multi-scale modeling of the effective chloride ion diffusion coefficient in cement-based composite...

364 Vol.27 No.2 SUN Guowen et al: Multi-scale Modeling of the Effective Chloride Ion Di...

Multi-scale Modeling of the Effective Chloride Ion Diffusion Coeffi cient in Cement-based Composite Materials

SUN Guowen1, 2, SUN Wei1*, ZHANG Yunsheng1, LIU Zhiyong1

(1.Jiangsu Key Laboratory of Construction Materials, Southeast University , Nanjng 211189, China; 2.School of Materials Science and Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China)

Abstract: N-layered spherical inclusions model was used to calculate the effective diffusion coeffi cient of chloride ion in cement-based materials by using multi-scale method and then to investigate the relationship between the diffusivity and the microstructure of cement-basted materials where the microstructure included the interfacial transition zone (ITZ) between the aggregates and the bulk cement pastes as well as the microstructure of the bulk cement paste itself. For the convenience of applications, the mortar and concrete were considered as a four-phase spherical model, consisting of cement continuous phase, dispersed aggregates phase, interface transition zone and their homogenized effective medium phase. A general effective medium equation was established to calculate the diffusion coeffi cient of the hardened cement paste by considering the microstructure. During calculation, the tortuosity (n) and constrictivity factors (Ds/D0) of pore in the hardened pastes are n≈3.2, Ds/D0=1.0×104 respectively from the test data. The calculated results using the n-layered spherical inclusions model are in good agreement with the experimental results; The effective diffusion coefficient of ITZ is 12 times that of the bulk cement for mortar and 17 times for concrete due to the difference between particle size distribution and the volume fraction of aggregates in mortar and concrete.

Key words: multi-scale; chloride diffusion coefficient; cement-based composite materials; general effective medium theory; composite spheres model; microstructure

©Wuhan University of Technology and SpringerVerlag Berlin Heidelberg 2012(Received: Sep. 16, 2011; Accepted: Nov. 2, 2011)

SUN Guowen (孙国文): Ph D; E-mail: [email protected]*Corresponding author: SUN Wei (孙伟): Prof.; E-mail:

[email protected]. cnFunded by the National Basic Research Program of China

(No.2009CB623203), the National High-Tech R&D Program of China (No.2008AA030794) and the Postgraduates Research Innovation in University of Jiangsu Province in China (No.CX10B-064Z)

DOI 10.1007/s11595-012-0467-6

1 Introduction

In practice, chloride ions penetrate into concrete and induce the corrosion of embedded reinforcing steel, which further leads to premature deterioration of concrete structure because of typical porous features of concrete. However, the initial time of steel corrosion largely depends on the chloride ion diffusion coefficient[1]. At present, the chloride ion diffusion coeffi cient is obtained by long-term chloride ponding test[2]. But this kind of method is time-consuming and can not meet practical requirements. The electrochemical accelerated testing methods[1,3] are often applied in practice, but different researchers have different results even for the identical materials

due to the differences in applied voltage and thickness of specimens tested or other experimental errors et al. If a reliable and simple model considering the microstructure information could be proposed to determine the chloride diffusivity coefficient, it is very useful for the estimation of durability. Only if the determination of the relationship between chloride diffusivity and the evolution of transport channels can timely realize chloride ions transport behavior and accurately evaluate the durable conditions of concrete. The transport channels of chloride ion diffusion are influenced by many factors. However, the most important influencing factors are the interfacial zone property and the cement paste itself (porosity and pore structure).Thus, it is important to build a model taken into account two factors mentioned above.

Concrete is a multiphase composite material with a random microstructure at different length scales ranging from the nanometer scale to the macroscopic decimeter scale. At the macroscopic scale, concrete consists of coarse aggregate embedded in a matrix of mortar, while the mortar consists of sand embedded in a matrix of hydrated cement paste at the mesoscopic scale. At the microscopic scale, the hydrated cement

Journal of Wuhan University of Technology-Mater. Sci. Ed. Apr.2012 365

paste consists of amorphous C-S-H and crystalline calcium hydroxide, containing an extensive network of capillary pores, plus unhydrated cement and submicroscopic calcium sulfoaluminate hydrate crystals. The problem is how to build the relationship between the overall chloride diffusion coefficient and the microstructure of cement-basted materials in terms of the characteristics at different scales. Fortunately, multi-scale modeling techniques[4] offer a promising solution to this problem. The microscopic information is put into the mesoscopic scale, in this way, the mesoscopic information is achieved, which is again put into the macroscopic scale. Finally, the relation between the overall properties and microstructure is built. However the microstructure information at nano-scale is still quite complex. For the convenience of applications, the mortar or concrete is treated as a four-phase sphere model, consisting of continuous cement phase, dispersed aggregates phase, interface transition zone and their homogenized effective medium phase in this paper. This simply multi-scale model is more suitable in practice.

