Multi-scale prediction of the effective chloride diffusion coefficient of concrete

Construction and Building Materials 25 (2011) 3820–3831

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Multi-scale prediction of the effective chloride diffusion coefficient of concrete Guowen Sun a,b, Yunsheng Zhang a,b, Wei Sun a,b,⇑, Zhiyong Liu a,b, Caihui Wang a,b a b

School of Materials Science and Engineering, Southeast University, Nanjing 211189, China Jiangsu Key Laboratory of Construction Materials, Southeast University, Nanjing 211189, China

a r t i c l e

i n f o

Article history: Received 9 October 2010 Received in revised form 6 January 2011 Accepted 1 March 2011 Available online 12 May 2011 Keywords: Concrete Chloride diffusivity Interfacial transition zone Multi-scale Composite spherical inclusions model Microstructure

a b s t r a c t The N-layered spherical inclusion theory is applied to develop a multi-scale model to predict the effective diffusion coefficient of chloride ion in concrete. The model treats concrete as four-phase composite materials consisting of matrix phase, aggregate phase, ITZ (interfacial transition zone) and their homogenization phase. With hardened cement pastes characterized by three parameters such as the porosity, tortuosity and constrictivity, the effect of the cement paste microstructures and ITZ on the chloride diffusivity in concrete is taken into account and the porosity distribution function and effective chloride diffusion coefficients of the ITZ are given in the model based on the cement particle distribution characteristics of the ITZ in concrete. To validate the proposed model, the diffusion coefficient of chloride ion by the steady-state migration test is measured on a series of mortar and concrete specimens and good agreement between the model and experiment is obtained. In addition, the model predicts that the chloride diffusivity of concrete composite materials depends on the chloride diffusion coefficient of the matrix and ITZ, volume fraction of the aggregate and ITZ, with the volume fraction of the ITZ influenced by aggregate size distribution, the volume fraction of aggregate and thickness of the ITZ. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Chloride ion diffusivity in a porous concrete material induces steel corrosion in steel-reinforced concrete and then leads to premature deterioration of concrete structures exposed to marine environment. The chloride ion diffusion coefficient is an important indicator for concrete durability. In general, it can be obtained by long-term chloride ponding tests [1] according to concrete service environment. However, this test method is too time-consuming to meet practical requirements in a timely manner. In practice, the electrochemical accelerated testing methods [2,3] are widely used, but discrepancies occur even for identical materials due to the difference in the applied voltage, sample thicknesses and other experimental conditions. The chloride diffusivity coefficient is also affected by the material microstructures, among which the interfacial transition zone (ITZ) between aggregates and bulk cement pastes as well as the microstructure of the cement paste itself (e.g. porosity and pore structure), are mostly dominant. A relatively reliable model to predict the chloride diffusivity coefficient should take into account concrete microstructural parameters at different scales ranging

⇑ Corresponding author at: School of Materials Science and Engineering, Southeast University, Nanjing 211189, China. Tel./fax: +86 025 52090667. E-mail address: [email protected] (W. Sun). 0950-0618/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2011.03.041

from nanometers to millimeters. Multi-scale modeling methods [4] offer a promising solution to this hard task. The objective of the current paper is to propose a simple model to predict the chloride diffusivity based on concrete microstructures at different length scales. The key feature of the model is to take into account the ITZ and the bulk cement paste through a multi-scale approach where concrete is treated as a four-phase composite material consisting of bulk cement paste phase, aggregate phase, ITZ and their homogenization phase at mesoscopic scale. A series of mortar and concrete specimens are tested to verify the proposed model. 2. Representation of the multi-scale microstructure in concrete 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 problems the microstructure can be broken down into three elementary scales in this paper, according to the published literature [5,6], as sketched in Fig. 1. – The microscopic scale (109–106 m) mainly takes into account the pore features of hardened cement paste, which is composed of amorphous C–S–H, together with unhydrated cement products, capillary pores, crystalline calcium hydroxide, grains of submicroscopic calcium sulfoaluminate hydrate crystals and

G. Sun et al. / Construction and Building Materials 25 (2011) 3820–3831

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Fig. 1. Scales partition for concrete microstructure.

macroporosity in the case of high water-to-cement ratio materials. At this scale, these pores can be characterized by three parameters such as the porosity, tortuosity and constrictivity, and therefore the effective diffusion coefficient of chloride ion (Deff) can be expressed as the following form:

Deff ¼ f ðD0 ; /cap ; s; d where D0 is diffusion coefficient of chloride ion in bulk water and /cap is capillary porosity, while s and d are tortuosity and constrictivity of pore structure. – The mesoscopic scale (106–103 m) corresponds to a ‘theoretically homogeneous’ material including cement paste and aggregates. At this scale, mortar may be considered as a fourphase composite material composed of aggregate phase, ITZ phase, cement matrix and their homogenization phase, corresponding to 1, 2, 3 and 4, respectively, in Fig. 1. So the effective diffusion coefficient of chloride ion (Deff) can be given as the following form:

Deff ¼ f ðDa ; V a ; DB ; DI ; V I Þ where V a and V I are volume fraction of aggregate and ITZ, respectively, while Da , DB and DI are chloride diffusion coefficient of aggregate, bulk cement paste and ITZ, respectively.

