Lattice Boltzmann simulation of the ionic diffusivity of cement paste

Transcription

1 Lattice Boltzmann simulation of the ionic diffusivity of cement paste M. Zhang 1, G. Ye 1, 2, and K. van Breugel 1 1 Microlab, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, 2628CN Delft, The Netherlands 2 Magnel Laboratory for Concrete Research, Department of Structural Engineering, Ghent University, Technologiepark-Zwijnaarde 904 B-9052 Ghent (Zwijnaarde), Belgium ABSTRACT: The lattice Boltzmann (LB) method is an efficient numerical approach which is widely utilized in the modeling of mass transport in porous media within a few decades because of the easy implementation of complex boundary conditions, inherent parallelism and high numerical stability of this approach. This paper presents the simulation of the effective ionic diffusivity of cement paste using the LB method. The entire simulation consists of three main steps. Firstly, the three-dimensional (3D) microstructure of hydrating cement paste is developed by the cement hydration model HYMOSTRUC3D. Cement paste is modeled as a composite material consisting of capillary pores, un-hydrated cement particles and hydration products (inner-, outer-, contact-). Secondly, the obtained 3D microstructure is digitized into a 3D array of voxels at a certain identical resolution (voxel size). Each voxel is assigned to be pore or solid depending on the position of its center. Finally, the diffusion process driven by the concentration difference between inlet and outlet is simulated using LB model. The operations such as collision and streaming are running until the steady-state diffusion condition is achieved. On the basis of the calculated steady-state molar flux across the whole domain, the effective ionic diffusivity can be estimated. Taking the diffusion of chloride ions for example, the simulated effective chloride diffusivities of cement paste are compared with the experimental results presented in literature. The simulation shows a good agreement with the experimental results. In addition, the influence of resolution on the simulated effective chloride diffusivity is studied. 1 INTRODUCTION The ionic diffusive transport in cement paste plays an important role in a wide variety of issues such as the long-term leaching and corrosion of embedded steel bars in the structure which is the predominant cause of deterioration and durability of reinforced concrete structures. Corrosion of reinforcing steel bars is accelerated by the presence of potentially harmful ionic species (e.g., chloride, sulfates, etc.) which may penetrate from the concrete surfaces to the reinforcing steel bars. This is particularly acute in a marine environment, in bridges and roads subjected to dicing salts, and in parking garages into which salt is transported from salted roads. When the concentration of chloride ions at the depth of embedded steel bars reaches a critical threshold value, the passive film on steel is destroyed by chloride ions and the corrosion of steel begins. To a great extent, the corrosion of reinforcement is dominated by the movement rate of chloride ions in cement paste due to the concentration gradient. In order to predict the service

