Coupled diffusion of chloride and other ions in saturated concrete

Abstract

Corrosion of reinforcing steel due to chloride ions is one of the severe deterioration problems in long-term performance of reinforced concrete structures. The deterioration process is frequently found in marine concrete structures, highway pavements, and bridges exposed to deicing salts. The diffusion of chloride ions is associated and strongly affected by other ions in the pore solution in concrete. In this paper, chloride penetration into concrete structures was mathematically characterized by the Nernst-Planck equation which considered not only diffusion mechanism of the chloride ions but also ionic interaction among other ions coming from externally applied deicers and within the Portland cement paste. Electroneutrality was used to determine the electrostatic potential induced by the ionic interaction. The material models of chloride binding capacity and chloride diffusion coefficient were incorporated in the governing equations. The governing equations were solved by using finite element method. A numerical example was used to illustrate the coupling effect of multi-ionic interactions and the effect of influential parameters. The numerical results obtained from the present model agreed very well with available test data.

Keywords

diffusion / chloride / concrete / Nernst-Planck equation / durability

Introduction

Chloride-induced corrosion in reinforced concrete structures is one of the severe problems affecting long-term performance of the structures. Chloride ions are usually from deicing chemicals used on the road in the winter. When chloride ions reach the critical value in concrete and the passivity layer of reinforcement steel is destroyed, the corrosion of steel takes place and the rust starts to form. The rust deposited around the interface between embedded steel and surrounding concrete has larger volume than the steel consumed to form the rust, resulting in large volume expansion. The volume expansion leads to cracking, spalling, and delamination of concrete cover in the reinforced concrete structures. The corrosion damage may significantly reduce the load-bearing capacity of the structural member by a reduction of cross-sectional area of steel bar. To understand the mechanisms of the chloride-induced corrosion and to predict the long-term performance of the structure, mathematical models characterizing the deterioration process need to be developed.

There are three stages in the deterioration process. The first one is the chloride penetration into concrete; the second is the rust formation and accumulation in the interface between rebar and concrete; and the third is the crack (damage) development in concrete. Among the three stages, the first one is the longest and thus the most important to ensure an excellent long-term performance of reinforced concrete structures. It is therefore very important to characterize the penetration process of chloride into concrete structures. During the penetration process of the chloride, there are other ions moving together with the chloride, and these ions have significant effect on the rate of penetration of the chloride, which is referred to as the coupling effect in multi-species diffusion in porous media. The coupling effect has not been studied extensively, and much research has been focused only on the diffusion of chloride in concrete. The main purpose of this paper is to develop a prediction model for simulating the penetration process of chloride into concrete structures based on multi-ionic transport theory taking into account the coupling effect among the diffusing species.

Sodium chloride, calcium chloride, and magnesium chloride are the major types of deicing salts commonly used on highway pavement and bridge decks. Frequently, different types of deicers are applied to the same area on the road for ice and snow control. For example, magnesium chloride solution is used before a snow storm for ice control, and calcium chloride solid salt is used during the snow storm for snow control. As a result, multiple components of salts co-exist on the surface and ingress into concrete simultaneously. Therefore, the penetration rate of chloride into concrete is affected not only by the concentration gradient of the chloride ions but also by the ionic interaction among the chloride and other ions. In addition to the other ions co-exist in deicers with chloride ions, there are many other ions existing in the concrete pore solution prior to the presence of deicers, such as hydroxyl ions (OH^-), sodium ions (Na⁺), and potassium ions (K⁺).

In recent years, many researchers have investigated the penetration of chloride into concrete structures. There have been two ways for formulating the mathematical models of chloride penetration into concrete. One is based on phenomenological model, i.e. Fick’s second law [1-4]. The other is based on the transport theory of multi-ions that is the Nernst-Planck equation taking into account the physical and chemical factors related to the diffusion of chloride and other ions in electrolyte diffusion in concrete. Mathematical models developed based on this approach can be used to describe and simulate the transport mechanism of multiple ions in cement-based materials.

