Multiscale Lattice Boltzmann-Finite Element Modelling of Transport Properties in Cement-based Materials

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T

ranspor

t Proper

ties in Cement-based Materials

Multiscale Lattice

Boltzman-Finite Element Modelling of

Transport Properties in

Cement-based Materials

Mingzhong Zhang

ISBN 978-90-90-27483-6

From mortar to concrete

From cement paste to mor

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Multiscale Lattice Boltzmann-Finite Element Modelling of

Transport Properties in Cement-based Materials

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 9 april 2013 om 10.00 uur

door

Mingzhong ZHANG

Master of Engineering aan de Wuhan University of Technology, China geboren te Xianyou, Fujian Province, China

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Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir. K. van Breugel

Copromotor: Dr. G. Ye

Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof.dr.ir. K. van Breugel Technische Universiteit Delft, promotor Dr. G. Ye Technische Universiteit Delft, copromotor

Prof.dr. D.A. Lange University of Illinois at Urbana-Champaign, USA Prof.dr. V. Baroghel-Bouny IFSTTAR, France

Prof.dr. J. Bruining Technische Universiteit Delft Prof.dr.ir. H.E.J.G. Schlangen Technische Universiteit Delft Prof.dr.ir. L.J. Sluys Technische Universiteit Delft

Prof.dr.ir. E.M. Haas Technische Universiteit Delft, reservelid

ISBN: 978-90-9027483-6

Keywords: Cement-based materials; Transport properties; Multiscale modelling; Lattice Boltzmann method; Finite element method; Durability

Printed by VSSD, the Netherlands

Cover design: J.F. Krook and Mingzhong Zhang

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without the prior written permission from the author.

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Acknowledgements

The research work reported in this thesis was sponsored by China Scholarship Council (CSC) and Delft University of Technology (TU Delft). This research was carried out within the Section of Materials and Environment at the Faculty of Civil Engineering and Geosciences at TU Delft. These institutions are greatly acknowledged for all kinds of support.

There are many people who contributes directly or indirectly to this research and my graduate studies. Although I certainly cannot fully thank them in few pages, I would like to recognize, at least briefly, their assistance and support.

I would like to express my true appreciation to my promotor Prof.dr.ir. Klaas van Breugel for giving me the opportunity to work with him and study at TU Delft. His guidance, support, insights and comments are the reasons that I was able to accomplish this work. My special cordial thank goes to my copromotor Dr. Guang Ye, for his guidance, patience, support, encouragement and friendship throughout the whole course of my study. I am particularly grateful to him for providing me with the opportunities to be actively involved in different research activities. I also thank Prof.dr. Zhonghe Shui at Wuhan University of Technology, who recommended me as a Ph.D. candidate to my promotors at TU Delft.

I am very thankful to the committee members for their insights, availability and willingness to serve on my defence committee. In particular, I sincerely thank Prof.dr. David A. Lange at University of Illinois at Urbana-Champaign and Prof.dr.ir. Bert Sluys at TU Delft for their valuable comments on the draft thesis. I would like to extend my appreciation to Prof.dr.ir. Erik Schlangen and Prof.dr. Hans Bruining at TU Delft for discussions about multiscale methods and fluid transport in porous media, respectively. I also gratefully acknowledge Prof.dr.ir. Rob Polder at TU Delft for his constructive comments and suggestions on Chapter 7 of the draft thesis, which help me to improve this thesis.

A special note of appreciation goes to Dr. Yongjia He at Wuhan University of Technology and Prof.dr. David A. Lange for sharing the X-ray micro-computed tomography images of cement paste and for their contributions to several relevant publications. I am also very much obliged to Prof.dr. Jianjun Zheng at Zhejiang University of Technology, Dr. Wei Chen at Wuhan University of Technology, Dr. Pietro Lura at EMPA and the anonymous reviewers for their helpful and constructive comments on the journal papers related to this thesis.

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I wish to thank Mr. Cees Timmers and Ms. Franca Post at the Center for International Cooperation and Appropriate Technology (CICAT) of TU Delft for their help and management support.

I would like to show my gratitude to all my friends, and my current and former colleagues at TU Delft for their friendship, support and funny discussions over years. Especially, I want to thank Dr. Junfeng Su not only for his valuable comments on the draft thesis but also for his helpful suggestions on the career development, Dr. Zhiwei Qian for giving me the Anm material model to generate the mesostructure of mortar with irregular-shaped sands and for all his help, Ms. Renée Mors for the translation of propositions and summary, Ms. Nynke Verhulst, Melanie Holtzapffel and Claudia Baltussen for their help in various kinds of documents. My special thanks also go to my best friends, who came to TU Delft to pursue a Ph.D. degree together, i.e. Quantao Liu, Yuwei Ma, Yue Xiao, Mingliang Li, Xu Jiang, Xuhong Qiang, Xun Gong, Xuefei Mei, Zhan Zhang, etc., for their help and encouragement during these years in the Netherlands.

I want to say an honourable thank to my parents for bringing me up, endless long-distance encouragement and continuous support throughout these years.

Last but not least, I would like to express my deepest gratitude to my fiancee, Sinuo, for her unconditional support and love. Without her, I would never have had such an enjoyable experience in the Netherlands. I am so grateful for having someone perfect like her, who is willing to grow old along with me.

Mingzhong Zhang Delft, March 2013

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List of Symbols

Roman lower case letters

ܿ௦ lattice sound speed [-]

dc critical pore diameter [µm]

ei microscopic velocity [-]

erf() standard error function [-] f1(w/c) effect of w/c ratio on the chloride diffusivity [-]

f2(t) effect of age on the chloride diffusivity [-]

f3(Cb) effect of chloride binding on the chloride diffusivity [-]

f4(Sw) effect of water saturation degree on the chloride diffusivity [-]

f5(Cf) effect of free chloride concentration on the chloride diffusivity [-]

f6(ITZ) effect of ITZ on the chloride diffusivity [-]

f7(߶஺) effect of aggregate content on the chloride diffusivity [-]

f particle distribution function [-] fi discrete particle distribution function [-]

fieq equilibrium particle distribution function [-]

fCH volume fraction of CH in hydration product [-]

fC-S-H volume fraction of C-S-H in hydration product [-]

