3D MODELING OF TRANSPORT PHENOMENON IN CEMENT BASED MATERIALS

Transcription

1 3D MODELING OF TRANSPORT PHENOMENON IN CEMENT BASED MATERIALS Neven Ukrainczyk University of Zagreb, Faculty of Chemical Engineering and Technology, Croatia Pregledni članak/ Subject review ABSTRACT: This paper presents an overview on a number of selected methods for modeling transport phenomenon in porous microstructure of cement based materials. The two main methodological approaches in modeling the microstructure evolution of cement based materials are: 1) digitalization of the experimental image of the real microstructure and 2) a continuous concept of growing spheres models. Representative microstructure obtained either by simulation packages (e.g. CEMHYD3D, HYMOSTRUC, MIKE and DuCOM) or experimentally (e.g. micro tomography) is a starting point for further analysis of transport properties. Rapid development of 3D numerical models has provided a number of methods to investigate the influence of microstructure on the evolution of transport properties of cement based materials. There are two main network approaches which may be employed, here termed as 1) direct and 2) indirect. In direct approach each pixel in the 3D microstructure is mapped into a node in either a finite element or finite difference analysis. The indirect approach constructs a convenient pore network of mostly larger pore elements that replicate the essential features of the pore space that are relevant to transport properties and thus tremendously reduce the computational costs. Key words: cement based materials, diffusion, transport properties, durability, concrete, porous microstructure, mathematical modeling. HDMT Hrvatsko društvo za materijale i tribologiju 502

2 1. Motivation: concrete structure durability Although the interest for numerical modeling of cementitious materials durability issues has increased rapidly in recent years, a comprehensive knowledge of their fundamental processes is still lacking. A deep understanding of the multi-ionic transport phenomenon through the porous (micro)structure of cementitious materials is of great fundamental and practical importance. Contrary to common belief, concrete is a complex composite material, whose structure and properties change over time [1-3]. Modeling challenges include difficulties in the overall description of this multi-scale random porous medium characterised by a wide range of sizes, from nanometer-sized pores to centimeter-sized aggregates (Figs 1 and 2). Indeed, the transport in cementitious materials plays a crucial role in both the degradation process of building materials (e.g. mortar and concrete) and the containing of hazardous wastes. For example aggressive external ions such as chloride ions and sulphates penetrate into the concrete s cover causing corrosion of reinforcing steel in concrete and cracking due to expansion mechanisms. Various diffusion based transport models are used to predict the time to initiation of deterioration when aggressive ions reach a critical value. A good quality of the concrete cover enhances the durability and reliability of concrete structures. Thus, the increasing focus on the reliability aspects requires a better understanding of the complex mass transport phenomenon through the evolving porous microstructure of cementitious materials. Although capillary absorption and advection (i.e. water movement due to pressure gradient) can in some certain cases be of significant importance, numerous studies have indicated that ions are mainly transported through the concrete pore structure by a diffusion process (e.g. [4]). This is why the mechanisms of diffusion in saturated cementitious materials have received most attention [5-11]. Still, much work is needed to obtain basic knowledge about the diffusive transport phenomenon in cementitious materials. Fig 1. Modeling challenges: multi-scale random porous medium that changes with time. 503