– The macroscopic scale (102–101 m) corresponds to concrete as a composite material composed of coarse aggregates embedded in a continuous homogeneous mortar matrix and an ITZ. At this scale, the estimation of effective diffusion coefficient is similar to the modeling of mesoscopic scale. The concrete is also treated as a four-phase composite material composed of aggregate phase, ITZ phase, mortar matrix and theirs homogenization phase, corresponding to 1, 2, 3 and 4, respectively in Fig. 1. 3. Transport model of chloride ion in concrete 3.1. Geometric model selection of concrete For the sake of solving the effective diffusion coefficient in concrete, a geometry model is firstly introduced into this paper and the geometry morphology of model is required as much as possible to approach that of actual materials. Generally speaking, the composite materials could not be simply described by a single geometry model. As the introduction mentioned above, the composition and the microstructure of concrete are quite complex at different scales, and therefore it can not be simply treated as two-phase composite materials. In this paper, the composite spheres geometry model proposed by Hashin [7] is adopted to model concrete structure as depicted in Fig. 2 (two-dimensional plane). It can be

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Fig. 4. Four phase model of cement-based material.

Fig. 2. Geometry model of composite material.

assemblage equals to the diffusion coefficient of the n + 1 phase of regions. In the case of chloride diffusivity in concrete, the analytic solution is expressed as:

Deff ðiÞ ¼ Di þ seen from Fig. 2 that the geometric model is composed of millions of continuously variable-size spherical inclusion phases and matrix phase and the advantages of the model are 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 phase and occupy the entire volume of the material. 3.2. Diffusion coefficient model at the mesoscopic scale In the case of the prediction of diffusion coefficient in a n-phase material, the (n + 1)-phase model as shown in Fig. 3, which has been already used in elasticity, viscoelasticity, elastoplasticity [8,9] and even in thermal and thermoelastic behaviors [10], is extended to the case of ionic diffusion [6]. So this model is also applied to the prediction of the diffusion coefficient of chloride ion in concrete in this paper. It is emphasized that all the initial phase are supposed to be isotropic and homogeneous. Based on generalized self-consistent theory, the analytic solution of composite spheres assemblage model has been given by Hervé [10]. The advantage of generalized self-consistent scheme can be adopted to 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 Ri1 and Ri. Accordingly, the effective diffusivity coefficient of n-phase constituted composite spheres

Di ðR3i1 =R3i Þ ðDi ðDeff ði1Þ

 Di ÞÞ þ ð1=3ÞððR3i  R3i1 Þ=R3i Þ

ð1Þ

where Di is effective diffusivity coefficient of the ith phase, Deff ðiÞ is effective diffusivity coefficient of the composite spheres assemblage made up of the first phase to ith phase and Ri and Ri1 are two concentric spheres with the radii Ri and Ri1. For the purpose of convenient application in practice, Eq. (1) is further simplified into a four-phase model to solve the problem of chloride ion diffusion in concrete as depicted in Fig. 4, where the aggregates are considered as spherical inclusions with the volume fraction Va and the diffusion coefficient Da ITZ coated aggregates with the volume fraction VI and the diffusion coefficient DI aggregate and ITZ treated as coated inclusions embed into the bulk cement paste with the diffusion coefficient DB. To obtain the effective diffusion coefficient of concrete, Eq. (1) is treated with as follows: (1) Aggregates considered as a homogeneous material:

Deff ð1Þ ¼ Da

ð2Þ

(1) 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:

Deff ð2Þ ¼ DI þ

DI ðR31 =R32 Þ ðDI =ðDa  DI Þ þ ð1=3Þ=ððR32  R31 Þ=R32 Þ

ð3Þ

where

R32  R31 R32

¼

VI Va þ VI

R31 R32

¼

Va Va þ VI

ð4Þ

(2) 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:

Deff ð3Þ ¼ DB þ

DB ðR32 =R33 Þ ðDB =ðDeff ð2Þ

 DB ÞÞ þ ð1=3Þ=ððR33  R32 Þ=R33 Þ

ð5Þ

where

R33  R32 R33 Fig. 3. n + 1 phase model of composite spheres.

¼ 1  Va  VI

R32 R33

¼ Va þ VI

ð6Þ

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

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Deff ¼ DB

N A

2

ð7Þ

ð9Þ

where

where

N ¼ 6DB DI ð1  V a ÞðV a þ V I Þ þ V I ð1 þ 2V a þ 2V I ÞðDI  DB Þ  ð2DI þ Da Þ þ 3Da ½V I DB þ V a DI ð1 þ 2V a þ 2V I Þ

 ð2DI þ Da Þ þ 3Da ½V I DB þ V a DI ð1  V a  V I Þ

4hR2 i 1  Va

ð9aÞ

d¼

4hRi 8pN V hR2 i2 þ 1  Va ð1  V a Þ2

ð9bÞ

g¼

4hRi 16pNV hR2 i2 hRi 64Ap2 N2V hR2 i3 þ þ 3ð1  V a Þ 3ð1  V a Þ2 27ð1  V a Þ3

ð9cÞ

ð7bÞ

When the diffusion coefficient of the aggregates is zero (Da ¼ 0), Eq. (7) can be given as follows:

6DB ð1  V a ÞðV a þ V I Þ þ 2V I ðDI  DB Þð1 þ 2V a þ 2V I Þ 3DB ð2 þ V a ÞðV a þ V I Þ þ 2V I ð1  V a  V I ÞðDI  DB Þ