2 life and assess the durability of reinforced concrete structures, it is necessary, therefore, to determine the diffusivity of water and ions in cement paste. The prediction of the ionic diffusivity of cement-based materials has been a challenging task to many researchers from all over the world, because this parameter is strongly dependent on the complex microstructure of cement-based materials. The ionic diffusivity of cement-based materials is a function of many factors such as porosity, the pore size distribution, the connectivity and tortuosity of the pores. The last few decades have seen a rapid development in the study of three-dimensional (3D) complex microstructures of cement-based materials, including modeling and image analysis techniques. In terms of modeling, several computerbased cement hydration models including CEMHYD3D [Garboczi and Bentz (1992)] and HYMOSTRUC3D [Ye (2003)] have been developed. In terms of image analysis, the 3D microstructure of cement paste can be reconstructed from thin-sections or high-resolution X-ray computed micro-tomography (XMT). Based on the 3D microstructure of cement paste, various approaches have been applied to predict the effective ionic diffusivity. Garboczi and Bentz (1992) computed the ionic diffusivity of cement-based materials by using random walker technique. This method didn t take into account the space-dependent diffusivity. The diffusivity of the ions in the fluid-filled capillary pores at different positions is considered to be identical. However, in the real field, due to the different moisture content at different positions, the diffusivity of the ions in the fluid-filled capillary pores at different positions is different and therefore, space-dependent. Zhang and Ye (2010) estimated the time dependency of chloride diffusion coefficient in cement paste using finite element method (FEM). FEM is based on continuum theory. Whether it is suitable for pore-scale simulation is still an open issue. Over the last two decades, one of the particularly suitable methods for simulating pore-scale transport phenomena, the Lattice Boltzmann (LB) method, has been developed based on kinetic Boltzmann equations [Qian et al (1992)]. The LB method provides insight into the internal concentration distribution of ions at pore-scale. Furthermore, it has several advantages such as the easy implementation of complex boundary conditions and high efficient computation through parallel compute. A more detailed introduction of LB method is illustrated in Section 2.2. The purpose of this paper is to present a pore-scale ionic diffusivity simulation of cement paste using the HYMOSTRUC3D model and the LB method. The 3D microstructure of cement paste consisting of capillary pores, hydration products (inner-, outer- and contact-) and un-hydrated cement particles are derived from the HYMOSTRUC3D model. Except capillary pores, other components are considered to be solid. On the basis of cluster-labeling algorithm, the connective pore structures are extracted. Then the diffusion process of ions in cement paste is simulated using the LB method. The effective chloride diffusivities of cement paste with various water-to-cement (w/c) ratios, i.e. 0.3, 0.4, 0.5 and 0.6, are calculated and verified with the experimental results obtained from literature. 2 THEORETICAL BACKGROUND 2.1 Governing equation Cement paste is a common heterogeneous porous media. The diffusion process of molecular species in a fluid imposed with a constant concentration gradient can be locally described by the diffusion equation (in one dimension) as follows: c = ( Di c) (1) t where c is the concentration of the molecular species, D i is the free molecular diffusivity in fluids which is equal to D 0 in the pore phase and D 1 in the solid phase. The diffusive transport is dominated by the large capillary pores as long as they percolate, i.e. form a continuous pathway

3 [Zhang and Ye (2010)]. This is the common situation which is referred herein. In this situation, the free molecular diffusivity in the solid phase is negligible and the solid phase is considered to be non-permeable. At length scales much larger than the typical pore size, diffusive transport can be generally described by the macroscopic diffusion equation (in one dimension) [Jeong et al (2008)]: C 2 = De C (2) t where C is the mean concentration of the molecular species, D e is the effective macroscopic diffusivity of molecular species through the porous media, which can be defined as follows: we De = D0 (3) w0 where w e and w 0 are the steady-state molar flux under an uniform concentration gradient across the heterogeneous porous media and the homogeneous media (without solid phase) with same physical dimension (length L and cross-section area A), respectively. 2.2 Lattice Boltzmann method The classical Boltzmann equation Eq. (4) provides an approximation to statistically describe the essential kinetics of a system by dealing with the dynamics of non-equilibrium processes and relaxation to an equilibrium state. The LB method is intrinsically a numerical scheme based on the dynamic evolution of statistical distribution of particles in a fluid. By discretizing the Boltzmann equation in spaceδx and timeδt, the LB equation can be obtained. In this study, the lattice Bhatager-Gross-Krook (BGK) model proposed by Qian et al (1992) is applied, in which the discretized LB evolution equation is defined as Eq. (5). A physical domain is discreted into lattices. Each lattice node is assigned to be pore or solid phase according to the corresponding position in the microstructure of cement paste. The space and time dependent averaged microscopic movements of molecular species at each lattice node are simulated using molecular populations called particle distribution function f i (x, t). This function represents the probability of finding a particle at location x and time t moving with a certain discrete velocity e i. f f 1 eq + ξ = ( f f ) (4) t x τ 1 eq fi( x + eiδtt, + δt) = fi( xt, ) fi( xt, ) fi ( xt, ) τ (5) Where f and f eq is the non-equilibrium and equilibrium particle distribution function, respectively,ξis the particle velocity,δt is time step,τ is the non-dimensional relaxation time and f eq i (x, t) is the equilibrium particle distribution function at the velocity direction i. Usually, a cubic lattice model D3Q19 (three dimensional lattice, 19 velocity directions) is used for 3D mass transport simulation. For D3Q19 model, the 19 discrete particle velocities e i are given as (0,0,0), (±δx/δt,0,0), (0,±δx/δt,0), (0,0,±δx/δt), (±δx/δt, ±δx/δt,0), (±δx/δt,0, ±δ x/δt), (0,±δx/δt, ±δx/δt) respectively, in which δx is the lattice. It has been presented that for the modeling of pure diffusive transport without convective term the lattice velocity directions may be reduced from 19 (D3Q19) to 7 (D3Q7) without degrading the accuracy [Jeong et al (2008)]. The remaining 7 lattice velocity directions are (0,0,0), (±δx/δt,0,0), (0,± δx/δt,0), (0,0,±δx/δt), as shown in Figure 1. For D3Q7 model, the non-dimensional relaxation timeτ is related to the molecular diffusivity D i according to the following expression [Jeong et al (2008)]:

4 7 δt 1 τ = Di + (6) 2 2 ( δ x) 2 The diffusivity of the ions D i is the same parameter listing in Eq. (1). If D i varies with position, one merely assigns differentτas a function of position according to Eq. (6). The equilibrium particle distribution function proposed by Li et al (2001) is used in the diffusive transport simulation which is described as follows: eq c fi ( x, t) = (7) a where a is the number of the discrete lattice velocity directions and the molecular concentration c is given by c= f ( x, t ) (8) i i Figure 1. Lattice velocity directions of D3Q7 model. 3 3D MICROSTRUCTURE OF CEMENT PASTE In this study, HYMOSTRUC3D is applied to simulate the 3D microstructure of hydrating cement paste. In HYMOSTRUC3D, the degree of hydration is simulated as a function of the w/c ratio, chemical composition of cement, particle size distribution (PSD) of cement and the reaction temperature. It has been shown that the size of the representative elementary volume (REV) for determination of diffusivity in cement paste was μm 3 [Zhang et al (2010)]. Accordingly, in this study, the cubic size of all samples is chosen as μm 3. The cement used in the simulation is Portland cement CEM I 42.5 N, the main constituents of which is listed in Table 1. The Blaine fineness of the cement is 420 m 2 /kg. A continuous particle size distribution (PSD) with the minimum size of 1 μm and the maximum size of 50 μm is used. The curing temperature is 20ºC. Table 1 Main constituents of Portland cement CEM I 42.5 N Phases C 3 S C 2 S C 3 A C 4 AF Weight (%) An example of the simulated microstructure and extracted pore structure of cement paste with w/c 0.50 at the age of 28 days is shown in Figure 2. It can be seen that the simulated cement paste is modeled as a composite material consisting of capillary pores and solid particles i.e. unhydrated cement particles, inner-, outer- and contact hydration product. In order to describe the position of these solid particles, six parameters are required in a 3D Cartesian coordinate

5 system, namely the particle center coordinates (x, y, z), the diameter of un-hydrated cement particle and the layer thicknesses of inner product and outer product, respectively. The entire microstructure is digitized into a three-dimensional array of voxels with an identical side length of 0.5 μm/voxel. The voxels are assigned to be pore or solid, depending on their corresponding positions in the system. Thus, the position of lattice nodes corresponds to the center of voxels. Since the isolated pores in the microstructure have no contribution to the diffusive transport, they will be removed in order to reduce the demand on core memory and improve the computational efficiency. On the basis of cluster-labeling algorithm [Hoshen and Kopelman (1976)], the connectivity of pore voxels is analyzed and the isolated pore voxels are removed. These isolated pore voxels are classified as solid. a) b) Figure 2. Simulated hardened cement paste with size of μm 3 (w/c=0.50, t=28 days): a) Microstructure; b) Its pore structure. 4 LATTICE BOLTZMANN SIMULATION The implementation of the above lattice Boltzmann method consists of three steps. Firstly, the initial particle distribution functions at all nodes are set to be equal to the initial equilibrium distribution function according to the initial concentration value and Eq. (7). Secondly, two operations namely collision and streaming are performed at each time step. The collision operation is to calculate the outgoing distribution function from each node at time t, i.e., the right-hand side part on Eq. (5) as f i * (x, t) = f i (x, t) - [ f i (x, t) - f i eq (x, t)]/τ. The streaming operation is to propagate the distribution function f i * (x, t) to their neighboring nodes at x+e i δ t to become a new distribution function according to f i (x+e i δt, t+δt) = f i * (x, t). Thirdly, the concentration c at each node is updated according to Eq. (8). In the particle streaming step, all components of the particle distribution function at each node are calculated except at the solid nodes. It is assumed that there is a wall between the respective pore and solid node. At this location, the so-called half-way bounce back boundary condition (as shown in Figure 3) is imposed, which means that the streaming particle distribution function from pore nodes to their neighboring solid nodes will be directly reflected back to the nodes from which they comes. In addition to these nodes, certain components of the distribution function of the inlet and outlet nodes are also unknown, where there is a concentration difference to drive the diffusive transport from inlet to outlet. In this study, the non-equilibrium extrapolation rule proposed by Guo et al (2002) is used to calculate these unknown components at the inlet and outlet nodes f i (x b, t) from the known components of distribution function at the nearest neighboring nodes f i (x n, t) and f i eq (x n, t) according to Eq. (9). In the other four boundary surfaces perpendicular to the inlet and outlet, the non-mirrored periodic conditions are imposed so that the nodes of opposite boundaries are considered as neighbors and the outgoing