The objective in this paper was to propose a simple model to estimate the diffusivity of cement-based materials. The key feature of this model proposed was taking into account the interfacial transition zone as well as the microstructure of the bulk cement paste itself based on a multi-scale approach. In addition, many series of concrete and mortar were tested to verify the proposed model on chloride ion diffusivities.

2 Representation of the multi-s c a l e m i c r o s t r u c t u r e i n cement-based materials

Concrete is a fairly complex heterogeneous composite material, with a random microstructure at different length scales ranging from the nanometer

scale to the macroscopic decimeter scale. For chloride ion diffusion problem, the microstructure can be broken down into three elementary scales[5], as sketched in Fig.1.

The microscopic scale (10−9-10−6 m) considers the pore features. These pores can be characterized by different parameters: porosity, tortuosity and constrictivity. At microscopic scale, the effective chloride diffusion coeffi cient (Deff) can be expressed as the following form:

where, D0 is diffusion coefficient of chloride ion in bulk water andcap capillary porosity, while τ and δ are tortuosity and constrictivity of pore structure.

The mesoscopic scale (10−6-10−3 m) corresponds to a ‘theoretically homogeneous’ material including cement paste and aggregates. At this scale, mortar or concrete is considered as a three-phase composite material composed of aggregate particle inclusions, interfacial transition zone (ITZ) and cement paste matrix, corresponding to 1, 2 and 3, respectively, in Fig.1. So the effective chloride diffusion coefficient (Deff) can be given as the following form:

where, Va and VI are the volume fraction of aggregate and ITZ, respectively, while Da, DB and DI the chloride diffusion coefficient of aggregate, bulk cement paste and ITZ, respectively.

The macroscopic scale (the order of magnitude is 10−2 m) corresponds to the size of specimens tested. At this scale, the effective diffusion coeffi cient (Deff) is directly determined by experiment.

3 Transport model of chloride ion in cement-based materials

3.1 Geometry model selected of cement-based materials


The cement-based composite materials are composed of millions of fine inclusions embedded in a matrix phase. At the mesoscopic scale, the transportation properties of the cement-based composite materials can be replaced with equivalent medium possessed the same diffusion properties at the macroscopic scale. The equivalent diffusion properties are also called the effective diffusivity coefficient of composite materials.

For the sake of solving the effective diffusivity coefficient in cement-based composite materials, a geometry model is fi rstly introduced in this paper. The geometry morphology of model is desired to approach actual materials as much as possible. As said in the introduction, the composition and microstructure of cement-based composite materials are quite complex at different scales, so that it can not be simply treated as two-phase composite materials. Therefore, the composite spherical geometry model proposed by Hasin[6] is adopted in this paper, as depicted in Fig.2 (two-dimensional plane). It can be seen from Fig.2 that the geometric model is composed of the millions of spherical inclusions with gradual change sizes phases and matrix phases. The advantage of composite spherical model is rather convenient for aspects of the solution and application of effective properties. It is noted that the composite spheres possess exactly the same volume fraction for each of the elementary phases and occupy the entire volume of the material.3.2 Transport model at the mesoscopic scale

In the case of prediction the diffusion coeffi cient, the (n+1)-phase model, as shown in Fig.3, used in elasticity, viscoelasticity, elastoplasticity[7,8] and even in thermal and thermoelastic behaviors[9], is extended to the case of ionic diffusion of cement-based composite materials in this paper [5]. It is emphasized that all the initial phases are supposed to be isotropic and homogeneous and so is the macroscopic behavior of the material. Based on the approximate method of

general self-consistent scheme, the analytic solution of composite spheres assemblage model have been given by Hervé[9]. The advantage of general self-consistent scheme can estimate the effective diffusivity coefficient composed of multi-size spheres. The process of estimation can be seen from Fig.3 that phase 1 constitutes the central core and phase (i) lies within the shell limited by the two concentric spheres with the radii Ri−1 and Ri. Accordingly, the effective diffusivity coeffi cient of n-phase constituted composite assemblage spheres equals to the diffusion coeffi cient of the n+1 phase of outer regions. In the case of chloride diffusivity in cement-based materials, the analytic solution is expressed as:

(1)

where,Di is the effective diffusivity coefficient of the ith phase, Di

eff the effective diffusivity coefficient of the composite assemblage spheres made up of the fi rst phase, i the phase and Ri and Ri1 the two concentric spheres with the radii Ri and Ri1.

For the sake of convenient application in practice, Eq.(1) is further simplifi ed four-phase model to solve the problem to chloride ion diffusion in cement-based materials as depicted in Fig.4. In Fig.4, aggregates considered as spherical inclusions with the volume


fraction Va and the diffusion coefficient Da; the interfacial transition zone (ITZ) coated aggregates with the volume fraction VI and the diffusion coefficient DI; aggregate and ITZ treated as coated inclusions are embedded into the bulk cement paste with the diffusion coefficient DB. To obtain the effective diffusion coeffi cient of cement-based materials, Eq.(1) is treated with as follows:

Aggregates are considered as a homogeneous material:

(2)

Aggregates and ITZ are treated as two-phase material where aggregate as inclusions embedded into the matrix of ITZ formed. In this way, the effective diffusivity coefficient of two-phase material is given by:

(3)

where

(4)

Aggregates surrounded by the ITZ as coated inclusions embedded into the matrix of bulk cement paste. The effective diffusivity coefficient of three- phase material is given by:

(5)

where

(6)

Finally, substituting Eqs.(2), (3), (4) and Eq.(6) into Eq.(5), the effective diffusivity coefficient of cement- based materials can be expressed as:

(7a)

where

(7b)

(7c)

Eq.(7) fully takes into account the materials with porous aggregates and ITZ, so it is also general

equation. Several special cases of cement-based materials can be considered.

When ITZ is ignored (VI=0) and aggregates are porous (Da≠0), there is the relation of DB=DI=Dp. In this case, Eq.(7) can be expressed as:

(8)

where, Dp is the diffusivity coeffi cient of cement paste.When there is no ITZ (VI=0) and aggregates

are not penetrability (Da=0), there is a relation of DB=DI=Dp. Eq.(7) is simplifi ed as follows:

(9)

When the diffusion coefficient of the aggregates is zero (Da=0), Eq.(7) can be given as follows:

( 10)

Assumed that aggregates are impermeable in this paper, Eq.(10) is used to predict the effective chloride diffusivity coeffi cient of cement-based. From Eq.(10), it can be seen that the evaluation of the effective diffuse properties for the material needs the knowledge of properties of each elementary phase and the factors influencing on chloride diffusivity are the chloride diffusivity of matrix, volume fraction of aggregate, the chloride diffusivity of ITZ and the volume fraction of ITZ. The chloride ion diffusion coefficient of matrix and ITZ depends on the mix proportions of cement-based materials. Therefore, the main parameters in Eq.(10) are four factors mentioned. Among them, the volume fraction of aggregate could be directly obtained by the mix proportion of cement-based materials, while the other three parameters will be discussed as follows.3.3 Determination of volume fraction of ITZ

Based on the statistical geometry of composites, Lu and Torquato[10] put forward nearest-surface distribution functions, which can be used to predict


the volume fraction of multi-size spherical particles packed. Shane[11] pointed out that when the volume fraction of aggregates in cement-based materials per unit volume exceeds 40%, the overlapping degree of ITZ between aggregate and bulk cement paste is very large. Fig.5 shows a schematic diagram in two dimensions. In Lu and Torquato′s theory, the overlapping layers between spherical particles packed was fully taken into account and Garboczi and Bentz[12] fi rstly applied the theory to predict the volume fraction of ITZ in concrete, where the concrete is modeled as three-phase composite materials at the mesoscopic scale, which is made of spherical aggregate particles, interfacial transition zones and the bulk cement paste.