c¼ ð7aÞ

A ¼ 3DB DI ð2 þ V a ÞðV a þ V I Þ þ V I ð1  V a  V I ÞðDI  DB Þ

Deff ¼ DB

V ITZ ¼ 1  V a  ð1  V a Þ exp½pNV ðctITZ þ dt ITZ þ gt 3ITZ Þ

ð8Þ

Assumed that aggregates are relatively impermeable in this paper, Eq. (8) is used to predict the effective chloride diffusivity coefficient of concrete in this paper. From Eq. (8), it can be seen that the evaluation of the effective diffuse properties for the material needs to know the properties of each elementary phase and the factors influencing chloride diffusivity such as 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 concrete (water to cement ratio of cement paste). Therefore, the main parameters in Eq. (8) are four factors mentioned above, and among them, the volume fraction of aggregate could be directly obtained by the mix proportion of concrete, while the other three parameters will be discussed respectively in the following sections. 3.3. Determination of the volume fraction of ITZ Based on the statistical geometry of composites, Lu and Torquato [11] put forward a theory of nearest-surface distribution functions, which can be used to predict the volume fraction of multi-size spherical particles packed. As Shane et al. [12] pointed out that the volume fraction of aggregates in concrete per unit volume is more than 40%, the overlapping degree of ITZ layers between neighboring aggregates is very large. Fig. 5 shows a schematic diagram in two dimensions. In Lu and Torquato’s theory [11], the overlapping layers of ITZ are fully taken into account. So, Garboczi and Bentz [4] firstly applied the theory to predict the volume fraction of ITZ in concrete, where the concrete is also modeled as a three-phase composite material at mesoscopic scale, which is made of spherical aggregate particle, interfacial transition zone and the bulk cement paste. Therefore, the ITZ volume fraction (VITZ) is expressed as:

where NV is the total number of particles per unit volume and Va is the volume fraction of aggregates in the concrete; hi indicates an average over the aggregate size distribution. A is a coefficient that can have different values (0, 2, or 3) according to the analytical approximation chosen in the theory [11]; c, d and g are determined in terms of the number averages of the particle radius and the particle radius squared. As can be seen from Eq. (9), the factors influencing the ITZ volume fraction are the aggregate gradation, the volume fraction of aggregate and the ITZ thickness. For a given concrete mixture, materials density and a sieve analysis, these variables are known or can be determined. It is noted once again that c, d, and g are determined in terms of number averages over the particle size distribution of the aggregates, not volume. The aggregates in actual mortar or concrete are often described as the mass fraction passing or retained by a certain sieve size. If the aggregates of different size distribution have the same density, the mass fraction is as much as volume fraction and the size distribution of aggregate is often expressed as the form of volume-based interval probability or volume-based interval cumulative probability. It is a key step that the volumebased interval probability of aggregates must be converted into the number-based probability function for Eq. (9) to quantitatively obtain the ITZ volume fraction. The method is given in detail as follows. It is assumed that FV(D), fV(D), and fN(D) is the volume-based accumulative probability function, the volume-based probability density function and the number-based probability density function of spherical aggregate particles, respectively, where D, V and N is the diameter, volume and number of arbitrary spherical aggregate particles, respectively. The mutual relationship of three functions mentioned above is as follows. fV(D) can be obtained by differentiating FV(D) with respect to D as follows:

fV ðDÞ ¼ F 0V ðDÞ ¼

dF V ðDÞ dD

ð10Þ

If the diameter of spherical aggregate particles varies from D  dD/2 to D + dD/2, and its volume-based probability density function is determined. Therefore, the number of spherical aggregate particles in the range of D ± dD/2 is given by:

nðDÞdD ¼

fV ðDÞdD

ð11Þ

p D3 6

And then, the total number of spherical aggregate particles is determined by integration as follows:

Z

Dmax

Dmin

Fig. 5. Overlapping diagram of ITZ.

nðDÞdD ¼

Z

Dmax

Dmin

fV ðDÞdD p D3

ð12Þ

6

So, the number-based probability function of spherical aggregate particles is given by:

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nðDÞdD D3 fV ðDÞdD fN ðDÞdD ¼ R Dmax ¼ R Dmax 3 nðDÞdD D fV ðDÞdD Dmin Dmin

ð13Þ

3.4. Prediction of effective diffusion coefficient of hardened cement paste at microscopic scale 3.4.1. Microstructure of the hardened cement paste The 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 material in saturated state. The first phase is the pore networks phase, and the second is the solid phase consisting of various hydration products as described in Fig. 6a. The pore networks of cement paste comprise interlayer porosity, gel pores, capillary pores and macropores, which are transport channels of aggressive medium intruded into 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 molecularrelated 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. 6b. It should be noted that the C–S–H gel pores are included in the solid phase, not in the pore phase. The main path of transport of hardened cement paste is known to be the capillary pore space while the gel pores play a minor role in transport [14], except at very low capillary porosity. At starting stage of cement hydration, the transport properties of medium are very well since all of the pores are fully connected. However, with the increasing of the de-

gree of cement, hydration products grow into the pore space. Consequently, the pores become smaller and only partially connected, so that the transport properties of medium decrease. The pore spaces in hardened cement paste can be classified as follows: continuous pores considering pores with tortuous effect, continuous pores with ink-bottle, dead-end pores with ink-bottle and isolated pores (Fig. 6b). The former two kinds of pores are also called the effective pores, which play an important role in transported channel for aggressive medium. The dead-end pores and isolated pores do not contribute to the transport properties. Usually, tortuous continuous pore can be expressed by the parameter of tortuosity or tortuosity factor, while a continuous pore with inkbottle can be described as the constrictivity of pore network, since the constrictivity is taken into account the interaction between pore structure and ion transport. 3.4.2. Effective media theory The effective diffusivity of a porous material can be expressed by the relationship with pore structure parameter [15].

Dp d ¼ /cap 2 D0 s

ð14Þ

where Dp is the effective diffusivity of a porous material (m2/s); /cap is capillary porosity; s and d are the tortuosity factor and constrictivity of the pore networks, respectively. D0 is the diffusivity of transport ion in bulk water, such as D0 = 2.03  109 m2/s for chloride ion at room temperature. However, the solid phase of the cement paste also has pores, i.e., C–S–H gel pores, and hence, is also diffusive [14]. So, Eq. (15) is more appropriate than Eq. (14).