݆_௫ x component of momentum [-] ݆_௬ y component of momentum [-] ݆௭ z component of momentum [-]

kc curing factor in DuraCrete model [-]

ke environmental factor in DuraCrete model [-]

kw saturated water permeability [m/s]

m(t) mass of liquid gained by the specimen at time t [kg] meq equilibrium moments [-] ݉_஼_య_஺ mass of C3A [-] ݉஼ర஺ி mass of C4AF [-] ݉_஼ௌு_మ mass of gypsum [-] p pressure a[N/m 2 ] pin pressure inside the droplet in LB units a[-]

pout pressure outside the droplet in LB units a[-]

q mass flux [kg/m2-s] ݍ௫ x component of energy flux [-]

ݍ௬ y component of energy flux a [-]

ݍ௭ z component of energy flux [-]

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vi List of Symbols ݏజ relaxation rate [-] ݏ௘ relaxation rate [-] ݏఌ relaxation rate [-] ݏ_௤ relaxation rate [-] ݏ_గ relaxation rate [-] ݏ௠ relaxation rate [-] t time [s] tex age of the sample at the start of exposure [s]

tgyp time at which the gypsum is completely consumed [s]

tref reference time [s]

ݑ_௭ volume-average velocity of flow [m/s]

v fluid velocity [m2/s] wi weight factor [-]

xd depth [m]

z electrical charge [-]

Roman capital letters

A cross-sectional area of the sample [m2] B proportionality factor [-] Bo Bond number [-] C concentration of species in solution [mol/l] C0 initial chloride concentration [mol/l]

ܥ௘ concentration in each element [mol/l] ܥ_௜௘ nodal concentration [mol/l] Ca capillary number [-] Cb bound chloride concentration [mol/l]

Cbm monolayer adsorption capacity [-]

Ccr critical chloride concentration [mol/l]

Cd drag coefficient [-]

Cf free chloride concentration [mol/l]

Cs surface chloride concentration [mol/l]

D diffusion coefficient [m2/s] D0 ionic diffusion coefficient for a given temperature [m2/s]

D(Cf) chloride diffusion coefficient at a given concentration Cf a[m 2

/s] D(t) time-dependent chloride diffusion coefficient [m2/s] D(T) chloride diffusion coefficient at a given temperature T [m2/s] Da apparent chloride diffusion coefficient through a porous medium [m2/s]

Dbp chloride diffusion coefficient in bulk paste [m2/s]

Dc chloride diffusion coefficient in concrete [m2/s]

Dcp,e effective chloride diffusion coefficient in cement paste [m2/s]

DCH ionic diffusion coefficient through CH phase [m2/s]

DC-S-H ionic diffusion coefficient through C-S-H phase [m2/s]

De effective chloride diffusion coefficient through a porous medium [m2/s]

Dhyd ionic diffusion coefficient through hydration product a [m 2

/s] DITZ chloride diffusion coefficient in the ITZ [m2/s]

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List of Symbols vii Dp ionic diffusion coefficient in the pore solution [m2/s]

Dref chloride diffusion coefficient at a reference condition a[m 2

/s] DRH chloride diffusion coefficient at a given humidity RH a[m2/s]

E electrical potential [-] Ea activation energy a[J/mol]

F Faraday constant a[-]

F0 dimensionless topology factor [-]

Fads fluid-solid adhesion force [-]

Fg external force including gravity effects aa[-] Fint fluid-fluid interaction force aa[-] g acceleration of gravity [Ns/m2] G coefficient of the attractive forces [-] J diffusion flux [m3/s] ܬ௘ diffusive flux through a porous medium in LB unit [-]

ܬ_௣ diffusive flux through a free pore space in LB unit [-] ܭ_଴ basic rate factor of the boundary reaction [-] L thickness of the sample [m] Lx length of the sample [m]

Ly width of the sample [m]

Lz height of the sample [m]

ܰ௜ shape function [-]

Q Volume of fluid passed in unit time [m3/s] R universal gas constant [J/mol-k] R∞ maximum pore radius [µm]

Req equivalent pore radius [µm]

RHc critical relative humidity [%]

Rm median pore size [m]

Rmin minimum hydraulic radius [µm]

Sa absorptivity [kg/m2-s]

Sw degree of water saturation [-]

T absolute temperature [K] V volume of the sample [m3] V0 volume of REV [m3]

V1,c critical volume fraction of capillary pore [-]

VC-S-H volume fraction of C-S-H [-]

Vk volume of phase k [m3]

ܺ_஼_మ_ௌ mass fraction of C2S [mass%]

ܺ஼య஺ mass fraction of C3A [mass%]

ܺ_஼_య_ௌ mass fraction of C3S [mass%]

ܺ஼ర஺ி mass fraction of C4AF [mass%]

XAFm mass fraction of AFm in cement paste [mass%]

XC-S-H mass fraction of C-S-H in cement paste [mass%]

Greek letters

α degree of hydration [-] ߙ஼మௌ degree of hydration for C2S [-]

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viii List of Symbols ߙ஼_య஺ degree of hydration for C3A [-]

ߙ஼యௌ degree of hydration for C3S [-]

ߙ஼_ర஺ி degree of hydration for C4AF [-]

γ

_{activity coefficient [-]} ε variation coefficient [-] σ2 standard deviation [-] ߪ଴ ionic conductivity of the free electrolyte [-]

ߪ௘ ionic conductivity of a saturated sample [-]

ߪ_௣ surface tension between water and gas [N/m]

ξ

particle velocity [-]

η

dynamic viscosity of the fluid [Ns/m2] ߠ _{contact angle [º]} κ intrinsic permeability [m2] ߢ௟௕ intrinsic permeability in LB units [-]

th

κ

theoretical intrinsic permeability [m2]

*

th

κ dimensionless intrinsic permeability [-] ߬ relaxation time [-]

u_{velocity of the fluid}[-] ݑ௣,ௐ average Darcy velocity of the wetting fluid [m/s]

ߥ _{lattice kinematic shear viscosity of the fluid}[-]

ρ

fluid density [kg/m3] ߩ଴ mean density of the fluid in the LB system [-]

ߩ௪ virtual density of the solid phase [-]

ߩ௣,ௐ density of water [kg/m3]

ߩ_௣,ீ density of gas [kg/m3] ߤ chemical potential [-] ߤ଴ standard chemical potential [-]