3 CSH pores Capillary pores Entrained air bubbles Cement particles 1E-3 0,01 0, Dimension, m Fig 2. A multi-scale range of the cement paste microstructure. One parameter that is of paramount importance is the effective (macroscopic) diffusion coefficient, D ef. A relevant and reliable method is needed in order to obtain D ef with an acceptable accuracy. Various experimental methods exist but they are either time consuming or have too many drawbacks. For example, one of the most important obstacles to the description of ions diffusion in concrete structures is its common partial water saturation state, as compared to specimens tested in water saturated laboratory conditions. It is challenging to experimentally study the influence of the degree of water saturation of cementitious materials upon ion diffusion while avoiding contributions of other mechanisms of transport, such as water movement [19] and/or ion-solid interaction. Representative microstructure obtained either by simulation packages (e.g. CEMHYD3D, HYMOSTRUC, MIE and DuCOM) or experimentally (e.g. micro tomography [16]) is a convenient starting point for further analysis of transport properties. In this paper, firstly, the simulation of 3D microstructure during cement hydration is reviewed in section 2. Secondly, the effective transport calculation within representative 3D microstructure is given in section Cement hydration models Rapid development of numerical models has provided novel methods to investigate the influence of microstructure on the evolution of the properties of cementitious materials of which the most famous are CEMHYD3D model [8-11] developed at National Institute of Technology (NIST) USA, the HYMOSTRUC model [5,6,12] developed at Delft University of Technology (TU Delft), and DuCOM model (Japan) [13]. Recently, a new modeling platform, called μic (mike) has been developed (Switzerland) [14] to speed up the simulations so as to allow modeling of more particles, and to allow its users more flexibility in most aspects of the complex hydration simulations. In these hydration models, the resulting microstructure can be simulated as a function of hydration degree, particle size distribution, chemical composition of the cement, the water/cement ratio, and reaction temperature. The two main methodological approaches in modeling the microstructure evolution of cement based materials are: 1) digitalization of the experimental image of the real microstructure (CEMHYD3D, Fig 3), and 2) a continuous concept of growing spheres models (HYMOSTRUC, μic, Fig 4). 504

4 Fig 3. 3D digitalization of the 2D experimental images in CEMHYD3D. In the two cement hydration modeling approaches, pixel based cellular-automaton (CEMHYD3D, Fig 3) vs. continuum based growing spheres models (HYMOSTRUC and μic), both the morphology of the hydration products and the location where they precipitate are different [5]. In the continuous growing spheres models (HYMOSTRUC, μic, Fig 4), cement particles are simulated as spheres and hydration products as concentrical spherical shells. The growing spherical shells are overlapping when cement hydration takes place. a) b) Fig 4. The 3D microstructure simulated by the HYMOSTRUC model: a) solid fraction; b) capillary porosity [15]. CEMHYD3D In the CEMHYD3D model (available for free in open source), the cement components are represented as different sets of digital pixels. The main advantage of this model is digitalization of the experimental image of the real microstructure (Fig 3) that considers the irregular shape of minerals. The hydration process is then simulated with a cellular automaton-type digital-image-based algorithm. The CSH gel forms mostly near anhydrous particles, while CH and other crystalline products are formed directly in the free pore space. In CEMHYD3D simulations only four sizes of particles were used, with diameters of 3, 9, 13 and 19 pixels, with resolution of 1 μm/ pixel [8]. The CEMHYD3D calculated that capillary 505

5 porosity percolation threshold changed from 24% to 18% to 11% as the digital resolution shifted from 1 to 0.5 to 0.25 μm/pixel in a cement paste with w/c ratio 0.30 [8]. This indicates that the pathways that would seem to be depercolated at lower resolution are percolated at higher resolutions (when the smaller pores can be resolved). One criticism of CEMHYD3D is that no clear boundaries of cement particles remain during simulation because of the cellular automata algorithm where cement particles voxels disintegrate into single voxels of only 1 μm size and diffuse in the pores. Moreover, the correct representation of ionic movements in the microstructure during hydration is not possible at 1 μm scale. HYMOSTRUC The HYMOSTRUC model employs a continuous particle size distribution, between 1 and about 70 μm, the image resolution is optionally, but typically 0.25 μm with the layer depth 1 μm. In previous studies, the capillary pore structure of hardening cement paste simulated with the cement hydration model HYMOSTRUC [5,6,12] were validated by mercury intrusion porosimetry and Scanning Electron Microscope image analysis. The crucial parameters of the simulated porous structure were deduced from a serial section analysis algorithm in combination with an overlap criterion [6,12]. Currently the HYMOSTRUC simulates only the growth of new layers of product on the cement particles. This limitation prevents the representation of products such as portlandite that tend to form novel clusters in the free capillary pore space, rather than depositing on the cement particles. Another major criticism of the HYMOSTRUC model is that it treats the entire microstructure only in a statistical manner (the neighbourhood of particles is not considered for the calculation of reaction rates), therefore the localized microstructure information is lost. Fig 5. Some possibilities for the description of the particles growth due to hydration reactions in the μic model [14]. μic The new modelling platform μic (stands for microstructure, pronounced Mike, available for free in open source) has been recently developed at Lausanne EPFL (by the Scrivener group) 506