Dp d ¼ V Tpor 2 D0 s

ð15Þ

V Tpor ¼ V cap þ V gel

ð16Þ

where Vcap and Vgel are the porosities of capillary and gel pores, respectively. According to Powers theory [16], the parameters of porosity in Eq. (16) can be calculated for Portland cement as follows:

(a) Component of hardened cement paste

V gel ¼

0:19a ðw=cÞ þ 0:32

ð16aÞ

V cap ¼

ðw=cÞ  0:36a ðw=cÞ þ 0:32

ð16bÞ

So, the total porosity can be obtained by Eq. (16a) and (16b) as follows:

V Tpor ¼

ðw=cÞ  0:17a ðw=cÞ þ 0:32

ð17Þ

where w/c is water to cement ratio and a is the degree of hydration. Eq. (17) illustrates that the developing of porosities can be determined by w/c and a. A continuous pore with tortuous characteristic can be represented by the parameter of tortuosity, whose meaning of tortuosity is sketched in Fig. 7. Tortuosity is defined as the ratio of the length

(b) Pore structure of hardened cement paste Fig. 6. Schematic diagram of the structure of hardened cement paste: (a) Component of hardened cement paste. (b) Pore structure of hardened cement paste.

Fig. 7. Definitions of the tortuosity.

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G. Sun et al. / Construction and Building Materials 25 (2011) 3820–3831

of actual ion transport pathways (le) to the corresponding length in the projected plane (l).

n ¼ le =l

ð18Þ

Tortuosity is an intrinsic characteristic of materials expressing the geometrical property of pore structure and it can often be assumed to be independent of the characteristics of ions. In fact, it is very difficult to measure tortuosity (n) of pore networks in hardened cement paste, so it is calculated by the materials permeability [17] as described in Eq. (19) or Eq. (20).

n¼

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼rc max q gg¼r g2 f ðgÞdg c min

ð19Þ

24Kð1 þ qV tot Þ

or

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2av g n¼ 4:24Kð1 þ qs V tot Þ

ð20Þ

where q, K and Vtot are the density (m3/kg), permeability (m/s) and total volume of porous materials, while qs and D2av g can be obtained by mercury intrusion porosimetry (MIP) measurements, where qs is apparent density of materials (m3/kg); D2av g is weighted average of pore diameter size (m) and is expressed as:

D2av g ¼ qs



X 2 1 1 I1 D21 þ Ii Di þ In D2n 2 2

 ð20aÞ

The relationship between the tortuosity and porosities and average pore diameter of porous materials can be determined by Eqs. (19) and (20), but the calculated process of tortuosity is still complicated. Therefore, tortuosity factor (s⁄⁄) is often used to describe the tortuous continuous pores, whose meaning is the ratio of the tortuosity (f) to the limited length (f), that is s = f/f. However, the results of tortuosity factor are also influenced by MIP test. So, a convenient and modified model put forward by Nakarai et al. [18] is adopted in this paper. Tortuosity factor is defined as a function of porosity and is given by:





s ¼ 1:5 tan h 8:0ðV Tpor  0:25Þ þ 2:5

ð21Þ

The relationship between tortuosity factor and porosity is shown in Fig. 8. It can be seen that From Fig. 8, with the growth of porosity, tortuosity factor decrease. When the porosity is more than 0.5, tortuosity factor is almost close to 1.0. These facts imply the pores in materials are almost connected.

Another important parameter is constrictivity which characterizes a continuous pore with ink-bottle. Constrictivity is formulated as an expression of the effect of pore size. Usually, porosity and pore size are taken into account the study of constrictivity. So pore dimensions are thought to be the main determinant of constrictivity. The parameter of constrictivity also adequately considers the interaction between pore structure and ion transport. If the cross-section of a pore space segment is straight, then constrictivity becomes unity, whereas if the segment is restricted at certain point, the value of constrictivity is less than unity. According to the pore characteristics in cement paste, the constrictivity of pores is defined with respect to the peak pore diameter by Eq. (22) [13].

n o d ¼ 0:395 tanh 8ðlogðr peak cp Þ þ 6:2Þ þ 0:405

ð22Þ

where d is the constrictivity of pores and r peak is the peak radius of cp capillary pores (m). The relationship between d and r peak is shown in Fig. 9. As can be cp seen from Fig. 9 that the value of constrictivity is about 0.8 for ordinary porous materials, however, the pores in cement paste are widely distributed over the nanometer-to-micro scale and pores with varying sizes are randomly connected, while the peak radius of capillary pores ranges from 20 nm to 120 nm, so, the value of constrictivity is the order of about 0.01 for fine pores in the sound cement paste, Of course, which becomes larger for deteriorated cement paste. Based on above analysis, only if the ratio of w/c, the degree of hydration and the peak radius of capillary pores measured by MIP are determined, the developing process of pore structures also can be simultaneously obtained. So, Eqs. (17), (21), and (22) are substituted into Eq. (15) to obtain chloride diffusivity of hardened cement paste in saturated state. 3.5. Prediction of effective diffusion coefficient of ITZ In normal concrete or mortar, the hydrated matrix surrounding the aggregate has different microstructures due to the w/c ratio gradient developed at the interfacial layer [3], which contains a higher w/c ratio (more porosity) than the bulk paste due to the inefficient packing of the cement particles, the so-called wall effect. Based on computer simulation, the relative diffusivity of ITZ proposed by Garboczi and Bentz [4] is adopted in this paper, where the relative diffusivity is defined as the ratio of the diffusivity of ions in a composite material to their diffusivity in bulk water. The equation is as follows:

Constrictivity

0.8 0.6 0.4 0.2 0.0 -10

-9

-8

-7

-6

-5

-4

-3

peak

log10(r Fig. 8. Tortuosity versus porosity.

/m)

Fig. 9. Constrictivity versus peak radius of capillary pores.