ߤ௣,ௐ dynamic viscosity of water [kg/m⋅s]

ߤ௣,ீ dynamic viscosity of gas [kg/m⋅s]

χ2 Chi-square coefficient a [-]

χt2 acceptable Chi-square coefficient [-]

ϕ

_{total porosity}[-] ߶_஺ aggregate content [-]

ϕ

c capillary porosity [-]

ϕ

e effective capillary porosity [-]

߶௦ volume fraction of solid phase [-]

߶௣ volume fraction of pore phase [-]

߶_{௦,௠௔௫} maximum volume fraction of solid phase [-] ߰ “effective mass” function [-] ߜݐ _{time step}[ts] ߜݔ _{lattice spacing}a [lu]

ߜ௧௥ transition thickness [µm]

ΔC concentration difference [mol/l] Δh difference of the hydraulic head across the sample [m] Δp pressure difference across the sample [N/m2]

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List of Symbols ix Δߜ௜௡,௫,௝ାଵ increase of the penetration depth [-]

Ω spatial domain of element [-] Ω(݂) collision function [-] Γ _{boundary [-]} Γ_ଵ_{inlet surface [-]} Γ_ଶ _{outlet surface}a[-]

Π(ܥ) functional equation of concentration [-]

Matrices and vectors

{C} concentration matrix [D] diffusion coefficient matrix {F} mass transport matrix M transformation matrix ܵመ collision matrix

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List of Abbreviations

w/b Water-to-binder ratio w/c Water-to-cement ratio AFm Monosulfate AFt Ettringite BCC / Body-centered cubic BGK Bhathagar-Gross-Krook BSE Backscattered electron C2S Dicalcium silicate 2CaO∙SiO2

C3A Tricalcium aluminate 3CaO∙Al2O3

C3S Tricalcium silicat 3CaO∙SiO2

C4AF Calcium ferroaluminate 4CaO∙Al2O3∙Fe2O3

CFD Computational fluid dynamics CH Calcium hydroxide

C-S Carnahan-Starling C-S-H Calcium silicate hydrates CT Computed microtomography D2Q9 Two-dimensional and nine-velocity D3Q19 Three-dimensional and nineteen-velocity DiffLBS Diffusivity lattice Boltzmann simulation EDX X-ray spectrometry

EOS Equation of state

ESEM Environmental scanning electron microscope FEM Finite element method

FHP Frisch-Hasslacher-Pomeau GEM General effective medium HCSS Hard core/soft shell HD High-density

HMM Heterogeneous multiscale method HP Hydration product

HTO Tritiated water

ITZ Interfacial transition zone LB Lattice Boltzmann

LD Low-density

LGA Lattice gas automata MD Molecular dynamics

MFEM Multiscale finite element method MFVM Multiscale finite volume method MIP Mercury intrusion porosimetry

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xii List of Abbreviations MRT Multiple-relaxation-time

PermLBS Permeability lattice Boltzmann simulation R-K Redlich-Kwong

R-KS Redlich-Kwong Soave P-R Peng-Robinson

PSD Particle size distribution RCM Rapid chloride migration

REV Representative elementary volume RH 1 Relative humidity

ROI Region of interest

SCMPLBS Modified Shan-Chen multiphase lattice Boltzmann simulation SEM Scanning electron microscope

SH Spherical harmonic SRT Single-relaxation-time UN Unhydrated cement

VMM Variational multiscale method VOI Volume of interest

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PART I

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Chapter 1 General Introduction

1.1 Research background

Concrete, the most popular and widely used construction material in the world, is made by mixing cement, coarse aggregate, fine aggregate or sand, and water together in appropriate proportions. In most European countries, about 50% of the total infrastructure budget is spent on the repair and maintenance of existing buildings and structures. These expenses are expected to increase even more in the future. A large percentage of these costs are due to problems related to the lack of durability of concrete structures. Therefore, the durability of concrete structures has become a main concern in recent years.

During service, concrete may interact with its environment. For instance, the constituent ions in the hydrated cement paste matrix may leach out from the concrete and cause the concrete structure to become weak. The aggressive chemical species (e.g. chloride, sulfates, carbon dioxide, etc.) from the environment can penetrate into the concrete, thereby leading to a series of physical or chemical interactions with the compounds of concrete. As a result, the degradation processes of the concrete itself or the reinforcement embedded in the structure may happen. The degradation processes of concrete structures can be briefly summarized in three main types, i.e. physical degradation process, chemical degradation process and mechanical degradation process. They consist of carbonation, leaching, frost action, alkali-aggregate reaction (AAR), sulfate attack, soft water attack, acid attack, chloride ingress, corrosion of reinforcement and so on, as shown in Figure 1.1.

As can be seen from Figure 1.1, the degradation processes strongly relate to the transport phenomena in concrete. The transport mechanisms involved in these degradation processes include the permeation of water or aqueous solutions, absorption of ionic contaminated water by capillary suction, and the diffusion of gaseous compounds through the concrete cover or ionic diffusion of dissolved constituents into or out of the concrete. Nowadays, it is generally accepted that concrete durability is to a large extent governed by the resistance of concrete to the ingress of aggressive agents. For example, chloride-induced corrosion of reinforcing steel is one of the main causes of deterioration of reinforced concrete structures. The time to corrosion initiation is considered dominant and usually defined as the service life that depends on the ingress of chloride ions into concrete. In such a case, transport properties in concrete, e.g. permeability and diffusivity, are usually considered as indicators to evaluate the durability and predict the service life of reinforced concrete structures.

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2 Chapter 1

DURABILITY

Figure 1.1 Relations between transport properties and durability of civil engineering structures

The transport properties in cement-based materials are dependent on the quality of cement-based materials represented by the cover depth and the microstructure of concrete, especially the pore structure, and environmental conditions, such as degree of water saturation or humidity, temperature, concentration of aggressive agents at concrete surface. Due to the continuous hydration of cement and physical/chemical interactions between the cement hydration products and the aggressive chemical species penetrating from environment, the microstructure of concrete cover changes with time. Accordingly, the transport rate of species in concrete is time-dependent.

