Diffusion-reaction model for alkali-silica reaction in concrete

XII International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XII E. Oñate, D.R.J. Owen, D. Peric and B. Suárez (Eds) DIFFUSION-REACTION MODEL FOR ALKALI-SILICA REACTION IN CONCRETE JOAQUÍN LIAUDAT*, CARLOS M. LÓPEZ AND IGNACIO CAROL ETSECCPB (School of Civil Engineering -Barcelona) UPC (Technical University of Catalonia) Jordi Girona, 1, Edif D2, E-08034 Barcelona Email: joaquin.l[email protected], [email protected], ignacio.caro[email protected] Key words: Concrete, ASR, Diffusion Reaction, Reactive transport, Numerical Modeling Abstract.A new diffusion-reaction model for the potentially deleterious Alkali-Silica Reaction (ASR) process in concrete is presented. The model involves three coupled diffusion processes, two in-going and one out-going from the aggregate viewpoint. Alkali (Na+and K+) and Calcium (Ca2+)ions diffuse “inwards”, from high molar concentration sites in the pores of the cement paste phase of the concrete specimen or at its boundaries, towards the aggregate-cement paste interfaces or the inner cracks of the aggregates. The OH-ions associated with alkali and calcium ions attack certain forms of silica in the aggregates (the “reactive silica”), dissolving it in the form of silicate ions which in turn diffuse back to the cement paste phase (“outwards”). The final potentially deleterious ASR precipitation process involves those silicate ions, plus calcium and alkalis. It takes place wherever the reactants are available by precipitating silicate hydrates of two kinds (Calcium-Silicate-Hydrates –CSH or Calcium-Alkali-Silicate-Hydrates –CASH) in a proportion depending on concentrations and temperature. The diffusion-reaction equations of this process are discretized in space and time using finite differences.An example of application in 1D is presented to illustrate the capabilities to reproduce realistically the ASR process, including some novel features not usually which are not considered in the available literature, such as the role of calcium in the development of the reaction and the inherent relationship between the reaction product composition and its swelling capacity. 1 INTRODUCTION Expansion due to Alkali Silica Reaction (ASR) is the second most common cause of concrete deterioration, after reinforcement corrosion. First reported by Stanton in 1940 [1],its reaction-expansion mechanism is not yet fully understood. It is accepted that certain types of metastable silica present in aggregates are attacked by hydroxyl ions in concrete pore solution producing its dissolution into silicate ions. These ions recombine with alkalis and calcium ions in anexpansive reaction forming a Calcium-Alkali-Silicate-Hydrates(CASH) of variable stoichiometry, molar volume and mechanical properties. It is