6 [14] to speed up the simulations so as to allow modeling of much more particles, and to allow its users more flexibility in most aspects of the simulations. Now, millions of cement particles can be simulated thus allowing a more realistic initial cement particle size distribution. Simulations in μic are designed for virtually all cementitious materials. Indeed, the design of the platform removes most of the hard-wired information that is generally hidden from the user, providing extra flexibility. A user can add and customise the properties of the materials, particles and reactions in the simulations. Furthermore, the user can customise the models for processes that control microstructural development using plugins [14]. The controlling of kinetics by user is determined individually for each particle at each step in the simulation. The number and the location of new particles formed during certain hydration reaction can also be flexibly defined using plugins (Fig 5). For example, new particles (hydration products) can be forced to grow on the free surface of another particle or in the pore space (Fig 5). DuCOM DuCOM is a multi-scale model of concrete developed at the University of Tokyo [13]. It employs a finite-element method to predict the durability of concrete structures based on semi-empirical correlations on the influence of a large number of process parameters and variables on the final properties of the concrete. The model depends almost entirely on empirical relations that have been calibrated to experimental results. Strictly speaking, this model is not a microstructural model, the pore sizes being simulated by a representative unit cells in a statistical manner using a single (median sized) initial particle of cement for hydration. 3. Effective transport properties A representative starting point for further analysis of transport properties in 3D cement based microstructure can be obtained either by: 1) simulation packages (e.g. CEMHYD3D, HYMOSTRUC, μic and DuCOM), or 2) experimentally (e.g. by micro tomography [16]). Simulating a microstructure evolution during hydration is an advantageous (fundamental) starting point to model the morphological nature of the effective diffusion (transport) coefficient. A virtual 3D microstructure created with an available hydration models provides a basis for the analysis of the morphological influence, including the porosity, tortuosity, constrictivity and the pore water content effect onto the effective diffusion coefficient. Such an approach contributes to a better understanding of the phenomenology and thus improves the predicting reliability of the coupled transport models. Universality of transport laws The calculation of the transport properties as described here can be viewed in a larger general aspect because of the analogy between the following four laws: Fourier's law for heat flow; Ohm's law for electric current (I=G*ΔU); Fick's law for diffusion (J=-D*Δc/Δx); and Darcy's law for liquid flow in materials (V=k Δp/η). The universality in these laws is that the flux is proportional to a driving force (difference in temperature, voltage, concentration or pressure) and inversely proportional to the resistance. Therefore an electric analogy approach may be used in calculation of the effective transport property. After obtaining the effective electrical conductivity (G) of the investigated random porous medium the desired effective property (e.g. diffusion coefficient, D or mass transfer coefficient, k) can be further calculated [6,9,10,16,17]. For example, the effective 507