During the past decades, a great deal of experimental and modelling work has already been conducted by researchers to take into account the effects of both age and environmental conditions. Due to the complexity in microstructural change and environmental conditions, it is very difficult to monitor or predict the durability and service life of concrete structures. Only modelling is possible to follow the change of the structure of concrete and environmental conditions, and to evaluate the transport properties in cement-based materials.

With respect to the modelling of transport properties in cement-based materials, a number of models have been proposed and developed in the past. A critical review on these models is given in Chapter 2. In principle, an ideal model for transport properties should be based on the direct measurements of the microstructure, especially the pore structure of cement-based

Transport mechanisms (permeation, diffusion, absorption)

Transport properties (permeability, diffusivity, absorptivity)

PERFORMANCE

Strength properties, Safety, Serviceability, Appearance Carbona-tion Oxygen diffusion Chloride ingress Water saturation Alkali ingress Sulfate ingress Ionic diffusion

Microstructure Environmental Cracking

Conditions Frost action Leaching Alkali attack Sulfate attack Corrosion of reinforcement

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General Introduction 3

materials [Garboczi and Bentz 1998]. The structural complexity of cement-based materials usually extends over a wide range of length scales from nanometer pores to millimeter aggregates. In most of the existing models, the transport properties are determined at the micro-scale or at the macro-scale, but not from a multiscale point of view. Most of existing models merely provide the results of transport properties, whereas, these models are unable to offer fundamental insights and understanding of fluid flow and ions transport processes in cement-based materials. In addition, transport properties in partially saturated cement-based materials are scarcely studied. In summary, the following modelling aspects are still inadequate:

- Fundamental insights and understanding of fluid flow and ions transport processes in cement-based materials.

- Time dependence of chloride diffusivity in cement-based materials due to the ongoing cement hydration and chloride binding.

- Transport properties in cement-based materials under unsaturated conditions.

- Effects of ITZ and aggregate on transport properties in cement-based materials.

- Microstructure-based prediction of transport properties in cement-based materials from a multiscale point of view.

In view of the current state of the modelling of transport properties in cement-based materials, the aim, the objectives, the scope and the strategy of this study are shown below.

1.2 Aim, objectives and scope of this study

The overall aim of this research is to improve the understanding of fluid flow and ions transport processes in cement-based materials and to develop an integrated multiscale modelling scheme from micro- to meso-scale to systematically estimate the transport properties in cement-based materials. The results of this study provide a better insight into the transport phenomena in cement-based materials, and offer a guidance and reference to engineers and researchers for assessing the durability and predicting the service life of reinforced concrete structures, and to designers for designing structures to meet their expected service life. The main objectives of this study are as follows:

- As in the existing models the transport properties in cement-based materials are usually investigated at one single scale, the first objective of this study is to develop an integrated multiscale scheme to simulate the transport properties in cement-based materials in order to cover the structural characteristics at each scale.

- The second objective of this study is to propose a new microstructure-based model that provides the capability to probe the behavior of fluids and ions at a microscopic level and enables one to detect the transport processes in cement-based materials, which cannot be captured from existing models.

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4 Chapter 1

- The third objective of this study is to quantitatively investigate the time-dependent chloride diffusivity in concrete as a result of ongoing cement hydration and chloride binding. Other influencing factors such as chloride concentration, interfacial transition zone (ITZ), aggregate content and aggregate shape are also taken into account.

- In view of the fact that transport properties in partially saturated cement-based materials are still not clear, the fourth objective of this study is to develop a suitable model to simulate the moisture distribution in cement-based materials and to study the influence of degree of water saturation on transport properties in cement-based materials in a quantitative manner.

Considering various transport mechanisms, this research mainly focuses on fluid permeability and chloride diffusivity in cement-based materials. The influencing factors taken into account in the chloride diffusivity prediction are listed as:

- W/c (w/b) ratio

- Age

- Chloride binding

- Degree of water saturation (humidity)

- Chloride concentration

- Interfacial transition zone

- Aggregate content

- Aggregate shape

1.3 Research strategy of this study

In order to accomplish these objectives, the following strategies are developed and applied:

- At micro-scale, the 3D microstructure of cement paste is obtained from the cement hydration model HYMOSTRU3D and X-ray computed microtomography test. Cement paste is simulated as a composite material consisting of capillary pore, hydration product and anhydrous cement grain. Percolation phenomenon of microstructure of cement paste is studied and the representative elementary volume (REV) of cement paste is determined. The lattice Boltzmann single-phase model and multi-phase model are applied to simulate the fluid permeability and ionic diffusivity in saturated and unsaturated cement paste, respectively.

- At meso-scale, the 3D structure of mortar is developed. Mortar is considered as a composite material consisting of sand, bulk paste and ITZ. The microstructure of ITZ is formed and its transport properties are determined by the approach at micro-scale. Finite element method (FEM) is then utilized to estimate the ionic diffusivity in mortar.

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- With respect to concrete, its 3D structure is modeled as a composite material made up of aggregate, mortar and ITZ. An upscaling procedure from mortar to concrete is carried out to estimate the ionic diffusivity in concrete.

Based on the above multiscale modelling strategy, the ingress of chloride ions into concrete cover can be simulated, the results of which can be directly used as input to predict the service life of reinforced concrete structures exposed to a marine environment, airborne salt and deicing salts.

1.4 Outline of this thesis

This thesis is subdivided into four parts. The flowchart of the structure is shown in Figure 1.2.

- Part I: Introduction (Chapters 1~ 2)

- Part II: Multiscale modelling: Methodology (Chapters 3 ~ 5)

- Part III: Multiscale modelling: Results and discussion (Chapters 6 ~ 7)

- Part IV: Conclusions and prospects (Chapter 8)

Chapter 1 is the general introduction. The background, the objectives, scope and research strategy are presented.

Chapter 2 is the literature review. The theoretical background of transport mechanisms is given. The state of the art of microstructure-based modelling of transport properties in cement-based materials and multiscale methods is presented based on a literature survey.

Chapter 3 focuses on the multiscale method. Based on a brief literature survey in Chapter 2, a multiscale lattice Boltzmann-finite element method is proposed. The multiscale modelling framework and procedure are described in detail. One of the upscaling approaches, i.e. volume averaging technique is also introduced, which is used for the upscaling procedure from cement paste to mortar, and mortar to concrete in this study.