a complex phenomenon that strongly depends on the dosage and composition of the concrete, as well as on its temperature and capillary pore humidity. ASR products are found filling pores and cracks both in the hydrated cement paste (HCP) Diffusion-reaction model for alkali-silica reaction in concrete 479 Joaquín Liaudat, Carlos M. López and Ignacio Carol 2 and the aggregates of concrete. However, the most expansive ASR products seem to be those with low content of calcium, formed in aggregate cracks or pockets were calcium from the HCP is less available. On the other hand, the presence of calcium appears to be essential for concrete expansion [2], as Portlandite (𝐶𝐶𝐶𝐶(𝑂𝑂𝑂𝑂)2)in the HCP acts as a buffer to maintain the 𝑂𝑂𝑂𝑂−concentration in the pore solution at high values, therefore allowing further dissolution of the reactive silica [3]. Some chemo-mechanical simulations of ASR have been reported in the last fifteen years [4, 5, 6]. To the best of the authors’ knowledge, only Poyet et al. [7] have considered the role of portlandite and calcium ions diffusion in the development of ASR. However, they have neglected the silicate ions diffusion by assuming that the ASR products only precipitate in the interfacial transition zone surrounding the reactive aggregate. In this paper, diffusion-reaction model currently under development is presented. It includes three diffusion processes: alkali and calcium ions, mainly from the HCP into the aggregate, and silicate ions, mainly from the aggregates towards the HCP. The main objective is to reproduce realistically some aspects of ASR which are not considered in the available models, such as the role of calcium in the development of the reaction and the inherent relationship between the reaction product composition and its swelling capacity. For the sake of simplicity, constant temperature and full water saturation of concrete are assumed at this stage of the formulation development. For validation purposes, the model has been implemented in 1D and compared with experimental results of free expansion tests at the level of a single cement-aggregate interfacetaken from an ongoing experimental campaign within the same study. 2 DESCRIPTION OF THE CHEMO-TRANSPORT MODEL 2.1 Chemical reactions involved Chemical reactions leading to ASR expansions occur in the pore water of the cement paste and aggregate phases in concrete. When pore solution of usually high pH is in contact with metastable silica present in the aggregate,an acid-base reaction takes place, leading to the dissolution of the silica into silicate ions [8]. This reaction involves the progressive dissociation of the silicate ions, and can be expressed in a simplified way [9] as: 𝑆𝑆𝑆𝑆𝑂𝑂2,𝑂𝑂2𝑂𝑂(𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠.) +𝑂𝑂𝑂𝑂−→𝑆𝑆𝑆𝑆𝑂𝑂4𝑂𝑂3−(𝐶𝐶𝑎𝑎) (1) Once in