6 components of distribution function from each surface are propagated to the opposite surface in the streaming operation. (, ) eq eq (, ) (, ) ( ) f x t = f x t + f x t f x t (9) i b i b i n i n, The lattice Boltzmann simulation is performed until the steady-state diffusion condition is reached. The steady-state criteria is defined such that the difference in the molar flux across the whole domain calculated in the current time and previous time step is less than Once the steady-state flux is known, the effective diffusivity of whole domain can be calculated according to Eq. (3). In this study, chloride ion is selected as a solute for example. Chloride binding is not taken into account. The diffusion coefficient of chloride ion in free water D 0 is m 2 /s at 25ºC [Bockris and Reddy (2002)]. In the diffusion experiment the concentration of NaCl or KCl solution is normally set to be 1 M (mol/l). Accordingly, in the LB simulation, the chloride concentration on the inlet and outlet surface is set to be 1 M and 0 M, respectively. In addition to this, the initial chloride concentration in cement paste is assumed to be zero. δx Pore Pore Solid Wall Figure 3. Half-way bounce-back boundary between pore and solid nodes. 5 RESULTS AND DISCUSSION 5.1 Effect of the resolution As mentioned above, in the LB simulation, it is necessary to digitize the microstructure derived from HYMOSTRUC3D into a number of lattice grids. The resolution (side length) of voxels in the digitized microstructure might affect the simulation accuracy. As we know, the higher the resolution is, the more accurate the solution is, especially when the porosity of samples is low. However, using higher resolution would significantly increase the demand on computer sources and computational time. In order to select a suitable and reasonable resolution, the influence of the resolution on the effective chloride diffusivity of cement paste should be estimated. The simulations on the samples with w/c ratio of 0.50 at the age of 28 days at five different resolutions, i.e., 1.0, 0.8, 0.625, 0.5 and 0.4 μm/voxel are investigated. Figure 4 shows the simulated effective chloride diffusivity of sample D e as a function of the resolution. As seen in Figure 3, D e increases with the increasing of resolution. The value increases from about m 2 /s to m 2 /s when the resolution changes from 1.0 μm/voxel to μm/voxel. However, the D e at a resolution of 0.5 μm/voxel is just 1.1 times as much as that at a resolution of 0.4 μm/voxel. This result implies that the resolution of 0.5 μm/voxel is not only suitable for the solution accuracy but also satisfies the computational efficiency. Therefore, 0.5 μm/voxel is chosen as resolution in the following simulations.