According to Lu and Torquato′s model, the ITZ volume fraction (VITZ) is:

(11a)

where

(11b)

(11c)

( 11d)

where, NV is the total number of particles per unit volume, tITZ is ITZ thickness and Va is the volume fraction of aggregates in the concrete; A is a coeffi cient that can have different values (0, 2, or 3) according to the analytical approximation chosen in the theory[10]; c, d and g are determined based on averages over the particle size distributions of the aggregates in terms of number.

During calculating the volume fraction of ITZ, the ITZ is modeled as a uniform property region. The ITZ thickness depends on the median size of the cement grains, and not on the aggregate size[12]. As can be seen from Eq.(11), the infl uencing factors infl uencing on the ITZ volume fraction are the aggregate gradation, the volume fraction of aggregates and the ITZ thickness. For a given concrete mixture, materials density and sieve analysis, these variables are known or can be determined. 3 . 4 P r e d i c t i o n o f e f f e c t i v e d i f f u s i o n

coeffi cient of hardened cement pasteThe hardened cement paste is composed of

unhydrated cement particles, hydration products and

pore networks occupied by the bulk water and air at normal temperature, which is also regarded as a two-phase composite in saturated state[14]. The first phase is the capillary pore phase, and the second is the solid phase consisting of various hydration products as described in Fig.6(a). The pore network of cement paste comprises interlayer pores, gel pores, capillary pores and macropore, which are transport channels of aggressive medium intruded in the hardened cement paste. Here, Maekawa et al[13] assumed that no ion is transported into the interlayer pores of the cement paste since the molecular-related size of the interlayer space is too small to allow substantial room of any ion. So, the latter three types of pores are taken into account in this paper, and schematic diagram of structural model of pores of hardened cement paste is depicted in Fig.6(b).

In this way, these pore spaces in the hardened cement paste in Fig.6(b) can be simplifi ed a continuous pore with tortuous effect, a continuous ink-bottle pore, a dead-end ink-bottle pore and an isolated pore, where the former two kinds of pores is also called the effective pores, which is played important role in transportation of aggressive medium. The dead-end pores and isolated pores do not contribute to the transport properties. Usually, continuous tortuous pore can be expressed by the parameter of tortuosity, while the continuous pore with ink-bottle can be described by the constrictivity, since the constrictivity take into account the interaction between pore structure and ion transport.


In terms of effective media theory, the effective diffusivity of a porous material can be expressed by Eq. (12)[14].

(12)

where, Dp is the effective diffusivity of a porous material (m2/s); cap the capillary porosity; τ and δ the tortuosity factor and constrictivity of the pore network, respectively. D0 is the diffusivity of transport ion in bulk water, such as D0=2.03×109 m2/s for chloride ion at 25 ℃.

τ and δ value depend on the choice of pore structure model and they are difficult to directly measure by experiment, therefore, it is not convenient for Eq.(12). In fact, general composite theories can be used to determine the pore structure parameter. The simplest one is probably Archie’s law, which is expressed as:

(13)

The effective medium theory like Bruggeman asymmetric medium theory, also proposes the similar form of relation to Eq.(13). According to Eq.(13), when the diffusivity of the dispersion medium is absolutely insulated, the empirical constant a corresponds to D0; when cap=1, Dp=D0. In the porous medium, the exponent n can be obtained by the constrictivity and tortuosity of the pore network.

According to percolation theory, when the concentration of dispersion phase in the mixture made up of dispersion phase and insulators is up to critical concentration, the clusters of percolation made of dispersion phase jointed each other are formed. In the case of hardened cement paste, cement particle initially mixed with water, which is dispersed in water. With the increase in the hydration degree of cement, cement hydrated products gradually contact each other and then form infi nite percolation clusters. According, the pore network also comes into being and provides the transport channels for liquid and air. Considering the percolation characteristic of the whole pore network, the effective diffusivity of a porous material can be expressed as:

(14)

where, the percolation threshold cri is the critical porosity and n also called the percolation exponent.

With regard to chloride ion transport in hardened cement paste, Eq.(14) illustrates that diffusion can

occur, when capillary pores exceed the percolation threshold (cri). So, (capcri) represents the fraction of open capillary porosity. Bentz and Garboczi[15] have confirmed that the percolation threshold of cement paste, with or without pozzolanic material addition, is about 0.18 from the numerical simulation by using three-dimensional image-based hydration model. As far as ordinary porous materials are considered, the percolation exponent n is in the range of 1.65 to 2.0. However, if the diffusivity phases have extreme geometries, higher n values are often encountered in practice[14].