the pore solution, the silicate ions react with calcium and alkali ions forming a Calcium-Alkali-Silicate-Hydrate gel (CASH) of variable stoichiometry depending on the relative concentration of the reactants [10, 11]. The variability of the stoichiometry seems to be an important issue,since a clear dependence of the molar volume and the mechanical properties of the CASH on its composition, particularly with the Calcium to silica ratio has been reported [11, 12]. In order to include this effect, the model considers the following two simultaneous reactions, one (2) forming a Calcium-Silicate –Hydrate (CSH) and the other (3) forming aCalcium-Alkali-Silicate-Hydrate (CASH), where 𝛼𝛼1,𝛼𝛼3,𝛼𝛼4and 𝛽𝛽2,𝛽𝛽3,𝛽𝛽4are stoichiometric coefficients to be fit with experimental data. The overall product of the reaction is dependent on the kinetics of CASH and CSH formation. 480 Joaquín Liaudat, Carlos M. López and Ignacio Carol 3 𝑆𝑆𝑆𝑆𝑂𝑂4𝐻𝐻3−+𝛼𝛼1𝐶𝐶𝐶𝐶2++𝛼𝛼3𝑂𝑂𝐻𝐻−+𝛼𝛼4𝐻𝐻2𝑂𝑂→𝐶𝐶𝛼𝛼1𝑆𝑆𝐻𝐻1+𝛼𝛼1+𝛼𝛼4 𝑤𝑤𝑆𝑆𝑤𝑤ℎ 𝛼𝛼3= 2𝛼𝛼1−1 𝐶𝐶𝑎𝑎𝑎𝑎 𝛼𝛼1>1 2 (2) 𝑆𝑆𝑆𝑆𝑂𝑂4𝐻𝐻3−+𝛽𝛽2𝐴𝐴++𝛽𝛽3𝑂𝑂𝐻𝐻−+𝛽𝛽4𝐻𝐻2𝑂𝑂→𝐴𝐴𝛽𝛽 2𝑆𝑆𝐻𝐻1+𝛽𝛽2 2+𝛽𝛽4𝑤𝑤𝑆𝑆𝑤𝑤ℎ 𝛽𝛽3=𝛽𝛽2−1 𝐶𝐶𝑎𝑎𝑎𝑎 𝑓𝑓𝑓𝑓𝑓𝑓 𝛽𝛽2> 1 (3) The prediction of the chemical composition of concrete pore water is a very complex problem [13]. However, in this study cement pore water is assumed to be a solution of 𝑁𝑁𝐶𝐶+, 𝐾𝐾+,𝐶𝐶𝐶𝐶2+ and 𝑂𝑂𝐻𝐻−ions, where portlandite (𝐶𝐶𝐶𝐶(𝑂𝑂𝐻𝐻)2)solubility controls 𝐶𝐶𝐶𝐶2+ concentrations [14]. Since 𝑁𝑁𝐶𝐶𝑂𝑂𝐻𝐻and 𝐾𝐾𝑂𝑂𝐻𝐻are strong bases, it can be assumed that all of it is dissociated, i.e. no solid 𝑁𝑁𝐶𝐶𝑂𝑂𝐻𝐻or 𝐾𝐾𝑂𝑂𝐻𝐻is in contact with the pore solution. Finally, only two dissociation reactions are considered: water self-ionization (4) and portlandite dissociation (5). 𝐻𝐻2𝑂𝑂+𝐻𝐻2𝑂𝑂↔𝑂𝑂𝐻𝐻−+ H3O+ (4) 𝐶𝐶𝐶𝐶(𝑂𝑂𝐻𝐻)2↔𝐶𝐶𝐶𝐶2++ 2(𝑂𝑂𝐻𝐻)− (5) Since 𝑁𝑁𝐶𝐶+and 𝐾𝐾+have similar effects on ASR, henceforth they are called generically 𝐴𝐴+. 2.2 Transport processes The preceding chemical reactions take place according to the availability of reactants in time and space which is basically determined by transport processes. In this model, only the diffusion of Ca2+,A+and SiO4H3 −are explicitly modeled. The OH−and OH3 +concentrations are obtained at each location by means of the chemical equilibrium equations above mentioned. These diffusion processes are formulated by means of Fick’s second law, which is expressed in 1D for ion SiO4H3 −in equation (6), where 𝑠𝑠− �𝑚𝑚𝑓𝑓𝑚𝑚/𝑚𝑚3 �is the ion concentration, 𝐷𝐷𝑠𝑠 [𝑚𝑚2/ℎ]is the overall diffusion coefficient (see section 2.3) and 𝑠𝑠󰇗−[𝑚𝑚𝑓𝑓𝑚𝑚 (𝑚𝑚3·ℎ) ⁄]is a well/sink term which accounts for the net production or consumption due to chemical processes of the ionic species 𝑠𝑠−. 𝜕𝜕𝑠𝑠− 𝜕𝜕𝑤𝑤 =𝜕𝜕 𝜕𝜕𝜕𝜕�𝐷𝐷𝑠𝑠𝜕𝜕𝑠𝑠− 𝜕𝜕𝜕𝜕�+ 𝑠𝑠󰇗− (6) Similar equations are assumed for ions 𝐶𝐶𝐶𝐶2+ (concentration 𝑐𝑐2+) and 𝐴𝐴+(concentration 𝐶𝐶+). The notation used for the independent variables is also stated in Table 1. 