Nguyen et al. [5] developed a mathematical and numerical model based on the Nernst-Planck equation to predict the chloride ingress into saturated concrete. Their model was developed based on an experimental study for determining the chloride diffusion coefficients and chloride binding capacity. The electroneutrality assumption was employed to calculate the balance of ionic interaction between the multi-ions in concrete pore solution. Then, the numerical results were validated by test data obtained from concrete samples under different exposure conditions. Li and Page [6] and Wang et al. [7] proposed a finite element model for characterizing the chloride removal technique. The model was developed based on the Nernst-Planck equation related to the ionic interaction of multi-ions in concrete pore solution and the adsorption and desorption of ions between the solid and liquid phases in concrete. The influences of boundary conditions, porosity, and tortuosity of concrete were taken into account in the model. Later on, Wang et al. [8] further extended their model by incorporating the convection of pore solution to predict the chloride penetration into concrete. Numerical results were compared with the solution obtained from Fick’s second law and test data. Samson et al. [9] presented a mathematical model for ionic diffusion in porous media. This model was developed based on the Nernst-Planck equation. Poisson’s equation was used to evaluate the electrical coupling among ions. These ions were assumed to be in an ideal solution (no chemical activity effects were considered). Numerical results were obtained by using a 1D finite element method. Then, Samson and Marchand [10] extended the model to account for the effect of chemical activity (ions in non-ideal solution). The mathematical model was formulated by taking into account the chemical activity in the governing equations. Lately, Samson and Marchand [11] proposed a transport model for ions in unsaturated cement-based materials, which was intended to model the leaching process of sulfate, calcium, or hydroxide. The mathematical model was developed based on a set of Nernst-Planck/Poisson equation by taking into account the diffusion, electrical coupling, chemical activity, and advection mechanisms. The model considered that the diffusion process took place under an isothermal condition.

In this paper, a comprehensive model of multi-ionic diffusion including the chloride penetration into saturated concrete was developed. The mathematical model was formulated based on the Nernst-Planck equation, and the interaction among the ions was dealt with by three methods: nil current, electroneutrality, and Poisson’s equation. The numerical results from the three methods were compared. The material parameters involved in the governing equations were not considered simply as constants as in previous studies but characterized by sophisticated material models, which were established based on the models for binding capacity and diffusion coefficient of chloride developed by Xi and Bazant [12]. The finite element method was used to solve the coupled governing equations. A numerical example was presented in which two different types of deicing salts, calcium chloride and sodium chloride, were applied as boundary conditions on the top surface of a concrete specimen. Ions in concrete pore solution, hydroxyl, sodium, and potassium ions, were also included in the example.

Basic formulation of governing equations

The flux of each ion transport in an ideal solution in porous media (i.e. concrete) based on the Nernst-Planck equation without considering the chemical activity effects can be described as:

(1)J_i=-D_i\nabla C_i-z_i{D}_i\left({\displaystyle \frac{F}{RT}}\nabla \phi \right)C_i,

where J_i is the flux, D_i the diffusion coefficient, C_i the concentration, z_i the charge number for the diffusing ion i, respectively; F is the Faraday’s constant, R is the gas constant, T is the temperature, Ф is the electrostatic potential, and index i represents i-th species. The mass balance equations for each ionic species can be expressed as:

(2){\displaystyle \frac{\partial (C_i+S_i)}{\partial t}}=-\nabla J_i.

By substituting Eq. (1) into Eq. (2), we have

(3){\displaystyle \frac{\partial (C_i+S_i)}{\partial t}}=\nabla \left(D_i\nabla C_i+z_i{D}_i\left({\displaystyle \frac{F}{RT}}\nabla \phi \right)C_i\right),

in which S_i is the bound ion concentration. Eq. (3) describes the transport process of each ion in an ideal solution, which can be interpreted in terms of the diffusion due to concentration gradient of free ions, as shown the first term in right hand side, and the migration or the electrical coupling between different free ions, as shown in the second term.

It is noticed from Eq. (3) that, in order to solve the Nernst-Planck equation, another equation accounted for electrostatic potential induced by ionic interaction is required. The electrostatic potential can be determined based on three different methods. Samson and Marchand [11] solved the electrostatic potential for ion transport in cement-based materials by using Poisson’s equation described as Eq. (4). As mentioned by Samson et al. [9], the Poisson’s equation can be used in more general cases. The nil current assumption was used by Li and Page [6], Wang et al. [7], and Wang et al. [8]. This method is based on the fact that there is no net current flow within the concrete pore solution, which is described by Eq. (5). Differing from those researchers, in this study, the electroneutrality approach was used to determine the electrostatic potential, which can be expressed by Eq. (6). The electroneutrality condition is based on the fact that the charges from all ions exist in the solution must be balanced out as zero.