However, the solid phase of the cement paste also diffusive[16]. This means that even if the capillary porosity is less than the percolation threshold, the diffusion can still occur. According to the effective medium equation and percolation theory, the diffusivity below the percolation threshold can be given by:

(15)

where, Ds is the diffusivity of low diffuse phase.However, the shortcoming of Eqs.(14) and

(15) is that they can not predict the diffusivity near the percolation threshold. As cap→cri, the effective diffusivity of composite materials approaches zero by Eq.(14), while infi nity result of that is obtained by Eq.(15) at cap=cri as described in Fig.7. In order to overcome the shortcoming of the Eq.(14) and Eq.(15), the general effective media (GEM) equation proposed by McLachlan[17] is introduced into the paper, which is expressed as:

(16)

where, Ds is the diffusivity of solid phase of cement paste; Dp the overall diffusivity of hardened cement paste and n the percolation exponent, which mainly depends on the morphology of diffuse phase, e g, n is equal to 1.5 for sphere particle[18]. Eq.(16) can solve the nonzero solid phase diffusivity problem and can predict the diffusivity even near the percolation threshold. Therefore, this equation is more suitable for predicting the diffusivity of hydrated cement paste considering the characteristics of its microstructure. Then, the analytic solution to Eq.(16) is derived according to the normalized diffusivities as follows:

(17a)


where

(17b)

where, Ds/D0 is the normalized diffusivity of the solid phase and depends on the complexity of the structure of hydrated solid matrix. It is noted here that Eq.(16) reduces to Eq.(14), when Ds=0. Of course, the theoretical solution of the Eqs.(14), (15) and (17) can be expressed intuitively in Fig.7. From Fig.7, it can be seen that Eq.(17) suitably predicts the diffusivity even near the percolation threshold.

3.5 Determination of effective chloride diffusivity of ITZ

Until now, it is difficult to directly measure the chloride diffusion coeffi cient of ITZ experimentally, so some results of ITZ were obtained by using regression analysis for experimental data of mortar or concrete, e g, the chloride diffusivity coefficient of ITZ was 6-10 times that of the bulk cement paste in terms of Delagrave et al’s[19] experimental data, while Breton et al[20] and Bourdette et al [21] obtained the effective diffusion coeffi cient of ITZ was 6-12 times greater than that of bulk cement paste. These two results coincide. In additon, Caré[5] pointed out that chloride diffusion coefficient of ITZ was 16.2 times that of the bulk cement paste; Zheng[22] used Yang’s mortar datum[3]

to give the chloride diffusivity coeffi cient of ITZ was 12.79 times that of the bulk cement paste. Therefore, the chloride diffusivity coefficient of ITZ is obtained by experimental results in this paper.

4 Effective chloride diffusion coeffi cient test

In order to validate the multi-scale model

proposed in present paper, the neat cement paste, mortar and concrete were cast and then chloride diffusion coefficient were measured by using steady-state accelerated test method. Experimental program was as follows.4.1 Experimental materials

The cement paste was prepared by using I type of Portland cement, which is produced by the pure cement clinker mixed with 5wt% of gypsum. The chemical composition of the cement is listed in Table 1.Three types of siliceous sand were used in cement-based materials, corresponding to coarse, medium and fine sand and their fineness modulus were 3.53, 2.61 and 1.80, respectively. The size distribution of siliceous sand is shown in Fig.8. The size distribution of crushed limestone was in the range of 5-20 mm. The densities of the cement, water, sand and stone were 3150, 1000, 2620 and 2650 m3/kg, respectively. The mix proportions of the neat paste, mortar and concrete are listed in Table 2.