Chapter 4 describes the details on the proposed micro-scale solver: lattice Boltzmann method, consisting of single-relaxation-time (SRT) model, multi-relaxation-time (MRT) model, modified Shan-Chen multiple phase model and diffusion model. The history, brief theoretical background, boundary conditions, conversion of units and implementations of lattice Boltzmann models are presented. The benchmark tests are carried out to validate each model.

Chapter 5 presents the 3D structures of cement paste, ITZ, mortar and concrete. The microstructure evolution of cement paste is investigated using HYMOSTRUC3D and X-ray computed micro-tomography images, respectively. The percolation analysis of cement paste is carried out. The formation and modelling of microstructure of the ITZ is also discussed. An algorithm is developed to simulate the 3D mesostructures of mortar and concrete as a three-phase composite material consisting of sand (aggregate), bulk matrix (mortar) and ITZ, respectively.

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6 Chapter 1

Chapter 6 shows the micro-scale study, i.e. transport properties in cement paste. The REV size for cement-based materials at the micro-scale is determined according to the numerical-statistical approach described in Chapter 3. MRT lattice Boltzmann model is used to simulate the water permeability in saturated cement paste. The simulations are verified with the experimental data from literature. After this, the modified Shan-Chen model and diffusion model are integrated to study the chloride diffusivity in saturated and unsaturated cement paste. The effects of w/c ratio, curing age, chloride binding, degree of water saturation and chloride concentration on the chloride diffusivity in cement paste are quantitatively analyzed. The results of this Chapter are used as inputs for the meso-scale study.

Chapter 7 deals with the meso-scale study, i.e. transport properties in mortar and concrete. Based on the micro-scale solver and microstructure of the ITZ, transport properties in the ITZ are estimated. Finite element method is applied to simulate the chloride diffusivities in mortar and concrete, respectively. The simulation results are compared with the experimental data obtained from literature. The influences of ITZ, aggregate content and aggregate shape upon chloride diffusivity are estimated in a quantitative manner.

Chapter 8 consists of conclusions and prospects. Some remarks and recommendations for future research are presented.

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Chapter 2 General Literature Review

2.1 Introduction

Cement-based materials are used in the construction of a wide range of structures. Many of these structures are exposed to a variety of aggressive environments during their service life. Their durability strongly depends on the transport of fluids and/or aggressive agents from the environment into cement-based materials. The transport of fluids and ionic species within the pore structure of cement-based materials can take place according to four basic mechanisms, i.e., permeation due to a pressure gradient, diffusion due to a concentration gradient, absorption due to capillary action and migration due to an electrical potential gradient. In many cases, the durability is controlled by the fluid permeability and ionic diffusivity in cement-based materials. Both of them are strongly related to the microstructure of cement-based materials. During the past few decades, a great deal of effort has been made towards developing microstructure-based models to predict permeability and diffusivity.

This chapter presents a critical review on permeability models and diffusivity models. The survey begins with a brief description of the four basic mechanisms of fluid flow and ion transport, and factors influencing chloride diffusivity in cement-based materials. Both the permeability models and diffusivity models derived from various approaches can be divided into three categories, i.e. empirical models, physical models and computer-based models, which are reviewed in different sections. It should be noticed that this review is strictly limited to the microstructure-based models. Emphasis is therefore placed on the most recent developments in this field. A number of multiscale methods have been proposed and developed in the past decades to investigate the mechanical behavior and transport properties in porous materials. In this chapter, a brief overview of the existing multiscale methods in general is also given.

2.2 Fluid flow and ion transport mechanisms

The transport of liquids, gases or ions in aqueous solutions into cement-based materials takes place through pore spaces in the cement paste matrix and interfacial transition zone and/or micro-cracks. A variety of different mechanisms, like permeation, diffusion, absorption and migration, may govern the transport process, depending on many factors, such as pore structure of cement-based materials, degree of fluid saturation in the pore system, local concentration of ions and environmental conditions.

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10 Chapter 2

2.2.1 Permeation

Permeation usually refers to the flow of a fluid (liquids or gases) through a material as a result of a pressure gradient. According to the pore structure and fluids viscosity, capillary flow may be grouped into two types, i.e. laminar flow and turbulent flow. The capillary pores in cement-based materials are generally in the order of 10-7 m. The capillary flow through cement-based materials is laminar flow. The intrinsic permeabilityκwith dimensions of area (m2) is a porous material characteristic describing the permeation of fluids through a porous material due to a pressure head, dependent purely on the characteristics of the porous material and independent on the fluid properties. It is the most rational concept of permeability and can be determined experimentally or numerically according to Darcy’s law:

QL A p

η

κ

=

∆ (2.1)

where Q represents the volume of fluid passed in unit time (m3/s), L is the thickness of the sample (m), A is the cross-sectional area of the sample (m2),

η

is the dynamic viscosity of the fluid (Ns/m2) and Δp is the pressure difference across the sample (N/m2). For practical purposes, a modified version of Darcy’s law can be specifically deduced for saturated porous materials: QL k A h = ∆ (2.2) where k is the saturated permeability with dimension (m/s) and is a function of the characteristics of the porous material and of the density and viscosity of the permeating fluid, Δh is the difference of the hydraulic head across the sample (m). If the fluid involved is water, the intrinsic permeability can be converted into saturated water permeability kw (m/s) by

multiplying by a conversion coefficient

ρ

g/

η (m

-1s-1), in which

ρ and g represent the fluid

density (kg/m3) and acceleration of gravity (9.81 m/s2) respectively. For water at a temperature of 20 ºC, this conversion coefficient equals 9.76×10-6 m-1s-1.

The intrinsic permeability of cement-based materials (κ) depends on the microstructural details of cement-based materials, such as volume fraction, size, shape, distribution, connectivity and tortuosity of pores. For this reason, κ depends on all factors that influence the pore structure of cement-based materials, such as water-to-cement (w/c) ratio, curing age, type of cement, temperature and so on.