(4){\nabla }^{\mathrm{2}}\varphi +{\displaystyle \frac{F({\displaystyle \sum z_i{C}_i+w})}{ϵ}}=\mathrm{0},

(5)I=F{\displaystyle \sum _{i=\mathrm{1}}^n{z}_i}{J}_i,

(6){\displaystyle \sum _{i=\mathrm{1}}^n{C}_i}{z}_i=\mathrm{0},

in which w is the fixed charge density, ϵ is the dielectric constant of the surrounding medium, and I is the external current density.

A study was conducted by Nguyen et al. [5] to compare among nil current, Poisson’s equation, and electroneutrality condition in modeling of chloride penetration into concrete. The study showed a good agreement of concentration profiles of chloride and other ions obtained by using the three different methods. In the present study for multi-ionic diffusion in concrete, the differences in concentration profiles of multiple ions obtained by using the three different methods were investigated (material models and numerical algorithms will be discussed later). A concrete sample was exposed to 1 mol/L of NaCl. The numerical results are plotted in Fig. 1. As noticed from Fig. 1, the concentration profiles of chloride, sodium, potassium, and hydroxyl ions obtained based on the nil current, electroneutrality and Poisson’s equation have a very good agreement.

Based on the comparative study, the electroneutrality method was used in the present study. One of the advantages of using electroneutrality condition rather than Poisson’s equation is the simplicity of the governing equations, which can significantly reduce the computation time. More importantly, a difficulty in computational schemes associated with the Poisson’s equation can be avoided. As shown in Eq. (4), F/ϵ ≈ 10¹⁶, which is a very large number often causing numerical difficulty in the computation [5,8].

Material parameters

To perform numerical simulations and obtain numerical results from the governing equations, the material parameters involved in the equations must be defined first. Among these parameters are chloride diffusion coefficient (D_Cl) and chloride binding capacity (\partial C_{\mathrm{l}\mathrm{f}}/\partial C_{\mathrm{l}\mathrm{t}}). Previous researchers described the value of chloride diffusion coefficient as a constant. In reality, it is not a constant but a function depending on many parameters, such as mix design parameters of concrete. In this study, we consider the chloride diffusion coefficient as a function of water-cement ratio, curing time, diffusivity of cement paste and aggregate. The effects of internal relative humidity, temperature, and free chloride concentration on the diffusion coefficient are considered by a material model proposed by Xi and Bazant [12].

Chloride diffusion coefficient

The diffusion coefficient of chloride ions in concrete can be estimated using the multifactor method as follows:

(7)D_{\mathrm{C}\mathrm{l}}=f_{\mathrm{1}}(w/c,t_{\mathrm{0}})f_{\mathrm{2}}(g_i)f_{\mathrm{3}}(H)f_{\mathrm{4}}(T)f_{\mathrm{5}}(C_{\mathrm{f}}),

in which f_{\mathrm{1}}(w/c,t_{\mathrm{0}}) is a factor accounting for the influence of water-cement ratio (w/c) and curing time of concrete (t_{\mathrm{0}}). The higher the water-cement ratio, the higher the diffusion coefficient. A formulation for f_{\mathrm{1}}(w/c,t_{\mathrm{0}}) was proposed by Xi and Bazant [12]:

(8)f_{\mathrm{1}}={\displaystyle \frac{\mathrm{28}-t_{\mathrm{0}}}{\mathrm{62500}}}+\left({\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}+{\displaystyle \frac{(\mathrm{28}-t_{\mathrm{0}})}{\mathrm{300}}}\right){\left({\displaystyle \frac{w}{c}}\right)}^{\mathrm{6.55}}.

The second factor, f_{\mathrm{2}}(g_i), is to deal with the effect of composite action of the aggregates and the cement paste on the diffusivity of concrete. This factor can be calculated by using the three phase composite model developed by Christensen [13]:

(9)f_{\mathrm{2}}(g_i)=D_{\mathrm{c}\mathrm{p}}\left(\mathrm{1}+{\displaystyle \frac{g_i}{(\mathrm{1}-g_i)/\mathrm{3}+\mathrm{1}/(D_{\mathrm{a}\mathrm{g}\mathrm{g}}/D_{\mathrm{c}\mathrm{p}}-\mathrm{1})}}\right),

where g_i is the volume fraction of aggregates in concrete, and D_{\mathrm{a}\mathrm{g}\mathrm{g}} and D_{\mathrm{c}\mathrm{p}} are the diffusivities of aggregates and cement paste, respectively. These two parameters can be determined by using the model proposed by Martys et al. [14]:

(10)D={\displaystyle \frac{\mathrm{2}(\mathrm{1}-(V_{\mathrm{p}}-V_{\mathrm{p}}^{\mathrm{c}}))}{S^{\mathrm{2}}}}{(V_{\mathrm{p}}-V_{\mathrm{p}}^{\mathrm{c}})}^{\mathrm{4.2}},

in which V_{\mathrm{p}} is the porosity, S is the surface area, and V_{\mathrm{p}}^{\mathrm{c}} is the critical porosity (the porosity at which the pore space is first percolated). When Eq. (10) is used for the diffusivity of cement paste, D_{\mathrm{c}\mathrm{p}}, V_{\mathrm{p}}, S, and V_{\mathrm{p}}^{\mathrm{c}} are considered as the parameters for cement paste. The critical porosity may be taken as 3% for cement paste [14]. Based on the study proposed by Xi et al. [15], the surface areas of cement paste, S, can be estimated by the monolayer capacity, V_m, of adsorption isotherm of concrete which is proportional to S. The porosity, V_p, can be estimated by adsorption isotherm, n(H,T) = W_sol/W_conc at saturation (H = 1). W_sol and W_conc are the weight of pore solution and concrete, respectively. More detail on adsorption isotherm can be found in Xi et al. [15,16]. The diffusivity of aggregates, D_{\mathrm{a}\mathrm{g}\mathrm{g}}, can be taken as a constant, and a proposed value is 1×10^-12 cm²/s.

The third factor, f_{\mathrm{3}}(H), is to take into account the effect of internal relative humidity on the chloride diffusion coefficient. A model proposed by Bazant and Najjar [17] can be applied for this factor:

(11)f_{\mathrm{3}}(H)={\left(\mathrm{1}+{\displaystyle \frac{{(\mathrm{1}-H)}^{\mathrm{4}}}{{(\mathrm{1}-H_C)}^{\mathrm{4}}}}\right)}^{-\mathrm{1}},

in which H_{\mathit{C}} is the critical humidity level at which the diffusion coefficient drops halfway between its maximum and minimum values (H_C = 0.75). Equation (11) is very important in terms of influencing on the coupling between the moisture and chloride diffusion.

The fourth factor, f₄(T), is to consider the effect of temperature on the diffusion coefficient of concrete. This can be calculated by using Arrhenius’ law:

(12)f_{\mathrm{4}}(T)=\exp \left[{\displaystyle \frac{U}{R}}\left({\displaystyle \frac{\mathrm{1}}{T_{\mathrm{0}}}}-{\displaystyle \frac{\mathrm{1}}{T}}\right)\right]

in which U is the activation energy of the diffusion process, R is the gas constant (8.314 J·mol^-1·K^-1), T and T₀ are the current and reference temperatures, respectively, in Kelvin (T₀ = 296 K). According to the studies by Page et al. [18] and Collepardi et al. [19], the activation energy of the diffusion process depends on water-to-cement ratio, w/c, and cement type which can be found in Table 1.

The fifth factor, f_{\mathrm{5}}(C_f), describes the dependence of the chloride diffusion coefficient on the free chloride concentration which can be expressed as follows:

(13)f_{\mathrm{5}}(C_f)=\mathrm{1}-k_{\mathrm{i}\mathrm{o}\mathrm{n}}{(C_f)}^m,

where k_ion and m are two constants, 8.333 and 0.5, respectively. k_ion and m were obtained by Xi and Bazant [12].

Chloride binding capacity

Generally, when chloride ions ingress into concrete, some of them will be bound to the wall of internal poresin concrete, and they are called bound chloride. The others called free chloride can diffuse freely through the porous media. The total chloride concentration is the summation of free chloride, C_f, and bound chloride, C_b, that is

(14)C_{\mathrm{t}}=C_{\mathrm{f}}+C_{\mathrm{b}}.

The chloride binding capacity is defined as the incremental ratio of the free chloride content and total chloride content:

(15){\displaystyle \frac{\mathrm{d}{\mathit{C}}_{\mathrm{f}}}{\mathrm{d}{\mathit{C}}_{\mathrm{t}}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\displaystyle \frac{\mathrm{d}{\mathit{C}}_{\mathrm{b}}}{\mathrm{d}{\mathit{C}}_{\mathrm{f}}}}}}.