4.2 Preparation of sample for microanalysisAll cement paste specimens were mixed with de-

aired distilled water and mixed with 1 min of mixing at low speed and then 2 min of mixing at high speed. The


pastes were poured into PVC tube with diameter 14 mm and vibrated secondly for 2 min to remove air bubbles. The samples were stored in a room temperature (20 ℃) for 24 hours and then the specimens were moved to fog room (temperature (20±3 ℃, relative humidity above 95%). The specimens cured for three days were removed and split into several parts. The middle part of the sample with height about 16 mm were taken out for measurement of porosities at the specified ages. Before testing, the specimens were immersed into ethanol to stop the cement hydration.4.3 Testing for pore structure

Mercury intrusion measurements were performed with Micrometrics AutoPore IV 9500, which the maximum pressure up to 415 MPa and determined pore sizes in the range of 3 nm to 360 μm. A measurement is conducted in two stages: a manual low pressure run from 0.003 MPa to 0.21 MPa and an automated high pressure run from 0.21 MPa to 242 MPa. Data is automatically collected by a computer program. After low-pressure testing, the penetrometer was removed and weighted. Then the high-pressure testing was initiated. The machine was set to equilibrate 30 seconds and contact angle of 130°.4.4 Steady-state accelerated chloride

diffusion testTo perform chloride diffusion test, cylinder

specimens with 10 cm in diameter and 3 cm in height were fi rst saturated for 48 hours by using vacuum water saturation instrument, and then the specimens were placed between two upstream and downstream cells. Two mesh electrodes were placed on two sides of the specimen in such a way that the electrical field was applied primarily across the test specimen. One of cells called anode was filled with 0.30 N NaOH solution and the other cell called cathode was with 3.0% NaCl solution by mass. The cells were forced to a 30 V DC Regulated Power Supply. During the test, the chloride ion concentration was determined from the solution in the anode cell by using titration method with 0.01 N AgNO3 standard solution. The cumulative chloride ion concentration in the anode cell was measured periodically. When the cumulative chloride ion concentrations versus time curves become quite linear, it illustrated the chloride flux reached steady state. And then the diffusivity coeffi cient of chloride ions for specimens, Dcl, was approximately calculated as:

(18)

where

(19)

where, Jcl is the constant flux of chloride in the downstream cell (mol/m2/s), T the absolute temperature (K), C the chloride concentration in the upstream cell, V/l the electrical fi eld (V/m), R the universal gas constant (8.30 J/mol/K) and F the Faraday constant (F= 96485.34 C/mol), Δt the interval time observed (s), Vcomp the solution volume of downstream cell (m3), ΔCS the change of chloride ion concentration in downstream cell and As the actual diffuse area of chloride ion transported in specimens (m2).

5 Results and discussion

In order to predict the chloride diffusivity coeffi cient of hardened cement paste at the microscopic scale, the specimens of hardened cement paste with w/c ratios of 0.23, 0.35 and 0.53 are cured for 60 d and then measured chloride diffusivity coefficient to validate the model proposed in this paper. Using Eq.(17a) and Eq.(17b) to predict the chloride diffusivity coeffi cient of hardened cement paste, where capillary porosities came from the results of MIP measurements as depicted in Fig.9 and pore diameters with more than 10nm are treated as capillary pore. The predicted and experimental results are listed in Table 3. From Table 3, it can be seen that the prediction of chloride diffusivity coefficient is in good agreement with experimental results. If the deviation is defined as the ratio of the difference between the experimental and predicted result and the experimental result, as can be seen from Table 3 that the maximum deviation of specimens of neat past with w/c ratio of 0.35 is 14.30%. Byung et al[14] point out that there exist some constant values of n and Ds/D0 in Eq.(17). As a rule, the realistic values


for n and Ds/D0 are obtained from the test result. In the case of I type Portland cement paste, it is found that n=3.2 and Ds/D0≈1.0×104 from Table 3. Of course, compared with the range of percolation exponent (n) of the traditional porous materials, 1.65–2.0, the value n for cement paste is rather higher. It also illustrates that the microstructure of the well-hydrated cement paste may be more complex than that of traditional porous composites.

It is noted that the value of Ds/D0 usually range from 2.0×106 to 1.0×103, e g, Byung[14] argued that Ds/D0=2.0×104 for ordinary portland cement and Ds/D0=5.0×105 for paste with silica fume addition, while Bentz et al[23] also calculated Ds/D0 to be 4.0×104, 3.0×104, 1.0×104 and 1.0×105 for portland cement pastes containing 0, 3%, 6% and 10% silica fume, respectively. Although the value of n and Ds/D0 is diversifi cation, after the determination of the type of cement, the changes of n and Ds/D0 are relatively small and can be estimate by experiment.