2.2.2 Diffusion

Diffusion is usually defined as the movement of free molecules or ions in the pore solution due to a concentration gradient, or more strictly speaking, chemical potential, from regions of higher concentration to regions of lower concentration of the diffusing substance. It has been presented that the motion of molecules or ions in a solution also follows the kinetic-molecular theory of ideal gases. At a specific temperature, molecules or ions move in straight lines from one location to another at a uniform velocity until they collide with each other or with the walls. As a result, the momentum is transferred from one particle to another and particles lose speed. Owing to the extremely difficulty of tracking the motion of each individual particle in

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General Literature Review 11

the solution, the movement of molecules or ions is commonly studied from a macroscopic point of view by measuring changes in concentration. The movement of the species of molecules or ions in the pore solution can be described in terms of their chemical potential ߤ that is equivalent to their free energy and can be given by [Tang 1996]:

( )

0 RTln C

µ µ

= +

γ

(2.3) where

μ

0 represents the standard chemical potential, R is the universal gas constant (J

mol-1k-1), T is the absolute temperature (K),

γ

is the activity coefficient and C is the concentration of species in solution. The chemical potential is directly associated with the concentration of species. A concentration difference between two points in the pore solution will lead to a gradient of chemical potential, which is the driving force of the molecular diffusion from the point of higher concentration to the point of lower concentration. Under a gradient of chemical potential, the rate of mass transfer through a unit area, the so-called diffusion flux J in one-dimensional situation along the gradient, can be described as [Tang 1996]: 0 ln ln 1 1 ln ln C C J BC BRT D x C x C x

µ

γ

∂  ∂ ∂  ∂ ∂ = − = − _ + _ = _ + _ ∂ _ ∂ _ ∂ _ ∂ _ ∂ (2.4)

in which B is the proportionality factor and assumed to be constant [Tang 1996]. Accordingly, the term D0 = BRT is a constant for a given temperature. Replacing Eq. 2.4 with the

well-known Fick’s first law of diffusion (steady-state diffusion):

C J D x ∂ = − ∂ (2.5)

we can obtain that the diffusion coefficient D (m2/s) also called diffusivity can be expressed as: 0 ln 1 ln D D C

γ

∂   = _ + _ ∂  (2.6)

The diffusion coefficient D is a function of both the activity coefficient

γ and the

concentration of the ionic species C. For an ideal solution, the activity can be considered to be equal to the concentration and the activity coefficient equals to 1. Thus, the diffusion coefficient D is constant, as D = D0.

The ionic species found in the pore solution in cement-based materials are so concentrated that the assumption of an ideal solution is not realistic and the activity is not equal to the concentration. Therefore, the activity coefficient that reflects the interaction between various ions present in solution should also be taken into account in the determination of diffusion coefficient. The activity coefficient is observed to be a complex function of the ionic concentrations in solution. Several semi-empirical expressions have been proposed for the quantitative evaluation of the activity coefficient. A comprehensive survey of these models can be found in [Bockris et al. 2002] and [Zhang and Gjørv 2005]. The concentration dependence of the ionic diffusion coefficient will be discussed in more detail afterwards.

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12 Chapter 2

In practical terms, Eq. 2.5 is only applicable to the steady-state diffusion process, i.e., there is no change in the concentration C at a location x with time. When diffusion solely occurs in a pore solution, the diffusion coefficient D only depends on the type of diffusing species and their concentrations. Under this condition, the diffusion coefficient is generally termed the self-diffusion coefficient (Dp). However, when passing through a porous medium,

like concrete, the diffusion process is not only affected by the pore structure of concrete but also influenced by the presence of other types of species in the pore solution. If the diffusing ions do not interact physically or chemically with the solid hydration products (which is the most cases an over simplification), the resulting diffusion coefficient is usually referred to as the effective diffusion coefficient De, which is a material characteristic describing the ability

of transfer for a given molecule or ion driven by a concentration gradient.

For a non-steady state process, the change of the ionic concentration C at a location x within a unit volume with time can be described by Fick’s second law:

C J t ∂ = −∇ ⋅ ∂ (2.7)

With respect to chloride ions, they are found in cement-based materials in two forms. One is bound chloride, and the other is free chloride. Bound chlorides are either the result of chemical binding, i.e. chloride ions are chemically bound by reacting with the cement hydrates or physical binding, i.e. chloride ions are physically adsorbed to the cement gel. The total chloride content in cement-based material is a sum of free and bound chlorides. It is only the free chlorides that can dissolve in the pore solution and diffuse in cement-based materials through the pore solution, reach the surface of steel rebars and induce the corrosion process. Therefore, the steel corrosion is related to the free chlorides. It is better to simulate the chloride diffusion in cement-based materials only in terms of free chloride concentration. The total chloride concentration Ct is the summation of free chloride concentration Cf and bound

chloride concentration Cb as Ct = Cf + Cb. Thus, the left-hand term in Eq. 2.7 can be written

as: 1 f f t b b f C C C C C t t t C t   ∂ ∂ ∂ ₌ ₊∂ ₌ ₊ ∂     ∂ ∂ ∂ _ ∂ _ ∂ (2.8)

By incorporating Eqs. 2.5 and 2.8 into Eq. 2.7, the governing equation of transient diffusion of chloride ions into cement-based materials can be rewritten as follows:

2 2 2 2 1 f e f f a b f C D C C D t _C x x C ∂ ∂ ∂ = = ∂  _∂  ∂ ∂ +    _∂    (2.9)

where Da denotes the apparent chloride diffusion coefficient that is a function of the effective

chloride diffusion coefficient De and chloride binding capacity ∂Cb/∂Cf. ∂Cb/∂Cf represents

the influence of chloride binding on the chloride diffusion coefficient and varies greatly with the free chloride concentration Cf. Therefore, the apparent chloride diffusion coefficient Da

should not be constant but relates to the effective chloride diffusion coefficient De and the free

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materials affected by factors including w/c ratio and age, and upon the environmental conditions including temperature and degree of water saturation.

With respect to simple geometry and boundary conditions, by assuming the apparent diffusion coefficient Da to be a constant, there exists a solution of the governing partial

differential equation Eq. 2.9. If the boundary condition is specified as C(x = 0, t > 0) = Cs (the

surface concentration is constant at Cs), the initial condition is specified as C(x > 0, t = 0) = C0

(the initial concentration in the specimen is C0) and the infinite point condition is specified as C(x = ∞, t > 0) = 0 (far enough away from the surface, the concentration will always be 0), an analytical solution to Eq. 2.9 was given by Crank [1975]:

( )

, 0

(

0

)

1 2 f s a x C x t C C C erf D t    = + −  − _ _  _ _   (2.10)

where erf() is the standard error function that can be expressed as:

( )

2 0 2 exp z erf z y dy

π

=

_∫

− (2.11)

The oversimplified Eq. 2.10 has been widely applied to describe the diffusion of chloride ions in cement-based materials. However, it is based on the assumption of constant chloride diffusion coefficient, which is inconsistent with the fact that the chloride diffusion coefficient is not constant but changes with time. As a result, the governing equation Eq. 2.9 is unable to be solved analytically and numerical methods are required to obtain an approximate solution.