2.3 Volume fractions of solid components During the development of the reaction the volume fractions of the solid components in the material are not constant. Portlandite and reactive silica are progressively dissolved reducing their volumes and increasing the capillary porosity of the material. The CASH formed occupies this empty volume, leading to a net volumetric expansion when the volume of CASH produced exceeds the available pore space. Eight different solid components are considered in this model: 1) Impervious remnant: comprises impervious and inert constituents in HCP and aggregate. There is no diffusion through it and it does not experiment any volume change. 2) Capillary pores 481 Joaquín Liaudat, Carlos M. López and Ignacio Carol 4 3) Reactive silica 4) Portlandite 5) Reaction product (CSH):product of reaction (2), 6) Reaction product (CASH):product of reaction (3) 7) Inert cement hydration products:comprises cement paste constituents that do not take part in the reactions above mentioned. It does not experiment any volume change due to ASR. 8) Inert aggregate constituents:comprises aggregate constituents that do not take part in the reactions above mentioned. It does not experience any volume change due to ASR. Components 1, 2, 4 and 7 are initially present in the HCP. Components 1, 2, and 8 are initially present in the aggregates. Components 5 and 6 precipitate both in the HCP and in the aggregates as the reaction takes place. At a given location in space x, the volume fraction of each component kis represented by a variable 𝑢𝑢𝑘𝑘(𝑥𝑥), so that the total volume V(x) is given by equation (7). 𝑉𝑉(𝑥𝑥) = �𝑢𝑢𝑘𝑘(𝑥𝑥) 8 𝑘𝑘=1 (7) The volume fractions 𝑢𝑢𝑘𝑘(𝑥𝑥)are calculated from the concentration variables by means of equations (8), (9) and (10), where 𝑣𝑣𝑘𝑘is the molar volume and 𝑐𝑐𝑘𝑘(𝑥𝑥)is the concentration of the chemical species associated to the component 𝑘𝑘(e.g. for portlandite 𝑘𝑘= 4,𝑐𝑐4=𝑐𝑐0). The superindex 0 applied to 𝑢𝑢𝑘𝑘(𝑥𝑥)stands for initial value. 𝑢𝑢𝑘𝑘(𝑥𝑥)=𝑢𝑢𝑘𝑘 0(𝑥𝑥) 𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘= 1,7,8 (8) 𝑢𝑢2(𝑥𝑥)= max �0 ; 𝑢𝑢2 0 (𝑥𝑥)−�[𝑢𝑢𝑘𝑘(𝑥𝑥)−𝑢𝑢𝑘𝑘 0 (𝑥𝑥)] 6 𝑘𝑘=2 � (9) 𝑢𝑢𝑘𝑘(𝑥𝑥)=𝑣𝑣𝑘𝑘·𝑐𝑐𝑘𝑘(𝑥𝑥)·𝑉𝑉𝑝𝑝𝑝𝑝(𝑥𝑥) 𝑓𝑓𝑓𝑓𝑓𝑓 𝑘𝑘= 3,4,5,6 (10) The volumetric distribution of the solid components in each location determines the volume of pore solution 𝑉𝑉𝑝𝑝𝑝𝑝(𝑥𝑥)in it by equation (11) and its overall diffusion coefficientsby equation (12) ( similar equations are posed for 𝐷𝐷𝑎𝑎(𝑥𝑥)and 𝐷𝐷𝑝𝑝(𝑥𝑥))where 𝜑𝜑𝑘𝑘and 𝑑𝑑𝑘𝑘are the intrinsic porosity and the diffusion coefficient of phase k, and 𝑚𝑚𝑝𝑝is a coefficients that introduces the effect of the ion type (electric charge, ionic radius).The intrinsic porosity is defined as the volumetric fraction of the phase occupied with physically bonded water through which the diffusion-reaction process can happen. 