The calculation on volume fraction of ITZ in cement-based materials is quite complicated. On the one hand, the volume fraction of aggregates pre unit mortar or concrete are over 40%[11], the overlapping degree of ITZ between aggregate and the bulk cement paste is rather large. On the other hand, the ITZ is not uniform. Nevertheless, for the sake of modeling, the ITZ is considered as a uniform region that has a certain thickness and aggregate morphology is treated as sphere in this paper. According to mixture proportions in Table 2 and the grading of aggregate, the volume fraction of ITZ is obtained by Eq.(11) and the results are listed in Table 3.

Assumed that the chloride diffusivity coefficient of ITZ is 7, 10, 12, 15, 17 and 20 times that of the bulk cement paste, respectively, and then the volume

fraction of aggregate and ITZ as well as the diffusivity coefficient of cement paste measured by experiment are substituted into Eq.(10) to predict the effective chloride diffusivity coeffi cient of mortar and concrete at the mesoscopic scale. Finally, predicted values of the effective diffusion coefficient are given in Table 3 according to different models at microscopic and mesoscopic scales. From Table 3, it can be seen that the difference between the test results and the predicted results is small. The chloride diffusivity coeffi cient of ITZ is 12 times of the bulk cement paste for mortar and is17 times that of bulk cement paste for concrete in this paper. The maximum deviation (deviation (a) in Table 3) is 18.60% for M50C, while is 20.50% for C25S specimen. The chloride diffusivity coefficient of ITZ is almost consistent with the results of Delagrave[19], Breton[20], Bourdette[21] and Caré[1]. The predicted results also illuminate the proposed model is suitable for the prediction of the effective diffusion coeffi cient of cement-basted materials.

If the predicted chloride diffusivity coefficients of cement paste are directly substituted into 10, the maximum deviation of mortar specimens (M50S) is 28.8% (Deviation (b) representation in Table 3), while the deviation of concrete specimens, C25S, C40S and C50S, is 5.42%, 4.63% and 18.7%, respectively. It is reasonable that the deviation is less than 30% for cement-based composite materials due to various kinds of uncertainties[1]. It shows that the deviation of concrete specimens comparatively reduces and further validates the reliability of predicted model in this paper. Of course, the model has to be improved, e g, the thickness and diffusivity coefficient of interfacial transition zone correspondingly carry out adjustment for practical applications.

6 Conclusions

Based on the multi-scale method to predict the effective chloride diffusivity coeffi cient, the following conclusions can be drawn:

a) The experimental results of hardened cement paste, mortar and concrete validate n-layered spherical inclusions model and it is suitable to model the effective chloride diffusion coefficient of cement-based composite materials, where mortar and concrete are treated as a four-phase sphere model, consisting of a cement continuous phase, an aggregates dispersed phase, an interface transition zone and their homogenization effective medium phase.


b) The microstructure including the interfacial transition zone (ITZ) between the aggregates and the bulk cement pastes as well as the microstructure of the bulk cement paste itself is introduced into the model to investigate the relationship between the diffusivity and the microstructure of cement-basted materials.

c) General effective medium equation is adopted in this paper to predict chloride diffusion coefficient of the hardened cement paste, and the realistic values for n and Ds/D0 obtained from the test data are n≈3.2, Ds/D0=1.0×104 for I type portland cement paste where n represents the tortuosity and constrictivity factors of pore structures in hardened cement paste.

d) The effective diffusion coefficient of ITZ is 12 times that of the bulk cement for mortar and is 17 times that of the bulk cement for concrete, due to the difference between particle size distributions and the volume fraction of aggregates in mortar and concrete.

e) The infl uencing factors of chloride diffusivity are the chloride diffusivity of matrix, volume fraction of aggregate, the chloride diffusivity of ITZ and the volume fraction of ITZ. The chloride ion diffusion coefficient of matrix and ITZ relies on the mix proportions of cement-based materials

f) The volume fractions of ITZ depend on the aggregate volume fraction, on the size distribution and on the ITZ thickness.

References[1] Caré s. Infl uence of Aggregates on Chloride Diffusion Coeffi cient into

Mortar [J]. Cement and Concrete Research, 2003, 33 (7): 1 021-1 028

[2] American Association of State Highway and Transportation Offi cials

T259. Standard Method of Test for Resistance of Concrete to Chloride

Ion Penetration[S].