2.2.3 Absorption

Absorption is the transport of liquids due to the capillary suction present in the pores in cement-based materials. When the pores come into contact with a wetting liquid, the liquid will invade as a result of capillary forces. The capillary absorption is influenced by both the characteristics of the liquid and solid, such as, fluid viscosity, surface tension, pore structure (radius, tortuosity and continuity of capillary pores) and surface energy. For example, the local capillary force is proportional to the surface tension and inversely proportional to the pore radius (smaller pores correspond to a larger capillary force). Capillary sorption of cement-based materials is generally defined by absorption rate, i.e. absorptivity Sa, which is

determined by the mass of the liquid absorbed per unit area and a function of the contact time

t using the following equation:

( ) a m t S A t

ρ

= (2.12)

where m(t) represents the mass of liquid gained by the specimen at time t, ߩ is the liquid density, A is the surface area of the specimen exposed to the liquid and the absorptivity Sa is

in units of length/time0.5. For field cement-based materials, the square-root-of-time dependency may not always be followed, as the exponent in the power law function may vary between 0.2 and 0.5 [Bentz et al. 1999].

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14 Chapter 2

2.2.4 Migration

Migration is the transport of ions in electrolytes due to the action of an electrical field as the driving force. In an electrical field positive ions will move preferentially to the negative electrode and negative ions to the positive one. Migration may result in a difference in the concentration of these ions in a homogeneous solution or may induce a species flux in the direction of concentration gradients. In general, the migration of a specific ion in electrolytes can be described as the sum of the diffusion component arising from concentration gradients, the migration component from potential gradients and the advection component by using Nernst-Planck equation [Tang 1996]:

C zF E J D DC CV x RT x ∂ ∂   = −_ + + _ ∂ ∂  (2.13)

where J is the mass flux, D is diffusion coefficient (is sometimes called migration coefficient),

C is the ionic concentration in pore solution, z is the electrical charge, F is Faraday constant, R

is gas constant, T is absolute temperature, E is electrical potential and V is the convection velocity of solution.

2.2.5 Test methods

Among the above mechanisms, permeation, diffusion and absorption are three primary ones used to describe the fluid flow and ion transport in cement-based materials, which are main interest in this research. Absorption is involved in the permeation and diffusion of ions in partially saturated cement-based materials. Permeability can be measured by gas permeability tests or water permeability tests. On the basis of steady state diffusion, non-steady state diffusion or electrical migration mechanisms, a number of test methods have been proposed during the past few decades to determine the chloride diffusion coefficient. These tests can be classified into two groups, i.e. natural diffusion tests and electrically accelerated tests. Different methods for measuring chloride transport properties in cement-based materials have been reviewed in [Tang 1996; Yuan 2009].

Natural diffusion tests generally refer to those simulating the natural process of chloride transport, such as diffusion cell tests and immersion tests with a chloride concentration similar to that in sea water.

In a diffusion cell test, two cells are separated by a thin slice of specimen where the upstream cell contains a specific chloride concentration and the downstream cell initially does not contain chloride ions. The increase in chloride ions in the solution of the downstream cell is monitored at a certain interval time. When a steady-state flow of the chloride ions is reached, the chloride diffusion coefficient is then calculated according to Fick’s first law as Eq. 2.5. The diffusion coefficient obtained from natural diffusion tests is actually the effective diffusion coefficient De. Since diffusion is a very slow process, natural diffusion tests are,

therefore, very time-consuming. It generally takes several months or years to obtain the test results.

In an immersion test, all surfaces of a specimen except one are sealed to prevent multi-directional penetration and form only one-dimensional penetration. The sealed

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specimen is immersed in a solution containing a constant chloride concentration. After a certain period of immersion, the chloride profile is measured by grinding the specimen successively from the exposed surface. The diffusion coefficient is then determined by curve-fitting the measured chloride profile to the error function in the analytical solution to Fick’s second law as listed in Eq. 2.11. The diffusion coefficient derived from the immersion test is actually the apparent chloride diffusion coefficient Da. The results from the immersion

test strongly depend on the immersion period and the chloride concentration in the bulk solution.

In order to accelerate the process of chloride transport, several rapid methods, such as Coulomb test, steady-state migration test, rapid immersion test, rapid chloride permeability test standardized by ASTM C1202 and rapid chloride migration (RCM) test have been proposed. Among these methods, an experimental setup and procedure for the RCM test recommended by Tang [1996] is widely utilized. The RCM test involves applying a potential of 30~40 V across a 50 mm thick specimen for a certain test duration, then splitting the specimen and measuring the penetration depth of chlorides by using a colorimetric method. The migration coefficient can be obtained by using following formula:

0 d d x a x RT D zFE t − = ⋅ (2.14)

in which parameters z, F, R and T are identical to those in Eq. 2.13 and xd is the penetration

depth from the colorimetric method. The factor a0 is given by:

1 0 0 2 2 RTL 1 Cd a erf zFU C −   = ⋅ _ − _  (2.15)

The diffusion coefficient derived directly from RCM test is actually the non-steady-state migration coefficient. A relationship between the migration coefficient and diffusion coefficient is required in order to estimate the chloride diffusion coefficient. The relations between different diffusion coefficients and migration coefficients were presented in [Tang and Nilsson 2002] and found to be strongly dependent on many factors, such as the effect of counter electrical potential, the ratio of cation velocity to anion velocity, the ionic activity coefficient, the friction coefficient and chloride binding capacity.

2.3 Factors influencing chloride diffusivity

Chloride diffusivity in cement-based materials may be influenced by several factors, such as w/c ratio, cement type and content, curing age, chloride binding, temperature, humidity or degree of water saturation, source concentration of chloride ions, ITZ, aggregate content and aggregate shape. In this section, the effect of each factor on chloride diffusivity is briefly reviewed.