𝑉𝑉 𝑝𝑝𝑝𝑝 (𝑥𝑥)=�𝑢𝑢 𝑘𝑘 (𝑥𝑥)·𝜑𝜑(𝑘𝑘) 6 𝑘𝑘=1 (11) 𝐷𝐷𝑝𝑝(𝑥𝑥)=𝑚𝑚𝑝𝑝 𝑉𝑉(𝑥𝑥)�𝑢𝑢𝑘𝑘(𝑥𝑥)·𝑑𝑑𝑘𝑘 8 𝑘𝑘=1 (12) 2.4 Chemical equations The set of chemical reactions described in section 2.1 is divided in two groups depending 482 Joaquín Liaudat, Carlos M. López and Ignacio Carol 5 on their kinetics. The first group includes water self-ionization (4) and portlandite dissociation (5), reactions that can be assumed to occur instantaneously. The second group involves slower reactions for which kinetic laws need to be established such as silica dissolution (1) and CSH/CASH formation (2) and (3). The equilibrium equations for the first group are: 𝐾𝐾𝑤𝑤={𝐻𝐻3𝑂𝑂+}{𝑂𝑂𝐻𝐻−}=𝛾𝛾ℎ+·𝛾𝛾𝑜𝑜ℎ−·ℎ+·𝑜𝑜ℎ− (13) 𝐾𝐾3𝑠𝑠𝑠𝑠 ={𝐶𝐶𝐶𝐶2+}{𝑂𝑂𝐻𝐻−}2=𝛾𝛾𝑐𝑐2+ ·(𝛾𝛾𝑜𝑜ℎ−)2·𝑐𝑐2+· (𝑜𝑜ℎ−)2 (14) in which 𝐾𝐾𝑤𝑤and 𝐾𝐾3𝑠𝑠𝑠𝑠are constants, and activity coefficients 𝛾𝛾ℎ+,𝛾𝛾𝑜𝑜ℎ−and 𝛾𝛾𝑐𝑐2+ are used instead of mere ion concentrations, for a more realistic representation. Activity coefficients are obtained from Davies equation (15), where 𝐴𝐴is the Debye-Hückel parameter, 𝑧𝑧𝑛𝑛is the electric charge of the ion and 𝐼𝐼is the ionic strength of the solution. The ionic strength is calculated from (16), where 𝑐𝑐𝑛𝑛is the molar concentration of the ion speciesn. log10𝛾𝛾𝑛𝑛=−𝐴𝐴𝑧𝑧𝑛𝑛 2�√𝐼𝐼 1 + √𝐼𝐼−0.2𝐼𝐼� (15) 𝐼𝐼=1 2�𝑐𝑐𝑛𝑛|𝑧𝑧𝑛𝑛| 2 𝑛𝑛 (16) Davies equation gives realistic values of activity coefficients for I<0.50. For greater ionic strength other method should be used (p.e. Pitzer equations). However, for the sake of simplicity, Davies Equation is used in the full range of ionic strength, even over 0.50. Additionally, electric charge balance (17) and calcium mass balance equations (18) need to be considered to obtain the equilibrium composition of the pore solution. �𝑐𝑐 𝑛𝑛 𝑧𝑧 𝑖𝑖 𝑖𝑖= 0 (17) 𝑐𝑐2++𝑐𝑐0−𝑐𝑐0 2+−𝑐𝑐0 0= 0 (18) For the second group of reactions, kinetics laws of order zero are assumed for silica dissolution (19), CSH formation (20) and CASH formation (21), where 𝑘𝑘1,𝑘𝑘4,𝑘𝑘5are kinetic constants. In equation (19), a term has been added to limit the silica dissolution to its saturation product 𝐾𝐾1𝑠𝑠𝑠𝑠.In turn, if oversaturation is reached, this term allows silica precipitation. 𝜕𝜕𝑠𝑠− 𝜕𝜕𝜕𝜕 =𝑘𝑘1·𝑠𝑠0·𝑠𝑠−·�1−𝑠𝑠− 𝑜𝑜ℎ−·𝐾𝐾1𝑠𝑠𝑠𝑠� (19) 𝜕𝜕𝑐𝑐𝑠𝑠ℎ 𝜕𝜕𝜕𝜕 =𝑘𝑘4·𝑠𝑠−·(𝑐𝑐2+)𝛼𝛼 1 ·(𝑜𝑜ℎ−)𝛼𝛼 3 (20) 𝜕𝜕𝑐𝑐𝐶𝐶𝑠𝑠ℎ 𝜕𝜕𝜕𝜕 =𝑘𝑘5·𝑠𝑠−·(𝐶𝐶+)𝛽𝛽 2 ·(𝑜𝑜ℎ−)𝛽𝛽 3 (21) The amount of reaction products and the pore solution composition for agiven time interval is obtained by posing the mass balance equations for 𝑠𝑠0,𝑠𝑠−,𝑐𝑐2+,𝐶𝐶+and 𝑜𝑜ℎ−.The 483 Joaquín Liaudat, Carlos M. López and Ignacio Carol 6 notation used for the independent variables is also stated in Table 1. Table 1: Summary of independent calculation variables. Notation Units Description Notes 𝑠𝑠0,𝑆𝑆0 − Molar concentration of reactive silica SiO2 (a), (c) 𝑐𝑐0,𝐶𝐶0 − Molar concentration of Portlandite Ca(OH)2 (a), (c) 𝑠𝑠− − Molar concentration of silicate ions 