7 (macroscopic) diffusivity coefficient is then obtained by utilizing the Nernst-Einstein relation [9,10,17] which states that the relative diffusivity (D e /D 0 ) is equivalent to the relative conductivity (σ e /σ o ), where the relative diffusivity is the ratio of the effective diffusivity (D e ) of an ion in the cement paste composite relative to its value when diffusing in bulk water (D 0 ) and ranges between 0 and 1. The calculation of the water permeability according to the Darcy s law is described in [6,16]. The morphological nature of the transport (diffusion) coefficient Macroscopic (i.e. effective) diffusivity coefficient corresponds to the diffusion coefficient at the macroscopic scale, the pore liquid phase being at the microscopic scale. A geometrical factor accounting for the complexity of the shape of the porous system (defined by its tortuosity, porosity and constrictivity) is termed as an effective transport coefficient. Relative diffusivity can be described as the ratio of the effective diffusivity (D e ) of an ion in the cement paste composite relative to its value when diffusing in bulk water (D 0 ) and ranges between 0 and 1. Based on microstructural modeling by CEMHYD3D, Garboczi NIST group [cited in 9,10] has shown that the relative diffusivity of cement paste is mainly a function of capillary porosity and derived a diffusivity-porosity (percolation based) relation for watersaturated cement paste. The relation resulted in a single universal curve for a variety of w/c ratios and degrees of hydration. The relative diffusivity values computed for cement pastes are needed as input into a multi-scale structural model for mortar or concrete to determine the effect of aggregates and their surrounding interfacial zones on the diffusivity of the structures. The multi-scale modeling approach presented by Garboczi (NIST) group [9,10] allow for the prediction of the diffusivity of a mortar or concrete at a specific hydration degree, thus providing rapid estimates for service life prediction models. D D ef G G ef 0 0 Fig 6. Steady state flux through the 3D simulated microstructure [Ye]. Numerical network calculation of an effective transport coefficient 508

8 The numerical network method employs (steady state) electrical analogy to derive an effective conductivity of the 3D simulated microstructure [6,9,10,16], Fig 6. The electric analogy steady state method is an established method for obtaining the macroscopic effective transport coefficient of a porous materials. An electric analogy approach is based on Ohms law (I=G*ΔU) which is equivalent to the steady state ionic flux equation (Fick's first law: J=- D*Δc/Δx) as well as Darcy s law (V=k Δp/η). The capillary porosity is assigned an ionic conductivity value as in aqueous solution (σ 0 ), while cement particles have zero conductivity since they contain no porosity. Fast (e.g. conjugate gradient relaxation) algorithm solves the complete electrical steady state problem of the voltage distribution in a random material across which a potential difference is applied (Fig 6). The output of the algorithm is the voltage at every node, from which the total current and thus equivalent conductance is calculated. The effective (macroscopic) diffusivity is then obtained by utilizing the Nernst- Einstein relation [9,10,17] which states that the relative diffusivity (D e /D 0 ) is equivalent to the relative conductivity (σ e /σ o ) Fig 6. Fig 7. Calculation of transport properties by direct approach: digitalization of the microstructure and employment of FEM or FDM [9,10]. There are two main network approaches which may be employed, here termed as: 1) direct (Fig 7), and 2) indirect (Fig 8 and 9). In direct approach each pixel in the 3D microstructure is mapped into a node in either a finite element or finite difference analysis (Fig 7). Each pixel in the 3D microstructure is mapped into a node in either a finite element or finite difference analysis [9,10]. Electrical conductivities are assigned to each phase comprising the microstructure. The indirect approach constructs a convenient pore network of mostly larger pore elements (Fig 8 and 9) that replicate the essential features of the pore space that are relevant to transport properties [6,16], and thus tremendously reduce the computational costs.. In other words, the indirect approach simplifies the simulated pore structure with an equivalent 3D pore network system which still captures its underlying physics. The obvious advantage of a direct method is that the calculation is done on a simulated microstructure directly as digitalized. However, extracting the crucial geometrical parameters of the simulated porous structure and creating larger unit elements for building the equivalent pore network (in indirect approach) tremendously reduces the computational costs. 509

