A coupled transport-reaction model for simulating autogenous self-healing in cementitious materials. Part I: Theory

A COUPLED TRANSPORT-REACTION MODEL FOR SIMULATING

AUTOGENOUS SELF-HEALING IN CEMENTITIOUS MATERIALS –

PART I: THEORY

Haoliang Huang (1), Guang Ye (1,2) and Denis Damidot (3)

(1) Microlab, Department of Civil Engineering and Geoscience, Delft University of Technology, the Netherlands

(2) Magnel Laboratory for Concrete Research, Department of Structural Engineering, Ghent University, Belgium

(3) Civil & Environmental Engineering Departement, Ecole Nationale Supérieure des Mines de Douai, Douai, France

Abstract

In this study, a coupled transport-reaction model was specifically developed for simulating autogenous self-healing in cementitious materials. The diffusion of ions between the solution in the crack and in the bulk paste is simulated with a transport model. The concentration of ions in the crack can be calculated. As the kinetics of chemical reactions is faster than diffusion, the chemical reactions taking place in the crack can be simulated by thermodynamic calculation. With the thermodynamic calculation, the amount of reaction products formed in the crack and the remaining ion concentrations after the chemical reactions can be determined. In this way, autogenous self-healing in cementitious materials is simulated iteratively by means of diffusion calculation then thermodynamic calculation at each time step.

Keywords: Autogenous self-healing, coupled transport-reaction model, crack, cementitious materials

1. INTRODUCTION

In the presence of water in cracks, autogenous self-healing of the cracks can take place and can prolong the service life of the concrete structures significantly [1]. It was found that the kinetics of self-healing in cementitious materials is strongly influenced by several factors, such as the amount of reactive material particles [2], the crack width [3] and the initial ion concentrations in the crack [4]. It is expensive and time-consuming to quantify self-healing by conducting experiments only. A model that can predict the self-healing kinetics under various conditions would be very helpful and will save time and money.

By now, there have been a few modeling work reported about self-healing in cementitious materials. Remmers et al. [5] modeled the recovery of strength and stiffness of cement paste due to self-healing. However, this approache have a phenomenological characteristic and lack

of detailed analysis on the physico-chemical process of self-healing [5]. It would mean a step forward to simulate the self-healing process based on the physico-chemical reactions and thus predict the amount of reaction products formed in cracks. He et al. [6] also used a hydration model to investigate the potential of self-healing due to further hydration of unhydrated cement clinker while simplifying cracks as planes intersecting the unhydrated cement particles. Lv and Chen [7] improved this method by taking into account the real patterns of cracks in the simulation. However, in these models the evolution of self-healing is simulated only by expanding the particles at the crack surfaces which originally represent the unhydrated cores. Thus, the simulated self-healing process is not consistent with the experimental observations presented in [2], where the reaction products are not only formed around the reactive material, but also at the adjacent crack surfaces. Moreover, the ion exchange between the solution in cracks and in the bulk matrix is not taken into account in the aforementioned models. In order to mimic the physico-chemical process of autogenous self-healing, a model that takes into account the chemical reaction and ion transport is needed. In this study, a coupled transport-reaction model is developed for simulating the physico-chemical processes of autogenous self-healing. The computational algorithm of the model is described. The related theories are presented. Validation of the model will be presented in Part II of this paper: A coupled transport-reaction model for simulating autogenous self-healing in cementitious materials - Part II : validation.

2. COUPLED TRANSPORT-REACTION MODEL

The physico-chemical processes involved during autogenous self-healing can be described as: diffusion of ions in the solution in the crack and in the bulk paste, dissolution of reactive material and precipitation of solid reaction products in the crack. When water penetrates into the crack, the reactive material, i.e. unhydrated cement and blast furnace slag present at the crack surfaces comes in contact with water and starts to dissolve (In this paper, reactive material refers to the material with high reactivity). At the same time, an exchange of ions takes place between the pore solution in the bulk paste and the solution in the crack. When the concentrations of ions in the crack reach the required supersaturation level for precipitation of reaction products, solid reaction products are formed in the crack.

In the coupled transport-reaction model, it is assumed that the crack is filled with water. The dissolution of reactive material, the diffusion of ions in the crack and the precipitation of reaction products in the crack are simulated. In this section, the computational algorithms for the simulation of the healing processes and the relevant theories are described in detail.