-2

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D ¼ 0:001 þ 0:07/ðxÞ2 þ 1:8  Hð/ðxÞ  /cri Þ  ð/ðxÞ  /cri Þ2 D0

ð23Þ

where D is the diffusion coefficient of chloride ion in a composite material, D0 = 2.03  109 (m2/s) is the corresponding diffusion coefficient in bulk water at 25 °C, /(x) is the capillary porosity at a distance x from an aggregate surface, H is the Heaviside function (if x > 0, H(x) = 1, and otherwise, H(x) = 0), /cri = 0.18 is the critical porosity at which the capillary pore space disconnected. It can be seen from Eq. (23) that once the porosity of ITZ, as a function of distance from the aggregate surface, x, is quantified, the diffusion coefficient of ITZ can be obtained. So it is very important to determine the porosity of ITZ. Usually, the porosity of the ITZ was measured by mercury intrusion porosimetry (MIP) [19] or microhardness [20]. Fig. 10 shows the plots of the capillary porosity versus distance from aggregates surface according to real specimens tested by experiment [21]. It can be seen from Fig. 10 that the porosity of ITZ is a variational value with the distance from the aggregate surface and w/c ratio. If every real specimen is tested by experiment, it is very timeconsuming and different researchers even found that the identical specimens have different results. Therefore, it is indispensable to build a model to predict the porosity of ITZ. Fortunately, cement particles volume density of ITZ, D(x, tITZ), which has been developed by Zheng et al. [22] in terms of computer simulation technology, is adopted in this paper to deduce the porosity distribution of ITZ. The process is as follows. D(x, tITZ) is given by [22]: k

Dðx; t ITZ Þ ¼ Dc ðx=t ITZ Þ½1kðx=tITZ Þ  ;

0  x  t ITZ

ð24Þ

where x is distance from the aggregate surface in ITZ layers, tITZ is the thickness of ITZ, Dc is cement volume density of bulk cement paste. k and k are two coefficients to be determined, among them k = 1.08. Dc is related to w/c ratio and the maximum diameter of cement particles in concrete (Dcem). It can be expressed as [22]:

1 Dc ¼ 1 þ 3:15w=c h  ð1:0482  105 D2cem þ 3:246  104 Dcem þ 0:0146Þw=c i  1:79  107 D2cem þ 5:0429  105 Dcem þ 1:00564 ð24aÞ

So, in terms of Eq. (24), the porosity distribution of ITZ can be expressed as:

( /ðxÞ ¼

1  ð1  /cap Þðx=tÞð1kðx=tÞ

k

Þ

0  x  t ITZ x  t ITZ

/cap

ð25Þ

where /cap is capillary porosity of bulk cement paste. The microstructure of cement paste in the vicinity of an aggregate in concrete or mortar differs from zones which is further away from the aggregate and also differs from which is in neat hydrated cement paste, therefore, the capillary porosity of bulk cement paste, /cap, is given by [23]:

/cap ¼ 1 

1 þ 1:31a 1 þ 3:2w=c

ð26Þ

where w/c is water to cement ration, a is hydration degree of cement. The modeling of porosity distribution of ITZ can be obtained by using Eqs. (24), (24a), (24b), (25), (26) as depicted in Fig. 11. Compared with Fig. 10, Fig. 11 is also relatively consistent with the porosity characteristic of real mortar or concrete. This implies that Eq. (25) deduced in this paper is reasonable. Although the ITZ is not uniform and its porosity is a variational value along with distance from the aggregate surface, for the sake of modeling, the ITZ is considered as a uniform region. If the thickness of ITZ is assumed to be tITZ mm, then the capillary porosity of ITZ can be estimated by averaging the capillary porosity from the aggregate surface distance tITZ mm:

R tITZ /ITZ ¼

0

/ðxÞdx t ITZ

ð27Þ

where /ITZ is the average capillary porosity of ITZ, x is the distance from the aggregate surface, and dx is the infinitesimal interval of the distance from the aggregate surface. Finally, the diffusion coefficient of ITZ is given by:

DITZ ¼ D0 ð0:001 þ 0:07/2ITZ þ 1:8  Hð/ITZ  ucri Þ  ð/ITZ  ucri Þ2 Þ ð28Þ

While k can be obtained by the following equation:

0

In a word, substituting Eqs. (9), (15), and (28) and the volume fraction of aggregates into Eq. (8), the effective diffusivity coefficient of concrete can be quantitatively calculated.

Fig. 10. Porosity distribution of ITZ (From Bentz and Garboczi’s paper [21]).

Fig. 11. Porosity distribution of ITZ by modeling.

125 ¼ ð125  t ITZ ÞDc þ 1 þ 3:15w=c

Z

t ITZ

Dðx; tITZ Þdx

ð24bÞ

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4. Effective chloride diffusion coefficient test In order to validate the multi-scale model proposed in present paper, mix proportions of various specimens of neat paste, mortar and concrete were cast and then chloride diffusion coefficients were measured by using steady-state accelerated test method. Experimental programs were as follows. 4.1. Experimental materials The specimens were prepared by using Portland cement type I, which is produced by the pure cement clinker mixed with gypsum 5 wt.%. Its chemical composition was listed in Table 1. Three types of siliceous sand were used in concrete, corresponding to coarse, medium and fine siliceous sand and their fineness modulus were 3.53, 2.61 and 1.80, respectively. The size distribution of siliceous sand was shown in Fig. 12. The size distribution of crushed gravels was in the range of 5–20 mm. The densities of the cement, water, sand and gravel were 3150 m3/kg, 1000 m3/kg, 2620 m3/kg and 2650 m3/kg, respectively. The mix proportions of neat paste, mortar and concrete were present in Table 2. 4.2. Microcosmic sample preparing and curing All cement paste specimens were mixed with de-aired distilled water and mixed with 1 min of mixing at low speed and then 2 min mixing at high speed. The pastes were poured into 500 ml plastic beaker, which were vibrated for 3 min in order to remove air bubbles. Finally, the paste was poured into PVC tube with diameter 16 mm and vibrated secondly for 2 min to reduce experimental error. The samples were placed in a room with temperature 20 °C for 24 h and then were moved to standard curing room (temperature (20 ± 3 °C, relative humidity above 95%). The specimens cured for 3 days were removed and split into several parts and then taken the middle part with height about 15 mm, finally, these samples were taken out for measurement of porosities of the specimens at the required curing age. Before testing, the specimens were immersed into ethanol to stop the cement hydration by removal of the free water.