𝑆𝑆𝑆𝑆𝑂𝑂4𝐻𝐻3− (b), (d) 𝑐𝑐2+ − Molar concentration of calcium ions 𝐶𝐶𝐶𝐶2+ (b), (d) 𝐶𝐶+ − Molar concentration of alkali ions 𝑁𝑁𝐶𝐶+and 𝐾𝐾+ (b), (d) 𝑜𝑜ℎ− − Molar concentration of hydroxyl ions 𝑂𝑂𝐻𝐻− (b), (d) ℎ+ − Molar concentration of hydroxyl ions 𝐻𝐻+ (b), (c) Notes: (a) Solid species, expressed in terms of pore solution volume (low case) and in terms of initial unitary volume of concrete (upper case). (b) Species in solution, expressed in terms of pore solution volume. (c) Local variable (d) Field variable. 3SUMMARY OF EQUATIONS AND NUMERICAL IMPLEMENTATION In sum,the overall system of unknowns and equations to be solved in the ASR model proposed is the following: -Three equations of diffusion/reaction (6) for c2+,𝐶𝐶+and s−. They are the only field variables of the problem. -For each point in space,the sink/well terms of Eqns. (6) and local concentration variables (𝑜𝑜ℎ−,ℎ+,𝑐𝑐0,𝑠𝑠0,𝑐𝑐𝑠𝑠ℎ,𝑐𝑐𝐶𝐶𝑠𝑠ℎ)are calculated by means of: oFour Chemical Equilibrium equations Electric charge balance equation (17) Water dissociation equilibrium equation (13) Saturation product of Portlandite dissolution (14) Molar mass balance of portlandite dissolution (18) oThree Chemical Kinetic equations Silica dissolution (19) CSH formation (20) CASH Formation (21) Additionally, the overall diffusion coefficients𝐷𝐷𝑎𝑎(𝑥𝑥),𝐷𝐷𝑐𝑐(𝑥𝑥),𝐷𝐷𝑠𝑠(𝑥𝑥) and the pore solution volume 𝑉𝑉𝑝𝑝𝑠𝑠(𝑥𝑥)are related to the volume fractions 𝑢𝑢𝑘𝑘(𝑥𝑥)by equations (11, 12), as well as the volume fractions 𝑢𝑢𝑘𝑘(𝑥𝑥)are related to the local concentration variables 𝑐𝑐0,𝑠𝑠0,𝑐𝑐𝑠𝑠ℎand 𝑐𝑐𝐶𝐶𝑠𝑠ℎ by equations (8, 9,10). 484 Joaquín Liaudat, Carlos M. López and Ignacio Carol 7 The solution scheme implemented is as follow: - Time is discretized in ∆𝑡𝑡=𝑡𝑡𝑗𝑗+1−𝑡𝑡𝑗𝑗 -Space is discretized in segments of variable length Δ𝑥𝑥𝑖𝑖=𝑥𝑥𝑖𝑖+1−𝑥𝑥𝑖𝑖 -For each time step,the initial concentrations 𝐶𝐶𝑛𝑛(𝑥𝑥𝑖𝑖)(expressed in relation to the initial unit volume) and volume fractions 𝑢𝑢𝑘𝑘(𝑥𝑥𝑖𝑖)at each point 𝑥𝑥𝑖𝑖are known. -With the volume fractions 𝑢𝑢𝑘𝑘(𝑥𝑥𝑖𝑖), the diffusion coefficients 𝐷𝐷𝑛𝑛(𝑥𝑥𝑖𝑖)and pore solution volumes 𝑉𝑉𝑝𝑝𝑝𝑝(𝑥𝑥𝑖𝑖)are calculated by equations (11, 12). - Concentrations are expressed in terms of pore solution volume by means of 𝑐𝑐𝑛𝑛(𝑥𝑥𝑖𝑖)=𝐶𝐶𝑛𝑛(𝑥𝑥𝑖𝑖)𝑉𝑉𝑝𝑝𝑝𝑝(𝑥𝑥𝑖𝑖)⁄ - Time derivatives of Eqns. (6) are evaluated at 𝑡𝑡=𝑡𝑡𝑗𝑗by the explicit finite difference method. Sink/well terms of Eqns. (6) are obtained by combining the equilibrium equations (13, 14, 17, 18) and the kinetics equations (19, 20, 21). The equilibrium equations are established at 𝑡𝑡=𝑡𝑡𝑗𝑗+1, while the kinetics equations are discretized in ∆𝑡𝑡 by amid-point integration scheme -The resulting equation system for concentrations at time 𝑡𝑡=𝑡𝑡𝑗𝑗+1 is solved by means of Newton-Raphson method. - Concentrations at 𝑡𝑡=𝑡𝑡𝑗𝑗+1 are saved for the next time step. 41D SIMULATIONS In order to validate the reaction mechanism