7 8 De ( m 2 /s) Resolution (μm/voxel) Figure 4. Effect of the resolution on the effective diffusivity D e. 5.2 Comparison of the simulated and measured chloride diffusivities Figure 5 illustrates an example of the concentration profiles of chloride ions in cement paste at the steady-state diffusion condition. The diffusion is driven by the concentration difference between inlet and outlet (x-axis). The effective diffusion pathways and their complexity can directly be seen from cross-sectional plots in the right-side. Figure 6 shows the estimated chloride diffusivity as a function of curing age. The chloride diffusivity of cement paste decreases with increasing of curing age. The decrease of chloride diffusivity is more pronounced during the first 14 days. With the increase of curing age, the cement hydration continuously proceeds and the capillary pores are filled by the hydration products. The capillary pores become finer and more tortuous. As a result, it is more difficult for capillary pores to form the connected paths for diffusive transport. Figure 7 shows the influence of w/c ratio on the effective chloride diffusivity of cement paste at 28 days. The results are also compared with the measured data from the laboratory diffusion cell test by Atkinson et al (1984). In their tests, the cement pastes were cured for 28 days in 100% relative humidity at 20ºC. The w/c ratio of cement paste was 0.20, 0.30, 0.40, 0.50, 0.60 and 0.70, respectively. The diffusion cell specimens with approximately 3 mm thick and 32 mm diameter were made and put in the 1 M KCl solution. As seen from Figure 6, the simulated chloride diffusivities show a good agreement with the measured data. The difference between simulation and experiments may be attributed to the fact that chloride binding is not taken into account in the simulation, which could result in an overestimation of the simulated results compared to the measurements. The influence of chloride binding will be taken into account in the future research.

8 Diffusion direction Figure 5. Distribution of chloride concentration (M) in cement paste at the steady-state condition ( μm 3, w/c=0.50, t=28 days). 1.E-08 1.E-09 De (m 2 /s) 1.E-10 1.E-11 1.E Time (days) Figure 6. Effective chloride diffusivity D e as a function of curing age (w/c=0.40). 1.E E-09 1.E-10 De (m 2 /s) 1.E-11 1.E-12 Simulation Measurement 1.E-13 1.E Water/cement ratio Figure 7. Comparison of simulated and measured chloride diffusivities D e (t=28 days).

9 6 CONCLUSIONS This work presented a pore-scale modeling of the effective ionic diffusivity of cement paste using the cement hydration model HYMOSTRU3D and lattice Boltzmann (LB) method. The HYMOSTRU3D model was used to simulate the 3D microstructure of the hydrating cement paste. LB simulation was then carried out to study the ionic diffusivity of cement paste on the basis of its microstructure. The simulation of the chloride diffusion process through cement paste is taken as an example. The chloride ionic diffusivities obtained from simulations were verified with experimental results. A good agreement was observed between the simulation and measurement. This study reveals that the LB method is a proper technique for modeling of the mass transport of solutes in the various pore media. REFERENCES Atkinson, A. and Nickerson, A.K. (1984) The diffusion of ions through water-saturated cement, Journal of Materials Science, 19: Garboczi, E. and Bentz, D. (1992) Computer Simulation of the Diffusivity of Cement Based Materials, Journal of Material Sciences, 27: Guo, Z.L., Shi, B.C. and Zheng, C.G. (2002) A Coupled Lattice BGK Model for the Boussinesq Equation, International Journal for Numerical Methods in Fluids, 39: Hoshen, J. and Kopelman, R. (1976) Percolation and Cluster Distribution I. Cluster Multiple Labeling Technique and Critical Concentration Algorithm, Physical Review B, 14(8): Jeong, N., Choi, D.H. and Lin, C.L. (2008) Estimating of Thermal and Mass Diffusivity in a Porous Medium of Complex Structure Using A Lattice Boltzmann Method, International Journal of Heat and Mass Transfer, 51: Li, Q., Wang, N.C. and Shi, B.C. (2001) LBGK Simulations of Turing Patterns in CIMA Model, Journal of Scientific Computing, 16(2): Qian, Y., Dhumieres, D. and Lallemand, P. (1992) Lattice BGK Models for Navier-Stokes Equation, Europhysics Letters, 17(6): Ye, G. (2003) Experimental Study and Numerical Simulation of the Development of the Microstructure and Permeability of Cementitious Materials, PhD Thesis, Delft University of Technology Zhang, M., and Ye, G. (2010) Modelling of Time Dependency of Chloride Diffusion Coefficient in Cement Paste, Journal of Wuhan University of Technology-Mater. Sci. Ed., 25(3): Zhang, M., Ye, G. and van Breugel, K. (2010) A Numerical-statistical Approach to Determining the Representative Elementary Volume (REV) of Cement Paste for Measuring Diffusivity, Materiales de Construcción, 60(300): 7-20