Fig. 12. Sand size distribution.

24 h. The samples were then transferred to a desiccator for 30 min, after they were weighed 1 gram for paste and the mass denoted as m0. Finally, the dried samples were placed in a 1050 °C furnace for 3 h and cooled in a desiccator and weighed again numbered as m1050°C. For each specimen, three samples were tested and the results were averaged. By now the degree of hydration was determined by the non-evaporable water content. The non-evaporable water content could be calculated as follows:

Wn ¼

L  Lc 1  Lc

ð29Þ

where L is ignition loss (g) and L = (m0  m1050°C)/m0, Lc ignition loss of raw materials (g), Wn the non-evaporable water content (%). According to Powers model, the formula of cement hydration can be expressed as:

a¼

100  W n 0:23

ð30Þ

where a is cement hydration (%). 4.3. Testing for pore structure 4.5. Steady-state accelerated chloride diffusion test Mercury intrusion measurements were performed with Micrometrics AutoPore IV 9500, whose maximum pressure can be up to 415 MPa and determined pore sizes in the range of 3 nm to 360 lm. The 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 0.21Mpa to 242 MPa. Data is collected and handled by a computer acting as a control module. After low-pressure testing, the penetrometer was removed and weighted. Then the high-pressure testing was initiated. The machine was set to equilibrate 30 s and contact angle of 130°. 4.4. Testing for degree of hydration The samples were crushed and ground into powder in an agate pot containing isopropyl alcohol. Then the powder was sieved through 0.08 mm sieve. Next the passed powder was placed in dried and weighed porcelain crucibles and dried at 105 °C for

To perform chloride diffusion test, columned specimens with 10 cm diameter and 3 cm height were first saturated for 48 h 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 filled with 3.0% NaCl solution. 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, titrated with 0.01 N AgNO3 standard solution by the potentiometric titration method. 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

Table 1 Chemical composition of the cement / wt.%. SiO2

Al2O3

Fe2O3

CaO

MgO

K2O

Na2O

TiO2

SO3

P2O5

BaO

ZnO

MnO

SrO

LOI

21.38

4.67

3.31

62.40

3.08

0.54

0.21

0.27

2.25

0.10

0.04

0.06

0.18

0.13

0.95

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G. Sun et al. / Construction and Building Materials 25 (2011) 3820–3831

Table 2 Mix proportions of specimens tested. No.

w/c

Type of sand

Volume fraction of sand

Volume fraction of gravel

P-23 P-35 P-53 M30S M40S M50S M50C M50F C25S C40S C50S

0.23 0.35 0.53 0.35 0.35 0.35 0.35 0.35 0.35 0.35 0.35

– – – Medium Medium Medium Coarse Fine Medium Medium Medium

– – – 0.30 0.40 0.50 0.50 0.50 0.18 0.26 0.35

– – – – – 0.51 0.44 0.35

diffusivity coefficient of chloride ions for specimens, Dcl, was approximately calculated as:

Dcl ¼

RT  J CF Vl cl

Fig. 13. Pore size distributions of hardened cement paste.

ð31Þ

where

J Cl ¼

DC Cl V comp A s  Dt

ð31aÞ

where Jcl is the constant flux of chloride in the downstream cell (mol/m2/s), T is absolute temperature (K), C is chloride concentration in the upstream cell, V/l is the electrical field (V/m). R is the universal gas constant (8.3 J/mol/K) and F is Faraday constant (F = 96485.3383C/mol). Dt is the interval time observed (s), Vcomp is the solution volume of downstream cell (m3), DCCl is the change of chloride ion concentration in downstream cell and As is the actual diffuse area of chloride ion transported in specimens (m2).

curing age (t), which be fitted with many datum published literatures, can be as: a = 0.716t0.0901 exp[0.103t0.0719/(w/c)]. The predicted results and experimental datum published in literatures are also listed in Table 3. It can be seen from Table 3 that the predicted results of the proposed model are almost in agreement with the literature results. Of course, it is also illustrated that when w/c ratio is more than 0.5, the deviations of specimens slightly become larger, which can be attributed to the bleeding effect of specimens due to high w/c ratio and other limitation of experimental conditions.