2.1 Computational algorithm

The simulation of the self-healing process in cementitious materials is performed in 2 dimensions (2D). A 2D microstructure of cement-based paste can be obtained from cement hydration and microstructure models, such as HYMOSTRUC3D model [8-10] and CEMHY3D model [11], or directly from backscattering electron (BSE) images. The 2D microstructure with a crack is represented with pixels. As illustrated in Figure 1 (a), the white pixels represent the crack and the grey pixels represent the hydration products in the bulk paste formed before cracking. The black pixels represent reactive material. It can be seen that in Figure 1 (b) the area of the crack surfaces can be classified into two groups: the area of reactive material (Acr,rm) and the area of hydrates (Acr,se). The hydrates at the crack surfaces

“seeding effect” [12]. In the coupled transport-reaction model, the seeding effect of the hydrates at the crack surfaces is taken into account.

To simulate the healing process, the period of self-healing is discretized into small time steps and the healing process is simulated iteratively. In each time step, the dissolution of reactive material is described as boundary conditions for simulating ion diffusion. With the information about the dissolution of reactive material and the initial concentrations of ions in the solution in the crack, the concentrations of ions in each micro-pixel before the formation of reaction products is determined and form input for the thermodynamic model. With the thermodynamic model the amount of reaction products and the remaining ion concentrations after chemical reactions are calculated. In the iteration process, the remaining ion concentrations after the formation of reaction products form the initial conditions for the next calculation step. Area of seed (non-reactive material) at the crack surfaces: Acr,se Area of reactive material at the crack surfaces: Acr,rm Crack surfaces: Acr = Acr,rm + Acr,se HP Boundary 1 Boundary 2 HP HP Boundary 1 Boundary 2 Boundary 1 Boundary 2 NM 5 0 µ m 5 0 µ m

(a) 2D sample of microstructure for the simulation (b) Scheme of the crack surface in 3D view UHC Crack Micro-pixels Acr,rm Acr,se 100 µm 1 0 µ m

Figure 1: Scheme of the sample for simulation of self-healing. UHC refers to unhydrated cement and other reactive material. NM refers to hydrates in the bulk paste formed before

cracking.

2.2 Reaction of reactive material present at the crack surfaces

2.2.1 Phase-boundary reaction

For the reaction of reactive material at the crack surfaces, because of the seeding effect of hydrates at the crack surfaces, the reaction products can nucleate and grow not only at the surfaces of reactive material, but also at "seed" surfaces (see Figure 2). This can change the rate of reaction of reactive material. Moreover, the amount of water in the crack for the reaction of reactive material is large. The induction period is, therefore, shortened remarkably [13]. Considering these effects, the rate of phase-boundary reactions can be described as:

where

δ

_P [m] is the penetration depth of the reactive material, which is the thickness of the layer of material that has reacted (as illustrated in Figure 2); R [m/s] is the basic dissolution ₀

0 2 1 R dt d _P ⋅ ⋅ =ε ε δ (1)

rate of reactive material, which depends mostly on tune value of the undersaturation of the aqueous phase;

ε

₁ [-] is the parameter used to describe the “induction period” of the reaction as affected by the large amount of water in the crack;

ε

₂ [-] is the parameter used to described the acceleration of the reaction due to the seeding effect of hydrates at the crack surfaces (Acr,se in Figure 1). P δ Inner product Ion diffusion Reaction product formed in

the crack Crack

Inner solid at crack surfaces (“seed” surfaces)

Reactive material

Figure 2: Scheme for reaction of reactive material present at the crack surfaces.

δ

_P is penetration depth, i.e. the thickness of the layer of material that has reacted. Inner product is

the reaction product that occupies the original space of the material that has reacted. Hence, the thickness of the layer of inner product is equal to

δ

_P.

2.2.2 Diffusion-controlled reaction

As mentioned above, because of the seeding effect of the hydrates at the crack surfaces, some reaction products also form at the “seed surface” in the crack. The increase of the thickness of the layer of outer product at the surface of inner product is much slower than without this seeding effect. Moreover, because of the large space in the crack, the outer products deposit very loosely. The change from a phase-boundary reaction to a diffusion-controlled reaction is, therefore, mainly determined by the thickness of the layer of inner product

δ

_P. The rate of diffusion-controlled reaction can be calculated by the formula:

(

P Ptr

)

P P R dt d , 6 0 2 1 10

δ

δ

δ

ε

ε

δ

_≥ × ⋅ ⋅ = ₍₂₎

where δ_P,tr [m] is the transition thickness particularly for the reaction of reactive material at crack surfaces.