[3] Yang CC, Su, JK. Approximate Migration Coefficient of Interfacial

Transition Zone and The Effect of Aggregate Content on the Migration

Coefficient of Mortar [J]. Cement and Concrete Research, 2002, 32

(10):1 559-1 565

[4] Garboczi EJ, Bentz DP. Multiscale Analytical /Numerical Theory of the

Diffusivity of Concrete [J]. Advanced Cement Based Materials, 1998,

8(2):77-88

[5] Caré S, Hervé E. Application of a N-phase Model to the Diffusion

Coefficient of Chloride in Mortar [J]. Transport in Porous Media,

2004, 56 (2):119-135

[6] Hashin Z. The Elastic Module of Heterogeneous Materials [J]. Applied

Mechanics, 1962, 29(1): 143-150

[7] Hervé E, Zaoui A. N-layered Inclusion-Based Micromechanical

Modeling [J]. International Journal of Engineering Science, 1993,

31(1):1-10

[8] Hervé E, Zaoui A. Elastic Behaviour of Multiply Coated Fibre-

Reinforced Composites [J]. International Journal of Engineering

Science, 1995, 33(10):1 419-1 433

[9] Hervé E. Thermal and Thermoelastic Behaviour of Multiply Coated

Inclusion-reinforced Composites [J]. International Journal of

Engineering Science, 2002, 39(4):1 041-1 058

[10] Lu BL, Torquato S. Nearest Surface Distribution Functions for

Polydispersed Particle Systems [J].Phys Rev A., 1992, 45(8):5 530-5 544

[11] Shane JD, Mason TO, Jennings H M, et al. Effect of the Interfacial

Transition Zone on the Conductivity of Portland Cement Mortars [J].

Am. Ceram. Soc, 2000, 83 (5):1 137-1 144

[12] Garboczi EJ, Bentz DP. Analytical Formulas for Interfacial Transition

Zone Properties [J]. Advn Cem Bas Mat, 1997, 6(3-4):99-108

[13] Maekawa K, Ishida T, Kishi T. Multi-scale Modeling of Concrete

Performance Integrated Material and Structural Mechanics [J].

Advanced Concrete Technology 2003, 1(2): 91-126

[14] Byung HO, Seung YJ. Prediction of Diffusivity of Concrete Based on

Simple Analytic Equations[J]. Cement and Concrete Research, 2004,

34(3):463-480

[15] Bentz DP, Garboczi EJ. Percolation of Phases in a Three-dimensional

Cement Paste Microstructural Model [J].Cement and Concrete

Research, 1991, 21(2-3):325-344

[16] Garboczi EJ, Bentz DP. Computer Simulation of the Diffusivity of

Cement-based Materials [J]. Mater. Sci., 1992, 27 (8):2 083-2 092

[17] McLachlan DS, Blaszkiewicz M, Newnham R E. Electrical Resistivity

of Composites [J]. Am. Ceram. Soc., 1990, 73(8):2 187-2 203

[18] Tang GB, Chen ZY, Shi ML. Modeling of Percolation Structure in

Cement Paste-graphite Mortar by DC Electrical Conductivity [J].

Journal of Building Materials, 2000, 3 (1):42-47(in Chinese)

[19] Delagrave A, Bigas JP, Ollivier JP, et al. Infl uence of Interfacial Zone

on the Chloride Diffusivity of Mortars [J]. Adv. Cem. Based Mater.,

1997, 5(3-4): 86-92

[20] Breton D, Ollivier JP, Ballivy G. Diffusivity of Chloride Ions in the

Transition Zone between Cement Paste and Granite. In: J C Maso (Ed.),

Interfaces between Cementitious Composites [M]. London: E & FN

Spon, 1992: 279-288

[21] Bourdette B, Ringot E, Ollivier J P. Modeling of the Transition Zone

Porosity [J]. Cem. Concr. Res., 1995, 25 (4): 741-751

[22] Zheng JJ, Zhou XZ. Prediction of the Chloride Diffusion Coeffi cient of

Concrete [J]. Materials and Structures, 2007, 40(7):693-701

[23] Bentz D P, Jensen O M , Coats A M, et al. Infl uence of Silica Fume on

Diffusivity in Cement-based Materials, I. Experimental and Computer

Modeling Studies on Cement Pastes[J]. Cem. Concr. Res., 2000, 30 (6):

953-962

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