The term \mathrm{d}{C}_{\mathrm{b}}/\mathrm{d}{C}_{\mathrm{t}} can be obtained experimentally. By using the model developed by Xi and Bazant [12], the chloride binding capacity can be expressed as:

(16){\displaystyle \frac{\mathrm{d}{C}_{\mathrm{f}}}{\mathrm{d}{C}_{\mathrm{t}}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\displaystyle \frac{A{\mathrm{10}}^B{\beta }_{\mathrm{C}-\mathrm{S}-\mathrm{H}}}{\mathrm{35},\mathrm{450}{\beta }_{\mathrm{s}\mathrm{o}\mathrm{l}}}}{\left({\displaystyle \frac{C_{\mathrm{f}}}{\mathrm{35.45}{\beta }_{\mathrm{s}\mathrm{o}\mathrm{l}}}}\right)}^{A-\mathrm{1}}}},

where A and B are two material constants related to chloride adsorption and equal to 0.3788 and 1.14, respectively [20]. The binding capacity depends on the two parameters, {\beta }_{\mathrm{s}\mathrm{o}\mathrm{l}} and {\beta }_{\mathrm{C}-\mathrm{S}-\mathrm{H}}.

The parameter {\beta }_{\mathrm{s}\mathrm{o}\mathrm{l}} is described as the relationship between the volume of pore solution and weight of concrete (L/g):

(17){\beta }_{\mathrm{s}\mathrm{o}\mathrm{l}}={\displaystyle \frac{V_{\mathrm{s}\mathrm{o}\mathrm{l}}}{w_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}}}={\displaystyle \frac{w_{\mathrm{s}\mathrm{o}\mathrm{l}}}{{\rho }_{\mathrm{s}\mathrm{o}\mathrm{l}}{w}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}}}}={\displaystyle \frac{n(H,T)}{{\rho }_{\mathrm{s}\mathrm{o}\mathrm{l}}}},

where V_sol is the volume of pore solution, w_sol is the weight of pore solution, w_conc is the weight of concrete, {\rho }_{\mathrm{s}\mathrm{o}\mathrm{l}} is the density of the pore solution (g/L) and is dependent on chloride concentration. To simplify the calculation, the parameter {\rho }_{\mathrm{s}\mathrm{o}\mathrm{l}} can be estimated by using the density of pore water. The weight ratio of pore solution to concrete (w_sol/w_conc) represents chloride adsorption isotherm which is related to relative humidity, H, temperature, T, and pore structure of concrete. Due to a lack of test data on chloride isotherm, n(H,T) may define as the isotherm of water adsorption instead of chloride isotherm. The adsorption isotherm of concrete can be described in terms of adsorption isotherm of cement paste and aggregate as follows:

(18)n(H,T)=f_{\mathrm{c}\mathrm{p}}{n}_{\mathrm{c}\mathrm{p}}(H,T)+f_{\mathrm{a}\mathrm{g}\mathrm{g}}{n}_{\mathrm{a}\mathrm{g}\mathrm{g}}(H,T),

in which f_{\mathrm{c}\mathrm{p}} and f_{\mathrm{a}\mathrm{g}\mathrm{g}} are the weight percentages of cement paste and aggregates, and n_{\mathrm{c}\mathrm{p}}(H,T) and n_{\mathrm{a}\mathrm{g}\mathrm{g}}(H,T) are the water adsorption isotherms of cement paste and aggregate.

The parameter {\beta }_{\mathrm{C}-\mathrm{S}-\mathrm{H}} can be explained as the weight ratio of C-S-H gel to concrete (g/g). This factor is used to determine the effect of the cement composition and age of concrete on the volume fraction of C-S-H gel which is written as:

(19){\beta }_{\mathrm{C}-\mathrm{S}-\mathrm{H}}={\displaystyle \frac{w_{\mathrm{C}-\mathrm{S}-\mathrm{H}}}{w_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}},

where w_{\mathrm{C}-\mathrm{S}-\mathrm{H}} and w_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}} are the weight of C-S-H gel and the total weight of concrete. The details of parameters n(H,T) and {\beta }_{\mathrm{C}-\mathrm{S}-H} can be found in the paper by Xi et al. [15].

The limitation of binding capacity based on the Freundlich isotherm, Eq. (16), is that the term \partial C_{\mathrm{f}}/\partial C_{\mathrm{t}}=\mathrm{0} when the free chloride concentration, C_f, is zero because of A<1. As a result, \partial C_{\mathrm{f}}/\partial C_{\mathrm{t}}=\mathrm{0} leads to \partial C_{\mathrm{f}}/\partial C=\mathrm{0}. This can be concluded that C_f is a constant at all time steps and equals to initial free chloride concentration. Thus, chloride diffusion never gets to start. To solve this problem, Tang and Nilsson [20] suggested that the Freundlich isotherm can be used when C_f is large (>0.01 mol/L), and the Langmuir isotherm is employed when C_f is small (<0.05 mol/L). For these reasons, in the present study, the chloride binding capacity is represented by Langmuir isotherm for initial free chloride concentration (C_f = 0), and while the free chloride concentration is more than zero (C_f>0), the chloride binding capacity can be determined by Eq. (16) based on Freundlich isotherm. The Langmuir isotherm is expressed as:

(20){\displaystyle \frac{\mathrm{1}}{C_{\mathrm{b}}^{\prime }}}={\displaystyle \frac{\mathrm{1}}{k^{\prime }{C}_{\mathrm{b}\mathrm{m}}}}{\displaystyle \frac{\mathrm{1}}{C_{\mathrm{f}}^{\prime }}}+{\displaystyle \frac{\mathrm{1}}{C_{\mathrm{b}\mathrm{m}}}},

where k^{\prime } is an adsorption constant, and C_bm is the bound chloride content at saturated monolayer adsorption [20]. Based on the adsorption test data, k^{\prime } and C_bm can be obtained by curve fitting. C_b^{\prime } and C_f^{\prime } are the bound and free chloride contents used in Eq. (20). The units of these two parameters are in milligrams of bound chloride per gram of calcium silicate hydrate gel (mg/g) and in free chloride per liter of pore solution (mol/L), respectively, which is different from C_b and C_f. In the numerical simulation, it is necessary to use the consistent unit. Therefore, C_{\mathrm{b}}^{\prime } and C_{\mathrm{f}}^{\prime } can be converted and correlated to the unit of C_b and C_f as follows:

(21)C_{\mathrm{b}}^{\prime }={\displaystyle \frac{\mathrm{1000}{C}_{\mathrm{b}}}{{\beta }_{\mathrm{C}-\mathrm{S}-\mathrm{H}}}},

(22)C_{\mathrm{f}}^{\prime }={\displaystyle \frac{C_{\mathrm{f}}}{\mathrm{35.45}{\beta }_{\mathrm{s}\mathrm{o}\mathrm{l}}}},

Substituting Eqs. (21) and (22) into Eq. (20), yields

(23){\displaystyle \frac{\mathrm{1}}{C_{\mathrm{b}}}}={\displaystyle \frac{\mathrm{1000}}{{\beta }_{\mathrm{C}-\mathrm{S}-\mathrm{H}}}}\left[{\displaystyle \frac{\mathrm{35.45}{\beta }_{\mathrm{s}\mathrm{o}\mathrm{l}}}{k^{\prime }{C}_{\mathrm{b}\mathrm{m}}}}{\displaystyle \frac{\mathrm{1}}{C_{\mathrm{f}}}}+{\displaystyle \frac{\mathrm{1}}{C_{\mathrm{b}\mathrm{m}}}}\right].

Equation (23) can be re-expressed in a simple form as

(24)C_{\mathrm{b}}={\displaystyle \frac{\mathrm{1}}{\beta +{\displaystyle \frac{\mathrm{1}}{\alpha C_{\mathrm{f}}}}}},

in which,

(25)\alpha ={\displaystyle \frac{k^{\prime }{C}_{\mathrm{b}\mathrm{m}}{\beta }_{\mathrm{C}-\mathrm{S}-\mathrm{H}}}{\mathrm{35450}{\beta }_{\mathrm{s}\mathrm{o}\mathrm{l}}}},

(26)\beta ={\displaystyle \frac{\mathrm{1000}}{{\beta }_{\mathrm{C}-\mathrm{S}-\mathrm{H}}{C}_{\mathrm{b}\mathrm{m}}}},

Derivative of Eq. (24) with respect to C_f yields

(27){\displaystyle \frac{\mathrm{d}{C}_{\mathrm{b}}}{\mathrm{d}{C}_{\mathrm{f}}}}={\displaystyle \frac{\mathrm{1}}{\alpha {(C_{\mathrm{f}})}^{\mathrm{2}}{\left(\beta +{\displaystyle \frac{\mathrm{1}}{\alpha C_{\mathrm{f}}}}\right)}^{\mathrm{2}}}}.

By substituting Eq. (27) into Eq. (15), the binding capacity based on Langmuir isotherm can be expressed as:

(28){\displaystyle \frac{\mathrm{d}{C}_{\mathrm{f}}}{\mathrm{d}{C}_{\mathrm{t}}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\displaystyle \frac{\mathrm{d}{C}_{\mathrm{b}}}{\mathrm{d}{C}_{\mathrm{f}}}}}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+{\displaystyle \frac{\mathrm{1}}{\alpha {\left(\beta C_{\mathrm{f}}+{\displaystyle \frac{\mathrm{1}}{\alpha }}\right)}^{\mathrm{2}}}}}}.