5.2. Predicted results of mortar and concrete 5. Results and discussion 5.1. Predicted results of hardened cement paste In order to predict the chloride diffusivity coefficient 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 were cured for 60 days and then chloride diffusivity coefficient was measured to validate the proposed model in this paper. The hydration degrees of hardened cement paste with w/c ratios of 0.23, 0.35 and 0.53 are 0.503, 0.602 and 0.731, respectively, and the corresponding pore size distributions are shown in Fig 13. It can be seen from Fig. 13 that the critical pore diameter of capillary pores is 53 nm, 62 nm and 77 nm corresponding to mentioned cement paste above. The hydration degree and critical pore diameter of each material are substituted into Eqs. (17), (21), (22), and (15) to predict the diffusion coefficient of chloride ion, once they are determined. And the predicted results are compared with the results of steady-state accelerated chloride diffusion testing as shown in Table 3. It can be observed from Table 3 that the predicted values are almost in accordance 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 neat paste specimens with w/c ratio of 0.53 is 20.70%. To further verify the model, the results in literatures [2,24,25] are compared predicted results. Caré0 [2]. Negala et al.[24] and Huang et al. [25] have measured the chloride ion diffusion coefficients of cement paste at different curing age and different w/c ratio. Since the hydration degree of cement were partly given in the literatures mentioned above, for the prediction of consistency, the relationship between the degree of hydration (a) and w/c ratio and

Chloride diffusivity coefficients of mortar and concrete are measured in terms of steady-state accelerated method to validate the proposed model at the mesoscopic. Experimental results are listed in Table 4. As can be seen from Table 4 that, compared with neat paste, the chloride diffusion coefficients in mortar increase with the increasing of the volume fraction of aggregate for the specimens, M30S, M40S and M40S, containing the same type of medium-size sand. The main reason is that the porous ITZ between the paste and aggregate increases transport property per unit volume of matrix. As Shah [26] pointed out that the influence of aggregate in the hydrated cement past is fourfold: dilution, tortuosity, ITZ and percolation. Generally speaking, the dilution and tortuosity effects reduce concrete diffusivity, while the interface transition zone (ITZ) and percolation effects increase diffusivity. However, the percolation effect of aggregates relates with the size distribution and volume fraction of aggregates. Yang0 s results [3] indicated that the chloride ion diffusion rate was little affected by the percolation. In the case of chloride diffusion coefficient of the specimens, M30S, M40S and M40S, the ITZ effect is more than the dilution and tortuosity effect of aggregate. Aiming at this point, the influence of aggregate on the pore structure of concrete can be observed intuitively as depicted in Fig. 14. Fig. 14a and b is total porosities and pore size distribution of the neat paste specimen P-35 and mortar specimens M30S, M40S and M40S. It can be seen that Fig. 14 that the presence of aggregates modifies the pore structure: on the one hand, the overall distributions are slightly shifted towards the finer pores, on the other hand, coarser pores that is more than 100 nm appear, which may be attributed to the presence of the ITZ. The results are in accordance with previously published results [2].Therefore, the bulk cement paste in mortars may be the more dense when ITZ concentration is high.

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G. Sun et al. / Construction and Building Materials 25 (2011) 3820–3831 Table 3 Comparison between predicted and experimental chloride diffusivities of cement paste. Researcher

w/c

Total porosity (%)

Tortuosity factor

Constrictivity

Predicted values (m2/s)

Experimental values (m2/s)

Deviation (%)

Author

0.23 0.35 0.53

0.26 0.37 0.48

2.35 1.38 1.08

0.01 0.01 0.01

9.70E13 3.92E12 8.41E12

1.03E12 4.12E12 1.06E11

5.89 4.79 20.7

Caré

0.45

0.41

1.23

0.01

5.46E12

5.65E12

3.40

Ngala

0.4 0.5 0.6

0.38 0.44 0.50

1.33 1.13 1.06

0.01 0.01 0.01

4.37E12 7.13E12 9.28E12

4.28E12 8.43E12 1.23E11

2.05 1.54 24.6

Huang

0.4 0.5 0.6

0.39 0.46 0.51

1.28 1.11 1.05

0.01 0.01 0.01

4.83E12 7.57E12 9.64E12

5.42E12 8.24E12 1.20E11

10.90 8.14 19.70

Table 4 Comparison between predicted chloride diffusivities and experimental results. No.

Volume fraction of ITZ (%)

Diffusion coefficients of ITZ (1011m2/s)

Predicted results (1012m2/s)

Experimental results (1012m2/s)

Deviation (%)

P-35 M30S M40S M50S M50C M50F C25S C40S C50S

– 11.54 14.71 17.27 10.60 24.61 8.56 9.23 11.23

– 4.20 4.20 4.20 4.20 4.20 7.74 7.74 7.74

4.70 5.49 5.96 6.31 4.66 8.15 6.72 7.48 8.76

4.12 5.12 7.23 8.10 4.28 10.0 6.19 6.35 7.06

14.1 7.22 17.60 22.10 8.98 18.50 4.66 14.70 22.70

For the mortars M50S, M50C and M50F containing the same volume fraction of aggregates, are made up of medium, coarse and fine sand and bulk cement paste, respectively. It can be seen from Table 4 that chloride diffusion coefficients of mortar mentioned above reduce with the growth of fineness modulus. The finer aggregates are, the larger diffusion coefficients of mortars are. This may be attributed to the dominant diffusion of the ITZ. Pore structures of the specimens M50F, M50S and M50C can also observed in Fig. 14c and d. Assumed that the volume fraction of aggregates and ITZ thickness are constant in concrete, the volume fraction of ITZ is approximate proportional to specific surface area of aggregates, so the finer aggregates are, the larger ITZ volume fraction is, so that total porosities increase accordingly too. The quantitative calculation of volume fraction of ITZ in concrete is quite complicated. On the one hand, the volume fraction of aggregates per unit mortar or concrete are over 40% [12], 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. In terms of the advanced scanning electron microscopy and backscattered electron imaging, the microstructure of ITZ in concrete have been widely investigated and shown the thickness of ITZ is usually in the range of 30–50 lm [2,3]. Therefore, assumed that the ITZ thickness is 30 lm, according to the size distribution of aggregate in Fig. 12 and mix proportions of specimens in Table 2, volume fraction of ITZ in mortar and concrete can be obtained by Eq. (9) and the results are presented in Table 4. From Table 4, it can be seen that the mortars containing the same type of sand, M30S, M40S and M50S, the volume fractions of ITZ increase with the growth of volume fractions of aggregates, and however, when volume fractions of aggregates are over certain limitation due to the overlapping degree of ITZ layers between aggregates and the bulk cement paste like the concrete specimens C25S and C40S, the volume fractions of ITZ will reduce. For the mortars M50S, M50C and M50F, which contains the same volume fraction of sand, the smaller the size

