Limits and possibilities of thermodynamic modelling of autogenous self-healing of concrete

. Autogenous self-healing of water retaining concrete structures is included in Eurocode 1992-3 as a possibility to heal cracks up to a width of 200 µm without additional repair. In this self-healing scenario water flow through a crack should result in a progressive closure of the fracture, mainly due to CaCO 3 precipitation, when certain hydraulic gradients are met, the pH of the water is > 5.5 and the concentration of CO 2 in the water remains < 40 mg*L -1 . The material composition is not further restricted by the regulation. However, despite standardization, the healing effect seems to be random in practice, which requires further research, while experiments aimed at quantifying autogenous self-healing are expensive and time-consuming. Thermodynamic models could support in estimating the effect of different environments such as groundwater or seawater exposure on autogenous self-healing. Moreover, adjusting the water chemistry according to the conditions of different construction sites and changing the material design could easily be considered. In this study thermodynamic models of a hydrated CEM I 52.5 R paste that is exposed to either simulated groundwater or seawater are discussed concerning the influence on autogenous self-healing and compared to experimental and literature data.


Introduction
Cracks in water retaining concrete structures such as water silos, basement-or tunnel-walls impair the functionality of the structure and are a severe durability concern. However, according to Eurocode 1992-3 cracks up to a width of 200 µm can be healed through autogenous self-healing. This effect relies on the permeation of water through the crack, whereas physical, mechanical and most importantly chemical processes lead to a crack closure [1]. Regarding the chemical processes, it was shown by many others that cracks exposed to tap-or fresh-water mainly show the formation of calcite (CaCO3) [1][2][3]. When seawater exposure applies brucite (Mg(OH)2) precipitates on the crack walls, whereas at later stages aragonite (CaCO3) deposits on the previously formed brucite layer [2]. It was reported by several crystallographic studies that aragonite is thermodynamically stabilized over calcite in the presence of Mg 2+ [4,5].
Interestingly, it was found by some authors that the exposure condition also influences the healing efficiency [6,7]. Accordingly, cracks up to a width of 150 µm could be healed for CEM I mortar specimen submerged in tap water, whereas cracks up to 600 µm could be healed in seawater submersion. For seawater submersion the authors observed an approx. 50 µm thick brucite layer on the crack walls through environmental scanning electron microscopy (ESEM).
In this study a thermodynamic model of a hydrated CEM I 52.5 R paste that is either submerged in tap water * Corresponding author: lahmannd@hsu-hh.de or seawater is developed and the predicted mineralogical changes are discussed concerning the chemical processes of autogenous self-healing and the healing efficiency. At first, the hydration of a CEM I 52.5 R paste was modelled according to Parrot & Killoh [8] to determine the degree of hydration at the point of subjection to a self-healing environment. The self-healing process was then modelled by a stepwise addition of either tap water or seawater to the hydrated paste and performing incremental thermodynamic equilibrium calculations. All calculations were performed with PHREEQC [9,10] considering the CEMDATA18 [11] database.
The aim of this study is to outline the limits and possibilities of a rather simple and flexible thermodynamic modeling approach of autogenous selfhealing as a support in estimating the mineralogical healing causes and healing efficiency in tap water and seawater exposure.

Kinetic modelling of the cement hydration
Cement hydration can be assumed to take place via dissolution and precipitation reactions [12]. According to Parrot & Killoh [8], the hydration of each clinker phase can be modelled through three rate equations considering nucleation and crystal growth, diffusion and the formation of a hydrate hem around the cement particles. The lowest rate is considered as the hydration rate controlling MATEC Web of Conferences 378, 09003 (2023) https://doi.org/10.1051/matecconf/202337809003 SMARTINCS '23 process. Furthermore, the hydration rate is corrected for the Blaine surface area of the cement, for curing temperature (293.15 K) with an Arrhenius term and the w/c-ratio of the mix. The relative humidity of the curing environment was assumed to be 100%. Therefore, correction for the relative humidity can be neglected. The equations are not issued in this study but can be found amongst others in Holmes et al. [9]. The kinetic parameters according to Lothenbach et al. [12] were chosen as input parameters for the rate equations.
The rate equations were implemented into PHREEQC using the RATES keyword to account for the kinetic reaction of the clinker phases. All other phases are considered as EQUILLIBRIUM_PHASES. The mineralogical composition of the used CEM I 52.5 R was quantified by an XRD/Rietveld analysis (Table 1) and transformed into molar concentrations per 100 g cement. The moles of each clinker phases are used for the corresponding initial molality m0 in the KINETICS data block. The concentration of the other phases is put into the EQUILIBRIUM_PHASES data block. All phases except for thaumasite, quartz, hematite, magnetite and dolomite were considered in the equilibrium calculations, whereas siliceous hydrogarnet was considered to be the main stable Fe-sink in the hydrated paste as experimentally proven by Dilnesa et al. [13]. The uptake of Na2O and K2O into the clinker phases was considered on the basis of the chemical (XRF) and mineralogical analysis (XRD/Rietveld) of the cement as described elsewhere [13,14]. The formation of C-S-H was modelled with an ideal solid solution of the CSHQ-model and implemented into the SOLID_SOLUTION data block [11]. The uptake of alkali into C-S-H was considered with the ECSH1-KSH and ESCH1-NaSH model that was added to the CSHQ solid solution [11]. The SOLUTIONS data block was then used to add 55 g of water (293.15 K, pH of 7.5) to 100 g of cement to account for a w/c-ratio of 0.55. The hydration was then modelled up to 1000 days with the CVODE numerical solver for ordinary differential equations and INCREMENTAL_REACTIONS set true. CVODE is a robust implicit solver for multiple rate equations in which the rates of the kinetic reactants vary strongly [15]. CEMDATA18 [11] database was used for the thermodynamic calculations.