Equation (28) is used to calculate the binding capacity when free chloride concentration tends to zero, then the binding capacity is \mathrm{1}/(\mathrm{1}+\alpha ). The parameter \alpha can be calculated by using Eq. (25) and is dependent on many factors. As a result, \alpha is definitely a nonzero number. The parameters in chloride binding capacity based on Langmuir isotherm can be obtained from Tang and Nilsson [20] paper that is , \mathrm{1}/C_{\mathrm{b}\mathrm{m}}=\mathrm{0.1849}, 1/\left(k^{\prime }{C}_{\mathrm{b}\mathrm{m}}\right)=0.002438, k^{\prime }=75.841, and C_{\mathrm{b}\mathrm{m}}=\mathrm{5.4083}.

Numerical model

The governing equations, Eqs. (3) and (6), can be solved by using finite element method. The finite element formulation will be briefly introduced in this study. The continuous variables, ionic species, in the system are spatially discretized over the space domain, \mathrm{\Omega }. The domain discretization can be described as:

(29)\mathrm{\Omega }=\underset{e=\mathrm{1}}{\overset{nel}{{\displaystyle \cup }}}{\mathrm{\Omega }}^e,

in which nel is the total number of elements in the space domain and {\mathrm{\Omega }}_e is a subdomain or an element. It is also defined \partial \mathrm{\Omega } as the boundary of space domain and \partial {\mathrm{\Omega }}^e the boundary of subdomain. By using isoparametric elements, the concentration of each species is defined in terms of nodal values, that is,

(30)C_i=[N]\{d_i\},

where [N] is the element shape function matrix and \{d_i\} is the nodal concentration vector of species i.

For the binding capacity of ions, the bound ion concentration is defined as S. In this study, the binding capacity is considered only for chloride ions so that the release rate of bound chlorides can be described in terms of the ratio of rates of the free to the bound chloride concentration, \lambda, as follows:

(31){\stackrel{•}{S}}_i=\lambda {\stackrel{•}{C}}_i,

in which \lambda =(\partial C_{\mathrm{b}}/\partial C_{\mathrm{f}}) which can be derived following Eq. (16). The nodal free ionic concentrations are solved by substituting Eqs. (30) and (31) into governing equations, Eqs. (3) and (6), and applying the Galerkin method to the weak forms of Eqs. (3) and (6), then the finite element matrix of transient mass diffusion can be obtained as follows:

(32)([M_i^S]+[M_i^C])\left\{\stackrel{•}{d_i}\right\}+([K_i^D]+[K_i^P])\{d_i\}=\{F_i\},

in which,

(33)[M_i^S]={\underset{\mathrm{\Omega }}{\int }[N]}^{\mathrm{T}}[{\lambda }_i][N]\mathrm{d}\mathrm{\Omega },

(34)[M_i^C]={\displaystyle \underset{\mathrm{\Omega }}{\int }{[N]}^{\mathrm{T}}[N]\mathrm{d}\mathrm{\Omega }},

(35)[K_i^D]={\displaystyle \underset{\mathrm{\Omega }}{\int }{[B]}^{\mathrm{T}}[D_i][B]}\mathrm{d}\mathrm{\Omega },

(36)[K_i^P]={\displaystyle \underset{\mathrm{\Omega }}{\int }{[N]}^{\mathrm{T}}[z_i][N]}\mathrm{d}\mathrm{\Omega }.

The matrices M_i^C, M_i^S, K_i^D, and K_i^P in above equations can be found in text for finite element analysis of transport problems, such as heat conduction. Basically, matrices M_i^C and M_i^S are similar to heat capacity matrix and K_i^D and K_i^P are similar to conductivity matrix in finite element equation of heat transfer.

Numerical results and discussions

As an example of the application of the present diffusion model, we consider two different types of deicing salts, NaCl and CaCl₂, penetrating into a concrete specimen from the top surface. The ions in concrete pore solution are K⁺, Na⁺, and OH^-. The material parameters and input data for the numerical simulation related to the governing equations are shown in Table 2. These are diffusion coefficients, initial concentrations at the top surface, initial concentrations of ionic species in concrete pore solution, water-cement ratio, and volume fraction of aggregate. The numerical simulations are performed for the concrete under the saturated condition. This means relative humidity inside and outside the concrete sample is equal to 100%. As a result, there is no effect from moisture gradient on multi-ions diffusion in concrete.

