International Journal of Polymer Science

Volume 2017 (2017), Article ID 8703736, 12 pages

https://doi.org/10.1155/2017/8703736

## Simulation of Chloride Ingress through Surface-Coated Concrete during Migration Test Using Finite-Difference and Finite-Element Method

Department of Civil Engineering, Kyonggi University, Suwon 16227, Republic of Korea

Correspondence should be addressed to Seyoon Yoon

Received 20 December 2016; Revised 21 March 2017; Accepted 21 March 2017; Published 17 May 2017

Academic Editor: Joao M. L. Reis

Copyright © 2017 Seyoon Yoon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Polymer surface coatings are commonly used to enhance the corrosion resistance of concrete structures in saline environments to ionic diffusivity; this diffusivity can be determined by migration tests. This paper presents the simulation of the effects of the surface coatings on migration tests by solving the Nernst-Planck/Poisson model using both finite-difference method and finite-element method. These two numerical methods were compared in terms of their accuracy and computational speed. The simulation results indicate that the shapes of ionic profiles after migration tests depend on the effectiveness of surface coatings. This is because highly effective surface coatings can cause a high ionic concentration at the interface between coating and concrete. The simulation results were also compared to homogenized cases where a homogenized diffusion coefficient is employed. The result shows that the homogenized diffusion coefficient cannot represent the diffusivity of the surface-coated concrete.

#### 1. Introduction

The corrosion of the reinforcing steel bars by chloride ingress is a significant problem for the marine concrete structures exposed to seawater. To delay this chloride ingress, the polymer surface coating has been applied to the reinforced concrete. Previous literature clearly shows that this surface coating improves the durability of reinforced concrete (RC) structures [1–3]. However, the polymer surface coating makes the evaluation of the service life of RC structures difficult, because the diffusivity of the surface-coated concrete as a composite material becomes more complicate than bulk concrete [4]. Therefore, the simulation of chloride ingress through surface-treated concrete requires further research.

Migration tests [5, 6] provide a convenient way to estimate diffusivity of cement-based systems. The classical mathematical treatment to analyze the experimental data of the migration test solves the movement of single species and approximates the electric field by a constant equal to the electric potential difference between intervals of the sample depth divided by the sample length. Recently, Glasser et al. [7–9] used the Nernst-Planck/Poisson (NPP) model to analyze data of the migration tests [5]. This approach improves upon the classical model by including multi-ionic interaction, distribution of electric potentials, and evolution of the electric field. The application of the NPP model explains why the diffusion coefficients of identical specimens calculated from the classical model depend on the different boundary conditions. Also, a slight difference in potential profiles leads to a large difference in chloride concentration profiles, so that the assumption of a constant electric field causes significant errors in the distribution of chloride across the entire sample [5]. Furthermore, even though most cases of ionic ingress in reinforced concrete structures dominantly occur by diffusion, in aggressive environments the ingress of multiple ions, such as chloride ingress, calcium leaching, and sulfate attack, can occur in concrete simultaneously. For this reason, the NPP model including the migration term can describe the interaction between different ionic species and accurately predict the distribution of multiple ions in reinforced concrete structures.

The NPP model is considerably more complicated than the classical single-species model. For the non-steady-state and surface-coated cases, a numerical scheme is necessary to compute the NPP model. As early as 1995, Kato [10] proposed the finite-difference method (FDM) to solve the NPP model for steady-state cases; however the finite-element method (FEM) has been recently employed preferably for the NPP model [5, 6, 8]. The NPP model was further improved by taking into account unsaturated conditions [11, 12] and binding isotherms [13]. The analytical solutions of the NPP model for the steady-state response and FEM solutions for the non-steady-state condition were presented by Rani et al. to simulate the migration test. Recently, it was reported that FDM computations achieve 2X to 3X speed-up in parallelization of FDM/FEM computations for large mesh sizes [14]. In fact, the surface coating requires increasing the mesh size because the surface coating is a thin section. Therefore, FDM takes advantage of computational speed to simulate the surface-coated concrete.