Thermodynamic modelling of the pastehealing environment interaction
Self-healing studies are often carried out on cracked mortar specimens that are submerged in a water container [3]. However, one must be aware that this test design is a simplification of the self-healing environment present in water retaining concrete structures which would require large concrete specimens and continuous water permeation through the crack [2]. Nevertheless, tap water submersion experiments have proven to show similar mineralogical sequences than tap water permeation experiments [1,3]. Therefore, in terms of predicting the thermodynamically controlled mineralogical changes as a result of autogenous selfhealing, it is assumed that the submersion test design can be used as a simplification of self-healing environments without the need to consider water flow.
The hydration of the CEM I paste was modelled according to section 2.1. The degree of hydration after 28 days was then determined as the mass fraction of remaining unhydrated clinker to the initial mass of the clinker phases. It is assumed that the hydration degree remains constant after 28 days. The moles of the clinker phases that hydrate within 28 days and the complete moles of the other phases in CEM I 52.5 R were implemented into PHREEQC with the EQUILIBRIM_PHASES data block. The same phases and C-S-H model as described earlier were considered in the equilibrium calculations. The hydration was then modelled by adding 55 g of tap water (293.15 K, pH of 7.5) ( Table 2) with the SOLUTIONS keyword to the cement to obtain a w/c ratio of 0.55 of the modeled paste. This approach gives the same phase assemblage as the kinetic approach (section 2.1) after 28 days of curing and is used as the starting point of modelling the interaction of the cement paste with the healing environment. A provided chemical analysis of the local laboratory tap water [16] and average seawater [17] was chosen as the chemical input for the healing environment. To model a successive penetration of the cement paste through the healing environment, increments to the hydrate assemblage with the REACTION data block and allowed to reach thermodynamic equilibrium. For more details reference is set to Loser et al. [18] and De Weerdt et al. [19] who chose a similar approach to model phase changes due to Clingress.

Self-healing experiment with mortar specimens
Mortar bars (40 x 40 x 160 mm³) with 450 g cement, w/cratio of 0.50 and 1350 g sand were prepared according to DIN EN 196-1. Metal plates of 500 µm thickness were cast into the mortar reaching 2 cm into the specimens. The plates were removed after 2 days of curing producing "ideal cracks" of rectangular geometry and approx. 500 µm in width. Ideal cracks have been assumed to be beneficial for reliable surface crack width measurements as a measure of self-healing. Moreover, it was considered advantageous that the cracks do not have to be initiated, for instance, through 3 or 4-point bending tests and the samples do not have to be reconnected with special retainers that allow adjustment of the crack width. A total of 9 mortar specimens (3 CEM I 52.5 R, 3 CEM II/A-LL 52.5 R and 3 CEM III/A 52.5 R) with ideal cracks were then put in separate containers and submerged in 2 L of tap water. 3 water samples of each container were taken after 0, 1, 3, 7, 11, 21 and 28 days and were diluted with 1 wt.% HNO3 in the ratio of 1 to 100. Moreover, the pHvalue of the water was recorded at the same time intervals and the "crack" was visually inspected for self-healing products. In order to determine the Ca 2+ and Mg 2+ concentration in the tap water samples, the 44 Ca and 24 Mgisotopes were measured with an Agilent 7800 ICP-MS. The ICP-MS was calibrated with CaCl2 and MgCl2 standards diluted in 1 wt.% HNO3. This approach aims to provide additional chemical information on the dissolving of phases and precipitation reactions during the experiments as a result of mortar-water interaction.