2.3 Diffusion of ions 2.3.1 Fick’s second law

In the healing process, ion diffusion takes place between the solution in the crack and the solution in the matrix. This ion diffusion can be described by Fick’s second law [14]:

( ) i i i c D c t ∂ =∇⋅ ∇ ∂ (3)

where c [mol/m_i 3] is the concentration of the ith ion and D_i [m2/s] is the diffusion coefficient in the solution in the crack for the ith ion.

In the paste sample used for the simulation, there are different phases, i.e. crack, reactive material and hydrates (see Figure 1). There are 3 types of interfaces between these different phases in the paste sample: interface between reactive material and the crack; interface

between hydrates material and the crack and interface between reactive material and the hydrates material in the bulk paste. In the model, there are 3 corresponding boundary conditions, which will be described in the following subsections.

2.3.2 Boundary condition for the interface between reactive material and the crack From the area of the reactive material (Acr,rm in Figure 1), the mole numbers of ions

released into the crack is related to the volume of reactive material that has been dissolved:

( )

t f

( )

V

( )

t

i =

θ

(4)

With the chemical composition of the reactive material, the number of ions (i.e. Ca2+, H2SiO42- etc.) that are released from the dissolved reactive material can be calculated. The released ions form rather dense inner products and more loosely packed products in the crack (see Figure 2). Hence the mole number

θ

_i

( )

t of ions that are released from the reactive material can also be expressed as:

( )

t _iinp

( )

t _icrp

( )

t

i θ, θ,

θ = − (5)

where θ_i,inp

( )

t [mol] is the mole number of different ion used for inner products ; θ_i,crp

( )

t [mol] for the reaction products formed in the crack.

In case that the reactive material is cement or slag, because the solubility of C-S-H, i.e. the supersaturation level for precipitation of C-S-H, is low [15], C-S-H tends to precipitate near the cement particles [16]. In the model, it is assumed that the reaction product occupying the original space of the reactive material, i.e. cement and slag, in the depth of δ_P consists of C-S-H only. This layer of C-C-S-H only fills the original space of the reactive material that has dissolved but not the space of the crack. Hence, the produced inner product, i.e. C-S-H, does not contribute to filling of the crack.

2.3.3 Boundary condition for the interface between hydrates and the crack

At the surface of hydrates (A_cr,se, see Figure 1) at the crack surfaces, ion exchange takes place between the solution in the crack and the pore solution in the bulk paste. The mole numbers of ions passing A_cr,se can be expressed as [17]:

( )

t

(

c c

)

dA dt

dθ_i,b−crack =−σ⋅ _i,crack − _i,bulk ⋅ _cr,se⋅ (6)

where σ [m2/s] is the coefficient of the ion exchange between the solution in the crack and pore solution in the bulk paste; c_i,crack [mol/m3] is the concentration of the ith ion in the solution in the crack (near the crack surfaces), while c_i,bulk [mol/m3] is the concentration of the ith ion in the pore solution in the bulk paste (near the crack surfaces); A_cr,se [m2] is the area of the hydrates at the crack surfaces. Because the microstructure is represented with micro-pixels, dA_cr,se =x⋅1 (the value of 1 in this equation represents the unit length perpendicular to the 2D microstructure). The exchange coefficient of the ion exchange between the solution in the bulk paste and in the crack is written as [18]:

        + = B i C i D D_{, ,} 1 1 1

σ

(7)

where D_{i C,} [m2/s] is diffusion coefficient of the solution in the crack for the ith ion, while

B i

2.3.4 Boundary conditions for the interface between reactive material and hydrates in the bulk paste

As shown in Figure 1 (a), some of the reactive material is embedded in the bulk paste. Before self-healing starts, reactive material has reacted for a certain period and the corresponding penetration depth is δ_P,0 . During the healing process, reactive particles continue to react and δ_{P a,} is the additional penetration depth of the reaction front in the reactive material. When self-healing of crack starts, the reaction of the embedded reactive material has reached the diffusion-controlled stage and the reaction rate of the reactive material during the healing process can be described as:

(

)

6 , 0 , 0 10 × + = a P P P R dt d

δ

δ

δ

(8) where R [m/s] is the basic dissolution rate of the reactive material, which depends on the ₀ chemical composition;

δ

_P,0 [m] is the penetration depth of the reactive material before self-healing;

δ

_{P a,} [m] is the additional penetration depth reached during self-healing.