diameters of sand are, the larger the aggregate volume fractions are. An estimation of effective chloride diffusion coefficient in mortars specimens adopts a four-phase composite sphere model composed of a cement paste matrix, sand particle inclusions, and an interfacial transition zone (ITZ) and a homogenization effective medium phase. The hydration degree of cement paste is 0.602, which can calculated by Eqs. (29) and (30) in terms of testing ignition loss by experiment and then hydration degree of cement paste is substituted into Eqs. (26) and (25) to obtain the capillary porosity of bulk cement paste and pore size distribution of ITZ as described in Fig. 15. From Fig. 15, it can be seen that capillary porosities distribution of ITZ, which is about 20 lm distance from the aggregate surface, are consistent with porosities of bulk cement paste. Average porosity of ITZ (27%) can be calculated by Fig. 15 and Eq. (27) and then is substituted into Eq. (28) to obtain the effective diffusion coefficient of ITZ. Finally, the volume fraction of aggregate and ITZ as well as diffusion coefficient of ITZ and bulk cement paste are substituted into Eq. (8) to predict the effective chloride diffusivity of mortar and results are listed in Table 4. As can be seen from Table 4 that predicted results are almost consistent with experimental results where the maximum deviation of mortar M50S and M50F is 22.10% and 18.50%, respectively. The prediction of effective chloride diffusion coefficient of concrete is similar to mortar. The gravels considered as inclusions embed in a continuous homogeneous mortar matrix. The effective chloride diffusion coefficients of mortar matrix in concrete C25S, C40S and C50S are firstly calculated and their values are 4.87  1012 m2/s, 5.29  1012 m2/s and 5.73  1012 m2/s, respectively. The average porosity of three types of mortar matrix is adopted and then is substituted in turn Eqs. (25) and (27) into Eq. (28) to predict effective chloride diffusion coefficients of ITZ in concretes. The results are listed in Table 4 and the maximum deviation is 22.70% for C50S specimens. It is reasonable that the deviation is less than 30% for concrete due to all kinds of uncertainties [2]. It shows that a multi-scale model proposed in this paper is suitable

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Fig. 14. Pore size distribution of hardened cement paste and mortar: (a) Total porosities of neat paste and mortar, (b) pores size distributions of paste and mortar, (c) total porosities of three types of mortar, (d) pores size distributions of three types of mortar.

Table 5 Comparison between predicted chloride diffusivities and Caré0 s experimental results. No.

Diffusion Predicted Experimental Deviation Volume coefficients of IT results results (%) fraction (1012 m2/s) (1012 m2/s) of ITZ (%) (1011 m2/s)

P – M1C 1.80 M2C 3.60 M2M 10.3 M1F 8.00 M2F 15.4

Fig. 15. Porosity distribution of interfacial transition zone.

to model the effective chloride diffusion coefficient in concrete based on N-layered spherical inclusions model.

– 5.91 5.91 5.91 5.91 5.91

5.43 4.49 3.65 5.95 6.50 7.54

5.65 5.40 4.80 7.40 8.10 9.50

3.89 14.81 19.82 13.65 16.15 13.76

In order to further validate the reliability of model proposed in this paper, Caré0 s experimental results [2] are introduced, where the hydration degree of cement paste with w/c ratio of 0.45 is 0.811, so the porosity distribution of ITZ can be obtained as depicted in Fig. 15. The calculated program is identical that mentioned mortar above. The predicted results are presented in Table 5. From Table 5, it can be seen that the predictions of chloride diffusion coefficient are almost consistent with Caré0 s experimental results [2]. The maximum deviation is 19.82% for M2C specimens. It illustrates that it is reasonable for the model. Of course, the model has to be improved, e.g., the thickness and diffusivity coefficient

G. Sun et al. / Construction and Building Materials 25 (2011) 3820–3831

of interfacial transition zone correspondingly carry out adjustment for practical applications.

6. Conclusion A multi-scale model is developed in which mortar and concrete are treated as a four-phase composite material consisting of matrix phase, aggregate phase, ITZ and their homogenization phase, and the following conclusions can be drawn. (1) The N-layered spherical inclusions model, as validated by the experimental results of neat paste, mortar and concrete, is suitable for predicting the effective chloride diffusion coefficient of concrete. (2) The interfacial transition zone (ITZ) between the aggregate particles 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 concrete. (3) Effective medium equation is adopted in this paper to predict chloride ion diffusion coefficient of the hardened cement paste. The pore structures of cement paste are characterized by three parameters: porosity, tortuosity and constrictivity. (4) Based on the cement particles volume density of ITZ, the porosity distribution of ITZ is developed and the average porosity and diffusion coefficient of ITZ estimated quantitatively. (5) The presence of aggregates in concrete modifies the pore structure, and the influence of aggregate on transport properties is that the dilution and tortuosity effects reduce concrete diffusivity, while the interface transition zone (ITZ) and percolation effects increase diffusivity. Comparatively speaking, the ITZ effect is larger than dilution and tortuosity effects. (6) The factors influencing chloride diffusivity are the chloride diffusivity of the matrix, volume fraction of the aggregate, the chloride diffusivity of ITZ and its volume fraction. The chloride ion diffusion coefficient of the matrix and ITZ depends on the water to cement ratio of the cement paste.

Acknowledgments Financial support from the National Basic Research Program of China with Grant (No. 2009CB623203) and that from the National High-tech R&D Program of China with Grant (No. 2008AA030794) is gratefully acknowledged.

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