, and 0.4, Dcon at h = 0.05 mm is larger than that at h = 0.01 mm by 2.77%, 5.69%, 8.02%, and 12.62%, respectively. It should be pointed out that, besides the diffusion of chloride ions, the random walk algorithm can also be applied to the transport of other ions, such as hydroxyl, sulphate, and sodium, and the water permeability of concrete. These will be investigated in future studies.

6. Conclusion A numerical algorithm has been presented for evaluating the chloride diffusion coefficient of concrete with crushed aggregates. The distribution of polygonal aggregates and ITZs has been realized by computer simulation within a square element subject to periodic boundary conditions. The random walk algorithm has been applied for the computation of the chloride diffusion coefficient of concrete and for the determination of the random walk radius and the number of simulations. Comparisons with experimental data obtained from the literature have verified the validity of the numerical algorithm. Several numerical examples have been given to examine the effects of the aggregate content, the ITZ thickness, and the chloride diffusion coefficient of ITZ on the chloride diffusion coefficient of concrete. It has been demonstrated that, the larger the aggregate content is, the smaller the chloride diffusion coefficient is. A larger chloride diffusion coefficient or thickness of ITZ leads to a larger chloride diffusion coefficient. It is concluded that the proposed numerical algorithm is effective in evaluating the chloride diffusion coefficient of concrete with crushed aggregates.

Conflict of interest There is no conflict of interest.

[1] P.D. Cady, R.E. Weyers, Deterioration rates of bridge concrete decks, J. Transp. Eng. 110 (1) (1984) 34–44. [2] K.C. Liam, S.K. Roy, D.O. Northwood, Chloride ingress measurements and corrosion potential mapping study of a 24-year-old reinforced-concrete jetty structure in a tropical marine environment, Mag. Concr. Res. 44 (160) (1992) 205–215. [3] N.R. Buenfeld, G.K. Glass, A.M. Hassanein, J.Z. Zhang, Chloride transport in concrete subjected to electric field, J. Mater. Civ. Eng. 10 (4) (1998) 220–228. [4] A. Delagrave, J.P. Bigas, J.P. Ollivier, J. Marchand, M. Pigeon, Influence of the interfacial zone on the chloride diffusivity of mortars, Adv. Cem. Based Mater. 5 (3–4) (1997) 86–92. [5] P. Halamickova, R.J. Detwiler, D.P. Bentz, E.J. Garboczi, Water permeability and chloride ion diffusion in portland cement mortars: relationship to sand content and critical pore diameter, Cem. Concr. Res. 25 (4) (1995) 790–802. [6] A.H. Asbridge, G.A. Chadbourn, C.L. Page, Effects of metakaolin and the interfacial transition zone on the diffusion of chloride ions through cement mortars, Cem. Concr. Res. 31 (11) (2001) 1567–1572. [7] C.C. Yang, J.K Su, Approximate migration coefficient of interfacial transition zone and the effect of the aggregate content on the migration coefficient of mortar, Cem. Concr. Res. 32 (10) (2002) 1559–1565. [8] S. Caré, Influence of aggregates on chloride diffusion coefficient into mortar, Cem. Concr. Res. 33 (7) (2003) 1021–1028. [9] E.J. Garboczi, L.M. Schwartz, D.P. Bentz, Modeling the influence of the interfacial zone on the DC electrical conductivity of mortar, Adv. Cem. Based Mater. 2 (5) (1995) 169–181. [10] E.J. Garboczi, D.P. Bentz, Analytical formulas for interfacial transition zone properties, Adv. Cem. Based Mater. 6 (3–4) (1997) 99–108. [11] S. Caré, E. Hervé, Application of a n-phase model to the diffusion coefficient of chloride in mortar, Transp. Porous Media 56 (2) (2004) 119–135. [12] J.J. Zheng, H.S. Wong, N.R. Buenfeld, Assessing the influence of ITZ on the steady-state chloride diffusivity of concrete using a numerical model, Cem. Concr. Res. 39 (9) (2009) 805–813. [13] J.J. Zheng, X.Z. Zhou, Y.F. Wu, X.Y. Jin, A numerical method for the chloride diffusivity in concrete with aggregate shape effect, Constr. Build. Mater. 31 (2012) 151–156. [14] G.W. Sun, Y.S. Zhang, W. Sun, Z.Y. Liu, C.H. Wang, Multi-scale prediction of the effective chloride diffusion coefficient of concrete, Constr. Build. Mater. 25 (2011) 3820–3831. [15] B. Šavija, J. Pacheco, E. Schlangen, Lattice modeling of chloride diffusion in sound and cracked concrete, Cem. Concr. Comp. 42 (2013) 30–40. [16] Q.F. Liu, D. Easterbrook, J. Yang, L.Y. Li, A three-phase, multi-component ionic transport model for simulation of chloride penetration in concrete, Eng. Struct. 86 (2015) 122–133. [17] L.Y. Li, D. Easterbrook, J. Xia, W.L. Jin, Numerical simulation of chloride penetration in concrete in rapid chloride migration tests, Cem. Concr. Comp. 63 (2015) 113–121. [18] X.L. Du, L. Jin, G.W. Ma, A meso-scale numerical method for the simulation of chloride diffusivity in concrete, Finite Elem. Anal. Des. 85 (2014) 87–100. [19] I.C. Kim, S. Torquato, Effective conductivity of suspensions of hard spheres by Brownian motion simulation, J. Appl. Phys. 69 (4) (1991) 2280–2289. [20] J.J. Zheng, C.Q. Li, L.Y. Zhao, Simulation of two-dimensional aggregate distribution with wall effect, J. Mater. Civ. Eng. 15 (6) (2003) 506–510. [21] Z.W. Wang, A.K.W. Kwan, H.C. Chan, Mesoscopic study of concrete I: generation of random aggregate structure and finite element mesh, Comput. Struct. 70 (5) (1999) 533–544. [22] I.C. Kim, S. Torquato, Determination of the effective conductivity of heterogeneous media by Brownian motion simulation, J. Appl. Phys. 68 (8) (1990) 3892–3903. [23] S. Torquato, Random Heterogenerous Materials: Microstructure and Macroscopic Properties, Springer-Verlag, New York, 2002. [24] J.J. Zheng, C.Q. Li, X.Z. Zhou, Thickness of interfacial transition zone and cement content profiles around aggregates, Mag. Concr. Res. 57 (7) (2005) 397–406. [25] X.Y. Lu, Application of the Nernst-Einstein equation to concrete, Cem. Concr. Res. 27 (2) (1997) 293–302. [26] J.J. Zheng, F.F. Xiong, Z.M. Wu, W.L. Jin, A numerical algorithm for the ITZ area fraction in concrete with elliptical aggregate particles, Mag. Concr. Res. 61 (2) (2009) 109–117.