9 In order to reduce the complexity of the 3D microstructure and transform the pore space into a transportation network, a number of 3D digital image processing algorithms may be employed [18] (Fig 10, 11). The TU Delft M&E group has recently presented [6] an indirect network model to predict the water permeability of cement paste from a HYMOSTRUC numerical simulation of its microstructure. The crucial parameters of the simulated porous structure were deduced from a serial section analysis algorithm in combination with an overlap criterion (Fig 11). The group at Institute of Building Materials, RWTH Aachen University has presented [16] an indirect network model to predict the water transport in cement pastes and mortars from a microtomographic images of its microstructure. The indirect transportation network was obtained by employing their modified version of the thinning algorithm [18] (Fig 9). Fig 8. Calculation of transport properties by indirect approach: construction of the simplified pore network. d 4 d 1 d 2 d 3 d n Fig 9. Construction of the network of the cylindrical pores; representation (transformation) the complex 3D pore network with 1D cylindrical pore elements having an effective diameter. Points are defined as intersection of pores. 510

10 a) b) Fig 10. Example of axis thinning algorithm (b) applied to the CT image of hand (a) [18]. Fig 11. Serial sectioning with an overlap criterion [6]. 4. Percolation of capillary porosity Transport properties of cement paste microstructure are dominated by the large percolating (i.e. continuous) capillary pores. If the capillary pores are discontinuous (i.e. depercolated), the transport is controlled by the properties of the much smaller C-S-H gel pores. The value of this critical porosity is called the depercolation threshold. Depercolation thresholds of capillary pores are fundamental for developing and tuning transport models for cementitious materials [5,8]. Although the percolation of cement paste is a well investigated subject of utmost importance [5,8], the percolation threshold of capillary porosity is still a controversial subject. There are large differences between the obtained percolation threshold values by different methods. NIST pixel based cellular-automaton hydration model CEMHYD3D [8], gave the critical values of capillary porosity to be between 15% and 20%. On the other hand, continuum based (growing spheres) models (HYMOSTRUC [5]) found much lower percolation thresholds of only a few percent. According to Ye [5] by simulated pore structure generated by HYMOSTRUC, the depercolation of capillary porosity is at about 5% porosity, for samples with w/c ratio 0.3, and no depercolation was found for pastes w/c > 0.3 during the first 28 days. Both results are in agreement with the experimental results [5]. The 511

11 pronounced differences in capillary percolation are due to dissimilarities in the hydration modeling methodologies and in the space discretization sizes of the various 3D numerical simulations. In the two models, pixel based cellular-automaton (CEMHYD3D) vs. continuum based growing spheres models (HYMOSTRUC), both the morphology of the hydration products and the location where they precipitate are different (as detailed in section 2). 5. The pore water concentration variations One of the most important obstacles to the description of ions diffusion in concrete structures is its common partial water saturation state, as compared to specimens tested in water saturated laboratory conditions. It is challenging to experimentally study the influence of the degree of water saturation of concrete upon ion diffusion while avoiding contributions of other mechanisms of transport, such as water movement [19]. Martys [17] studied theoretically the diffusion in partially saturated porous materials by applying a Lattice Boltzmann cellular automaton-type simulation. In his work the porous microstructure was simulated by a simplified model consisting of inert spheres with only two sizes. The modeling of phase separating binary mixtures in porous media is a great research challenge. The phase separation of a fluid mixture directly on the 3D chaotic pore space, can be calculated either by numerically solving the Navier Stokes equation [20] or by applying a Lattice Boltzmann simulation [17], however at considerable computational expense. 6. The chemical nature of the diffusive transport It is a well known fact that the modeling of the ions transport through cementitious materials by a plain (unmodified) Fick s law is inappropriate [4,7]. Ions in solution are subjected to various types of interaction that readily complicate the mathematical treatment of the problem. Many advanced approaches are proposed that capture the important underlying mechanisms and provide more reliable predictions. Some models are based on a single-ion approach but incorporate nonlinear chemical ion-solid relationships, humidity and temperature variations ( [4,7]). Recently, multicomponent ionic approaches were developed to provide a more accurate prediction of the complex interaction between ions [4,7]. Multicomponent ionic transport is described by the Nernst-Planck equation. Further advancement in numerical modeling is made by considering the intricate reactions with the solid phase. These advanced models incorporate a more detailed transport but require a larger number of input parameters and the calculations are still restricted to 1D. Much work is needed to feed such advanced numerical models with a wide range of confident input parameters. 7. Conclusion The two main methodological approaches in modeling the microstructure evolution of cement based materials are: 1) digitalization of the experimental image of the real microstructure and 2) a continuous concept of growing spheres models. Representative microstructure obtained either by simulation packages (e.g. CEMHYD3D, HYMOSTRUC, MIKE and DuCOM) or experimentally (e.g. micro tomography) is a starting point for further analysis of transport properties. A virtual 3D microstructure created with an available hydration models provides a convenient basis for the analysis of the morphological influence, including the porosity, tortuosity, constrictivity and the pore water content effect onto the effective diffusion coefficient. Such an approach contributes to a better understanding of the phenomenology and thus improves the predicting reliability of the coupled transport/durability models. 512