The present study evaluates the effect of surface coatings on the migration test using these numerical simulations. Since FEM and FDM have different advantages in accuracy and computational speed, it is necessary to discuss the strength and limitations of the methods for simulating the effect of surface coatings. Furthermore, the present study attempts to simulate the effects of the surface coatings by solving the NPP model during migration tests. The simulation results are discussed, and the different shapes of ionic profiles produced by different effectiveness of the surface coatings are compared. In addition, homogenized cases employing a homogenized diffusion coefficient are presented and compared to discretely simulated ones having two diffusion coefficients of surface coating and concrete.

#### 2. Model Description

In general, the range of pore size in hardened concrete varies from a nanometer to a millimeter. Some pores connect to others, forming continuous paths for ionic transport in a saturated system. These diffusion paths in concrete and a surface coating are tortuous compared to those in free liquid. For this reason, the term “tortuosity” was introduced to account for this complex pore structure [15, 16]. The diffusion coefficient of each ionic species has the following relation with tortuosity [15]:where is the tortuosity of the saturated material and is the diffusion coefficient of the species in free water. Different tortuosity values were used to distinguish between concrete and surface coating to account for their different pore structure. An interfacial resistance may develop when two materials having different pore structures are connected. Zhang et al. [17] found no significant evidence of a strong interfacial resistance between surface treatment and cement-based materials; therefore interfacial resistance was not included in the present mathematical model. The ionic profiles of the simulations are presented in terms of ionic concentration of the pore solution to estimate possibility of chemical reactions at the interfacial region.

Migration tests are commonly performed in saturated conditions and constant temperature. Most studies that simulate the migration test assume that chemical reactions are negligible [7]. However, recent research shows that this assumption can overestimate the tortuosity of the material [18, 19]. The objective of the present study is to show the trend of ionic behavior in the surface-coated concrete during migration tests so that the chemical reactions are not included.

##### 2.1. Nernst-Planck/Poisson (NPP) Model

The NPP model involves a separate mass balance equation for each of the ionic species present. The flux of an ionic species in solution is given bywhere is the concentration, is the diffusion coefficient, is the valence number of the species, is the Faraday constant, is the perfect gas constant, is the temperature, and is the electrical potential. The mass conservation for the species is expressed asSubstituting (2) into (3) results in the complete Nernst-Planck equation:The Nernst-Planck equation must be solved simultaneously with the Poisson equation that directly relates the electric potential to the electric charge:where is the total number of ionic species, is the absolute permittivity, and is the fixed charge density. Here, we assume that the fixed charge density is negligible because the fixed charge density is not a relevant parameter for most porous materials [8].

#### 3. Numerical Schemes

The finite-difference procedure for the surface-coated concrete is newly developed here. The mathematical treatment of the FEM follows the work of Yoon et al. [4], but here the emphasis is on developing a strategy in how to include the coating and have an efficient simulation.

##### 3.1. Finite-Difference Procedure

For many cases of practical relevance, the diffusion coefficient of NPP model is assumed to be a constant value, which can then be extracted from the parenthesis. If the object consists of two different materials, it must be dealt with differently. As the first step, the governing equation of the unidirectional case is expanded asEquation (6) is derived by applying the product law of the differential equation. However, for the simulation of surface coatings, is problematic because of the discrete (step) function of along the -axis. The diffusion coefficient function cannot be directly differentiated by . For this reason, it must be converted to a differentiable form. Since the original step function reveals no change except at the interface, the differentiated function is equal to zero everywhere but the interface must have infinite increase. This is the Dirac-delta function. Instead of the differentiated function (), this study used an approximate delta function, which is also known as normal distribution function.

The original step function can be also reexpressed by the integration of the approximate delta function. This simpler expression is similar to the Heaviside step function and suggested in (8). Then (7) and (8) are plotted in Figure 1 for different values of