proposed, an ongoing experimental test similar to the one proposed by Schlangen and Çopuroglu [15] is simulated. The test has been set up to measure ASR expansions at the level of single interfase between HCP and aggregate. Cylindrical specimens of 33 mm of diameter and 64mm of height, with a borosilicate glass disc in the middle as reactive aggregate and HCP in the extremes (see Figure 1, left)are exposed to a 1M NaOH solution at 60ºC for a period of 14 days during which expansions are measured regularly. Figure 1:Experimental ASR expansion test at the level of a single aggregate-HCP interface: specimen after 14 days imersion in 1M NaOH solution at 60ºC (left image), and length change from experiment (circles) and model (line) (right diagram). 485 Joaquín Liaudat, Carlos M. López and Ignacio Carol 8 The geometry and discretization for the 1D numerical analysis are is shown in Figure 2. A total of 200 nodes were used with a constant separation of 5.0E-06m. The boundary and initial conditions as well as the parameters and coefficients used in the simulations are summarized in Tables 2, 3 and 4. A symmetric boundary condition is imposed on the right end. Figure 2:Model geometry and discretization (not represented at exact scale). Table 2:Constants and coefficients used in the simulation. 𝐾𝐾𝑤𝑤 9.43E-14 𝑘𝑘1 2.00E+02 𝑚𝑚𝑐𝑐 0.75 𝛼𝛼1 0.50 𝐾𝐾1𝑠𝑠𝑠𝑠 1.00E+05 𝑘𝑘4 2.00E+03 𝑚𝑚𝑐𝑐 1.25 𝛼𝛼2 0.00 𝐾𝐾3𝑠𝑠𝑠𝑠 4.60E-06 𝑘𝑘5 2.00E+00 𝑚𝑚𝑠𝑠 1.00 𝛽𝛽1 1.00 A 5.49E-01 𝛽𝛽2 0.00 Table 3: Intrinsic properties of the solid phases (SP). Unit SP 1 SP 2 SP 3 SP 4 SP 5 SP 6 SP 7-8 𝒅𝒅𝒌𝒌 m2/h 0.00E+00 1.00E-07 1.00E-12 1.00E-11 1.00E-09 1.00E-09 1.00E-11 𝝋𝝋𝒌𝒌 -- 1.00E-01 1.00E+00 0.00E+00 1.00E-03 1.00E-01 1.00E-01 1.00E-01 𝒗𝒗𝒌𝒌 10-3m3/mol N/A N/A 2.87E-02 3.32E-02 5.68E-02 8.56E-02 N/A Table 4:Initial conditions u 2 𝑺𝑺𝟎𝟎 𝑺𝑺− 𝑪𝑪𝟎𝟎 𝑪𝑪𝟐𝟐+ 𝑨𝑨+ Unit m3/m3 103mol/m3 103mol/m3 103mol/m3 103mol/m3 103mol/m3 HCP 0.18 1.00E+00 0.00E+00 3.3800E+00 1.93E-06 4.43E-01 GLASS 0.00 2.78E+01 0.00E+00 0.0000E+00 0.0000E+00 0.00E+00 Left boundary 0.18 0.00E+00 0.00E+00 3.3800E+00 1.93E-06 4.43E-01 0.40mm 0.20 mm 0 1 … … HCP GLASS 159 160 161 162 199 200 486 Joaquín Liaudat, Carlos M. López and Ignacio Carol 9 The overall expansion curve obtained with the model is plotted together with the experimental results in Figure 1. The main trend of experimental behavior is caught by the model. The volumetric solid phase distribution in the area near the HCP-glass interface, for different exposure times,is represented in Figure 3. There, the progressive glass and portlandite depletion near the interface becomes apparent, as well as the CASH precipitation, first filling the capillary pores and finally producing a volumetric expansion. Figure 3: Volumetric solid phase distribution in the interface zone of HCP-glass for different exposure times. 487