Degree of hydration
The modelled degree of hydration of a CEM I 52.5 R paste with a w/c-ratio of 0.55 after 28 days of curing is 75%. Assuming a paste-water interaction of further 28 days the hydration degree would rise to 80%. Patel et al. [20] found slightly higher degrees of hydration with 75% after 14 days and 90% after 90 days of curing at 100% relative humidity through XRD/Rietveld and TG analysis. Therefore, neglecting the influence of further clinker hydration in the thermodynamic modelling of the pastewater interaction is considered as an acceptable simplification. However, it is desirable to validate the modelled hydration degree with experimental data.

Thermodynamic modelling of the pastehealing environment interaction
Modelling the tap water penetration of a cured (28 days) CEM I 52.5 R paste leads to several mineralogical changes of the original hydrate assemblage (Fig. 1). One can see that at first portlandite (CH) dissolves by the addition of up to approx. 10 L of tap water. This reaction leads to a release of Ca 2+ and OHions and the precipitation of calcite (CaCO3) is predicted. At tap water additions greater than 10 L, C-S-H starts to dissolve, whereas further calcite precipitates and the predicted amount of ettringite (Et) and mono-carboaluminate (Mc) starts to increase up to the addition of approx. 40 L of water. This is due to further carbonate-(HCO3 -, CO3 2-) and sulfate-ions (SO4 2-) that are brought into the modeled system and also lead to a slight increase of the total volume of the paste. Mg 2+ is compensated in hydrotalcite (Ht) that increases over the whole range of tap water addition and is predicted to precipitate parallel with calcite and portlandite dissolution. The predicted mineralogical changes for tap water additions greater than 10 L are considered unrealistic since calcite precipitates would form a layer on the crack walls, which would act as a barrier for further interaction of the mortar with the tap water [3]. Moreover, no self-healing study reported the formation of phases such as straetlingite and natrolite. Furthermore, one must be aware that with the chosen thermodynamic approach it is assumed that all ions in the tap water and paste interact simultaneously and instantaneously reach a thermodynamic equilibrium. Time and space dependent reactions and processes such as diffusion through layered reaction products are not considered [18,19]. However, the expected formation of calcite could be predicted by the model. Further experimental studies should aim to determine a Mg 2+ sink in self-healing products formed in tap water exposure. Modelling the seawater penetration of a cured (28 days) CEM I 52.5 R paste leads at first, similar to the tap water model, to a dissolution of portlandite (Fig. 2). However, portlandite is completely consumed by lower seawater addition levels compared to tap water. Parallel to the portlandite dissolution a strong increase of ettringite is predicted due to the high amount of sulfate that is brought into the system (Table 2). Moreover, monocarboaluminate is destabilized and transformed to ettringite through this process.

Fig. 2. Predicted mineralogical changes for a CEM I 52.5 R paste penetrated by seawater.
Overall, the ettringite formation up to an addition of approx. 10 L of seawater leads to a strong increase of the total volume of the paste and therefore could explain the higher healing-efficiency of CEM I in seawater submersion compared to tap water submersion that was observed by Palin et al. [7]. However, Maes [21] found no experimental prove that high sulfate concentrations would increase the self-healing efficiency due to ettringite formation. The latter author states that the formation of ettringite is too slow compared to CaCO3 precipitation to improve the healing efficiency. Nevertheless, Maes could prove the formation of ettringite, gypsum Friedel´s salt (Ca4Fe2Cl2(OH)12(H2O)4), sodium and magnesium chloride in the cracks of his specimens in terms of XRD and SEM analysis. In contrast to this literature data Kuzel´s salt (Ca4Al2Cl(SO4)0.5(OH)12(H2O)6) is predicted as a chloride binding phase in the thermodynamic model. Sodium and magnesium chloride as well as gypsum were not predicted by the model.
A further difference of seawater and tap water healing environments is the much higher concentration of Mg 2+ in seawater ( Table 2). In the literature [6,7], this is attributed to lead to the formation of a brucite layer directly on the crack's walls, whereas brucite is overlayed by an aragonite layer as soon as portlandite is completely consumed. Interestingly, the thermodynamic model predicts the formation of calcite parallel to the dissolution of portlandite. This is probably due to the reason that the uptake of Mg 2+ into calcite is not considered in the model. Additionally, hydrotalcite is predicted as a Mg-sink, whereas brucite is only stable over a small range of seawater addition from approx. 3 L to 10 L. Thus, with this model the predicted mineralogical sequence would be a mixture of calcite, hydrotalcite, ettringite and Kuzel´s salt that is overlayed by brucite. This difference suggests a kinetic influence in the formation of brucite that is not considered in the thermodynamic model. However, it can be seen from the model that the formation of Mgcontaining phases contributes to the volume increase of the cement paste and could therefore be the reason for the experimentally observed higher self-healing efficiency of CEM I in a seawater environment [6,7]. The predicted mineralogical changes for seawater addition > 10 L are not further discussed due to the afore mentioned limits of the modelling approach.