12 The two main network approaches which may be employed for 3D effective transport calculation, are termed as 1) direct and 2) indirect. In direct approach each pixel in the 3D microstructure is mapped into a node in either a finite element or finite difference analysis. The indirect approach constructs a convenient pore network of mostly larger pore elements that replicate the essential features of the pore space that are relevant to transport properties and thus tremendously reduce the computational costs. Different simplifications (assumptions) used in the hydration modeling methodologies and in the space discretization sizes (resolution) of the various 3D numerical simulations result in pronounced differences in calculated capillary percolations. To accurately describe the diffusion in cement based materials an electrochemical migration term has to be introduced to account for electro neutrality where diffusion coefficients of charged species differ. The more advanced multi-component species transport models that incorporate the ion-ion and/or ion-solid interactions require a larger number of input parameters and the calculations are still limited to 1D solutions of a homogenized medium. 8. Acknowledgment The authors acknowledge support from the Croatian Ministry of Science, Education and Sports under project s no Development of Hydration Process Model and thank LGM d.o.o (Zagreb) for providing SBR latex samples. 9. References 1. N. Ukrainczyk, V. Ukrainczyk, A Neural Network Method for Analyzing Concrete Durability, Magazine of Concrete Research, 60 (2008), 7; N. Ukrainczyk, P.I. Banjad, N. Bolf, Evaluating Rebar Corrosion Damage in RC Structures Exposed to Marine Environment Using Neural Network, Civil Engineering and Environmental Systems, 24 (2007) 1; Structures and performances of cements, 2end ed., Eds. J. Bensted, P. Barnes, London E. Samson, J. Marchand, Modeling the transport of ions in unsaturated cement-based materials, Computers and Structures 85 (2007) G.Ye, Percolation of capillary pores in hardening cement pastes, Cement and Concrete Research 35 (2005) G. Ye, P. Lura, K. van Breugel, Modelling of water permeability in cementitious materials, Materials and Structures 39 (2006) O. Truc, J.-P. Ollivier, L.-O. Nilsson, Numerical simulation of multi-species transport through saturated concrete during a migration test - MsDiff code, Cement and Concrete Research 30 (2000) E.J. Garboczi, D.P. Bentz, The effect of statistical fluctuation, finite size error, and digital resolution on the phase percolation and transport properties of the NIST cement hydration model, Cement and Concrete Research 31 (2001) E.J. Garboczi and D.P. Bentz, Multi-scale analytical/numerical theory of the diffusivity of concrete, Journal of Advanced Cement-Based Materials 8 (1998) D.P. Bentz, E.J. Garboczi, and E.S. Lagergren, Multi-Scale Microstructural Modeling of Concrete Diffusivity: Identification of Significant Variables, Cement, Concrete, and Aggregates, 20 (1998) E.J. Garboczi, D.P. Bentz, The effect of statistical fluctuation, finite size error, and digital resolution on the phase percolation and transport properties of the NIST cement hydration model, Cement and Concrete Research 31 (2001)

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