2.3.1 Effect of w/c ratio

Page et al. [1981] found that the chloride diffusivity became 4 to 6 times larger when the w/c ratio increased from 0.4 to 0.6. They also found that the diffusivity increased with the increase of temperature.

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16 Chapter 2

Dhir et al. [2004] studied the effects of w/c ratio and cement content on chloride diffusivity in concrete. Three mixes made with CEM I, CEM II/B-V, CEM III/A cement with w/c ratios of 0.45, 0.55 and 0.65 were prepared and examined. The results showed that the chloride diffusivity increased with increasing w/c ratio and for the type of cements tested, reduction in cement content at a given w/c ratio had only a minor effect on chloride diffusivity.

2.3.2 Time-dependent chloride diffusivity

Takewake and Mastumoto [1988] are probably the first who stated the dependency of chloride diffusivity on time. They proposed a purely empirical equation (with a power law function) to describe the decrease of diffusivity with time:

0.1 0 ( ) D t = ⋅a t− (2.16)

where D(t) is the time-dependent diffusivity, t is the curing age of cement-based materials, a0

is an empirical constant.

From the rapid diffusivity test, Tang and Nilsson [1992] found that the measured chloride diffusivity varied with age and the time-dependent diffusivity D(t) was mathematically expressed using the following equation:

( ) n ref ref t D t D t   = _⋅ _   (2.17)

where Dref is the diffusivity at some reference time, tref (i.e. usually 28 days), n is a constant

and normally being referred to as the aging factor, which is dependent on the variables such as the type of cement used and the mix proportions.

Eq. 2.17 is the most common equation used to predict changes of diffusivity with age of cement-based materials [Tang and Gulikers 2007]. However, values of n for various cement-based materials have not yet been fully obtained although some preliminary values have already been published [DuraCrete 2000]. Further research on the determination of this parameter will be helpful to accurately predict the time-dependent chloride diffusivity.

Another question regarding the above models is that the exposure duration was not taken into account. A direct application of these models will overestimate the rate of chloride transport and inaccurately evaluate the risk of chloride-induced corrosion. In 2007, Tang and Gulikers [2007] provided an improved mathematical expression to describe the time-dependent diffusivity, in which a given exposure duration is considered.

1 1 ( ) 1 1 n n n ref ex ex ref D t t t D t n t t t − − _ _ _ _ _{ } _ = ₋  +  −               (2.18)

where Dref and tref are a pair of known diffusivity and age, tex is the age of the sample at the

start of exposure, t is the duration of exposure and n represents the aging factor. A large number of experiments are required to determine the value of n, which is a very time-consuming task.

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2.3.3 Effect of chloride binding

When chloride ions penetrate into cement-based materials, some of them can be captured and immobilized by the hydration products. This process of interaction between the chloride ions and cement hydrates is known as chloride binding. Although the mechanisms of chloride binding are not quite clear, it is a common view that two types are involved in chloride binding: one is the physical binding by calcium silicate hydrate (C-S-H) gel, and the other is the chemical binding by some of the hydration products, in particular the calcium monosulfoaluminate hydrate (AFm: 3CaO∙ Al2O3∙ CaSO4∙ 12H2O) to form Friedel’s salt

(3CaO∙Al2O3∙CaCl2∙10H2O), which is insoluble in water. Chloride binding may change the

pore structure of cement-based materials and lead to a decrease in the chloride diffusivity in cement-based materials. The effect of chloride binding on the chloride diffusivity in cement-based materials can be quantified in terms of 1/(1+∂Cb/∂Cf), as described in Eq. 2.9.

In order to estimate the apparent chloride diffusivity Da, it is essential to establish the

quantitative relationship between the concentrations of bound Cb and free chlorides Cf over a

range of chloride concentrations at a given temperature, which is referred to as the chloride binding isotherm. In the literature, four types of chloride binding isotherms, i.e. linear, Langmuir, Freundlich and BET binding isotherms, have been commonly used to mathematically describe the relationship.

Arya and Newman [1990] proposed a linear binding isotherm, which can be expressed as follows:

b f

C =aC + b (2.19) where Cb and Cf are the concentrations of bound and free chlorides, a and b are constants. The

linear binding isotherm is an oversimplification and seems to be applicable within a limited range of chloride concentrations [Tang 1996]. It was found that the linear binding isotherm overestimates the amount of bound chlorides at high chloride concentrations and underestimates chloride binding at low chloride concentrations [Yuan 2009].

The Langmuir isotherm for chloride binding can be expressed as follows:

1 f b f aC C bC = + (2.20) where the binding constants a and b change with the chemical composition of cement-based materials, which can be derived from the statistical analysis of a great deal of experimental data and do not have any physical meaning. Sergi et al. [1992] utilized the Langmuir isotherm to investigate the chloride binding and obtained the values of a and b as 1.67 and 4.08, respectively based on the linear regression analysis of data from cement paste samples with w/c ratio of 0.5. It was found that the Langmuir isotherm fit the experimental data very well when the free chloride concentration in the pore solution is lower than 0.05 mol/l [Tang 1996].

Tang [1996] presented that monolayer adsorption takes place at low levels of chloride concentration and can be described better by Langmuir isotherm. However, the adsorption becomes more complex when the free chloride concentration is high and can be described better by the Freundlich isotherm, which can be given by:

Multiscale Lattice Boltzmann-Finite Element Modelling of Transport Properties in Cement-based Materials

Multiscale Lattice

Boltzman-Finite Element Modelling of

Transport Properties in

Cement-based Materials

Mingzhong Zhang

Mingzhong Zhang

ISBN 978-90-90-27483-6

From mortar to concrete

From cement paste to mor

Multiscale Lattice Boltzmann-Finite Element Modelling of

Transport Properties in Cement-based Materials

Acknowledgements

Table of Contents

Table of Contents

List of Symbols

γ

ξ

η

κ

ρ

ϕ

ϕ

ϕ

List of Abbreviations

PART I

Chapter 1

General Introduction

Chapter 2

General Literature Review

η

κ

η

ρ

η (m

ρ and g represent the fluid

( )

µ µ

γ

μ

γ

µ

γ

γ

γ

γ and the

( )

(

)

( )

( )

π

∫

ρ

_∫