During the healing process some of the dissolved ions are consumed and form reaction products occupying the original space of the reactive material that has reacted. The remaining part of the dissolved ions is released to the pore solution. This remaining part of dissolved ions forms reaction products in capillary pores or in the crack. In case that the reactive material is cement or slag, because the solubility of C-S-H, i.e. the supersaturation level for precipitation of C-S-H, is low [15], C-S-H tends to precipitate near the cement and slag particles [16]. In the model, it is assumed that the reaction product occupying the original space of the reactive material, i.e. cement and slag, in the depth of

δ

_{P a,} consists of C-S-H only. The mole numbers of remaining ions released into the pore solution can be calculated:

( )

t _ia

( )

t _iinpa

( )

t

bulk r

i,

θ

,

θ

, ,

θ

₋ = − (9)

where

θ

_i,r−bulk

( )

t [mol] is the mole number of different ion diffusing into the pore solution in the bulk paste;

θ

_i,a

( )

t is the total mole number of different ion released during self-healing from the dissolving cement, which can be calculated with Equations 4;

θ

_i,inp,a(t) the mole number of different ions forming inner product, i.e. C-S-H, that occupies the original space of the material that has reacted.

2.4 Thermodynamic calculation of chemical reaction in the crack

After knowing the concentrations of different ions in the solution in the crack, the precipitation of solids in cracks can be simulated with a thermodynamic calculation considering thermodynamic equilibrium, mass balance and charge balance.

The system contains N number of chemical species at the initial stage (non-equilibrium). The system becomes in equilibrium after l number of independent chemical reactions take place. After equilibrium has been reached, there are M number of types of chemical species in the system. The chemical species include ions and solid. The charge in the system should be in balance, which is written as [19]:

1 1 N M i i j j i j Z c Z c = = =

∑

∑

(10)

where Z_i [-] is the valence (including sign) of the ith species at the initial stage (For example,

i

Z would be -2 for H2SiO42-) and ci [mol/m

3

initial stage. Z [-] is the valence (including sign) of the jj

th

species and c [mol/mj

3

] is the concentration of the jth species after the reactions.

The mass of each chemical element before and after chemical reaction should be in balance as well [19]: , , 1 1 N M e i i e j j i j b c b c = = ⋅ = ⋅

∑

∑

(11)

where b [-] is the number of atoms of element e i, e in the chemical formula of the i th

species before reactions, while be j, [-] is the number of atoms of element e in j

th

species after reactions. For example, in the mass balance equation for the element Ca, b_e should be 2 for the species (Ca(OH)2)2(SiO2)2.4·(H2O)2.

The equilibrium equation for the kth chemical reaction can be expressed as [19]:

1 1

log log log 0

N M N M k ki i ki i i i K ν β ν c + + = = − +

∑

+

∑

= (12)

where K [-] is the equilibrium constant for the k_k th chemical reaction ( k≤l); v [-] is the _ki coefficient before the ith species in the kth chemical reaction (for example, v is 2 for OH_ki - in the chemical equation: Ca(OH)2 ↔ Ca2+ + 2OH-); βi [-] is the activity coefficient of the i

th

species, which can be calculated by the extended Debye-Huckel equation [19]. The set of Equations 10, 11 and 12 can be generalized as:

( )

c_i 0

Ψ = (13)

Equation 13 can be solved iteratively by using Newton-Raphson iteration method [20]. In this research, the geo-chemical modeling code JCHESS [21] is used for the thermodynamic calculation. By using JCHESS, the concentration of the solution at equilibrium stage and the amount of precipitated solids can be calculated.

3. CONCLUSIONS

In this paper, a coupled transport-reaction model was specifically developed for simulating autogenous self-healing in cementitious materials. The diffusion of ions between the solution in the crack and the pore solution in the bulk paste is described by Fick’s second law. Chemical reactions in the crack during the healing processes are simulated by thermodynamic modelling.

The dissolution rate of reactive material at the crack surfaces is faster than the dissolution rate of cement mixed with water or a solution to generate a bulk paste. The change from a phase-boundary reaction to a diffusion-controlled reaction is delayed. This is due to the seeding effect of the crack surfaces. The reaction products can be formed not only at the surface of reactive material, but also at the “seed” surface, i.e. hydrates or other solids at the crack surfaces. This seeding effect should be taken into account for simulating autogenous self-healing of the crack.

As some parameters of the model have to be determined experimentally, the second part of this paper will present the determination of these parameters and the validation of the model for the first step of autogenous self healing that occurs just after the opening of the crack and thus before carbonation.

ACKNOWLEDGEMENTS

The authors would like to thank the National Basic Research Program of China (973 Program: 2011CB013800) and the China Scholarship Council (CSC) for the financial support. REFERENCES

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