The geometry of concrete sample used in the numerical simulation is shown in Fig. 2. It is a 3 cm by 5 cm concrete cylinder. The concrete sample is exposed to 0.5 mol/L NaCl and 0.5 mol/L CaCl₂ solutions on the top surface. The other boundaries are assumed to be insulated. The concrete sample is divided into 400 elements and 451 nodes by using isoparametric elements for the finite element model.

The relation between free chloride concentration and the depth at different times of exposure is shown in Fig. 3. The depth of penetration is measured from the top surface of concrete sample. It can be seen from the Fig. 3 that the free chloride concentration decreases with increasing depth from the top surface. At a fixed depth, the longer the exposure time, the higher the free chloride concentration. Figure 4 shows the relationship between free chloride concentration and exposure time at different depths from the top surface. The trend shows that, at the fixed exposure time, free chloride concentration is higher at the depth close to the exposed surface.

The trends of sodium and calcium concentration are similar to the free chloride concentration because they have similar initial conditions. The concentration gradients of chloride, sodium, and calcium are in the same direction, from the outside (the top) to inside concrete sample. The plots of sodium and calcium concentration with the depth at different times of exposure are shown in Figs. 5 and 6, respectively.

In contrast to the concentration gradients of chloride, sodium, and calcium, the concentration gradients of potassium and hydroxyl ions are from inside to outside of the concrete sample because the inside concentrations of these ions are higher than the concentrations at the exposed surface, and in fact, there are no potassium and hydroxyl ions at the top (exposed) surface. The variations of potassium and hydroxyl concentration with depth at different times of exposure are shown in Figs. 7 and 8, respectively. As noticed, the concentrations of potassium and hydroxyl ions increase with the increasing depth from the top surface.

Comparison with experimental results

For validation, the numerical results obtained from the present model were compared with the test data conducted by Damrongwiriyanupap [21]. The experimental results are the chloride concentration profiles obtained from a 30-day ponding test with three different types of chloride solution, 3% NaCl, 3% CaCl₂, and 3% MgCl₂. The concrete samples have a 0.55 water-cement ratio. The comparisons are shown in Figs. 9, 10, and 11 for 3% NaCl, 3% CaCl₂, and 3% MgCl_2, respectively. As illustrated, the total chloride concentrations predicted by the present model agree very well with the test data.

Conclusions

1) Chloride ions from deicing salts penetrate into concrete structures, which is the main cause of the corrosion of steel bars. The penetration of the chloride ions is associated with the diffusion of other ions in the deicing chemicals and in concrete pore solution. There are coupling effects among the diffusing ions, which need to be considered in the model for chloride diffusion. A comprehensive mathematical model is developed based on the Nernst-Planck equation for multi-ionic diffusion in saturated concrete. The Nernst-Planck equation incorporates an electrostatic potential to consider the coupling effect among the diffusing ions.

2) Among the available methods to solve for the electrostatic potential, the electroneutrality method is used in the present study. The accuracy of the electroneutrality method is compared with other methods.

3) Material models for transport parameters related to chloride diffusion in concrete are included in the mathematical model. These parameters are chloride diffusion coefficient and chloride binding capacity. The material models are based on microstructural variation of cement paste and composite theory, and they take into account various influential parameters such as water-cement ratio of concrete, type of cement, curing time, and temperature.

4) The mathematical model is solved numerically by using the finite element method which is an effective tool and widely used for solving coupled partial differential equations for mass transport in porous media such as concrete. The finite element solution can give ion concentrations in concrete at any depth and at any time.

5) Comparing with previous models that only consider chloride diffusion, the present multi-ionic diffusion model is a more comprehensive mathematical model to characterize the sophisticated transport phenomena in concrete. The model includes material models for transport parameters of concrete and considers the ionic interaction between ions not only from the exposed surface but also exist in the concrete pore solution.

6) The numerical results obtained by the present model are compared with available test data from a chloride ponding test of three different chloride solutions, 3% NaCl, 3% CaCl₂, and 3% MgCl₂ at 30 days of exposure. The numerical results agree very well with the test data. Thus, the present model can be used to predict satisfactorily the penetration of multi-ionic species such as deicing salts into saturated concrete.

7) The present model can be applied to simulate the saturated concrete structures subjected to other aggressive chemicals. The framework of present model can also be extended to simulate the multi-species deicing salts ingress into non-saturated concrete structures by taking into account the moisture effect.

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