Experimental data
Changes in the pH-value of the water in which mortar specimen have been immersed provide information about dissolution and precipitation reactions that consume or release hydroxide ions. An increase of the pH-value of the tap water in the mortar experiments confirmed the predicted dissolution of portlandite (Fig. 3). The pH-value decreased to a local minimum after 3 days of submersion, followed by an asymptotic increase to a maximum of approx. 12 for all submersion experiments. The decrease of the pH-value indicates the removal of hydroxide ions from the solution. This could be explained by the reaction of Ca 2+ with atmospheric CO2 at the interface of water and atmosphere according to equation (1).
The formation of dendritic crystals at the wateratmosphere interface was observed by visual inspection. However, a determination of these crystals by XRD would be desirable to support the suggested process but was out of the scope of this study. An alternative explanation for the kink in the pH curve could be the uptake of hydroxide ions into minerals such as hydrotalcite (Mg6Al2(OH)16CO3 4H2O) or brucite (Mg(OH)2) that were both predicted by thermodynamic modelling to form parallel to the portlandite dissolution and calcite precipitation reactions. However, ICP-MS measurements show that within 1 day of submersion Mg 2+ is almost completely consumed from the tap water (Fig. 4). Thus, neither hydrotalcite nor brucite precipitation could be responsible for the decrease of the pH-value at 3 days.
However, the previously discussed kinetic influence on the brucite formation in seawater submersion selfhealing experiments is supported by the ICP-MS measurement of Mg 2+ , since even in the tap water submersion experiments of this study, Mg 2+ is rapidly removed from the solution. For future studies it is desirable to determine the mineralogical sink of Mg 2+ in tap water submersion experiments for instance by electron microprobe measurements of the self-healing products in the specimens. Furthermore, in contrast to the theoretical prediction, the ICP-MS measurements could not prove an increase of the Ca 2+ ions in the tap water that was expected due to the dissolution of portlandite (Fig. 5). Like the pH-value, the Ca 2+ concentration decreases to a minimum of almost 0 ppm after 3 days of submersion. One can conclude that in the beginning, the portlandite dissolution that is taking place at the interface of mortar to water, releases hydroxide ions into the solution, whereas the released Ca 2+ ions must be bound immediately. Literature data [1,3] and thermodynamic modelling suggest that Ca 2+ is bound into calcite according to equation (2).
Moreover, the experimental data suggests that until 3 days of submersion the calcite precipitation rate must be higher than the portlandite dissolution rate to explain the depletion of Ca 2+ ions in the solutions. This is consistent with literature data reporting the highest self-healing rate through CaCO3 precipitation within the first days of submersion [22]. Calcite nuclei formed at the mortarwater or water-atmosphere interface must then act as crystallization sites and attract Ca 2+ ions out of the solution to form CaCO3 until the solution is almost completely depleted of Ca 2+ .
After 3 days of submersion the Ca 2+ concentration in the tap water increases again, whereas the observed curves show large scatter and seem to depend on the cement type. However, the high standard deviation (error bars) of the measurements of samples taken after 3 days must be contributed to artefacts probably due to the uptake of small floating CaCO3 crystallites during sampling. Therefore, in future studies samples must be filtered before analysis. The visual inspection of the "ideal cracks" in the CEM I and CEM II specimens showed no noticeable formation of self-healing products up to 28 days of submersion. Only in the cracks of CEM III specimens' little fringe like crystallites could be observed. These fringes have formed within 1 day of submersion, whereas no further growth could be observed up to 28 days of submersion (Fig. 6). This is probably caused by the slower hydration process of CEM III compared to CEM I and CEM II. However, the fringes need further mineralogical characterization. The lack of self-healing products in the "ideal cracks" of CEM I and CEM III could be caused by the relatively smooth surface that was produced by the cast-in metal plate. The micro-roughness of a realistic crack surface would produce numerous surfaces, edges and corners that provide crystallization sites and cause a reduction of the surface energy of forming calcite nuclei [23]. Therefore, crystallization in real cracks is thermodynamically favorable compared to the "ideal cracks" that were produced by the cast-in metal plates. Moreover, microstructural changes induced through a realistic cracking process such as the exposure of the yet unhydrated interior of cement grains are not possible with the experimental approach that was chosen in this study. For future studies it is therefore absolutely necessary to induce real cracks into specimens that should heal through autogenous self-healing.

Conclusion
The potential of thermodynamic models to predict mineralogical changes of a hydrated CEM I 52.5 R paste due to tap water and seawater penetration were discussed within this study regarding the chemical process of autogenous self-healing. The results showed that dissolution processes and most precipitation products can be correctly predicted up to a water addition of approx. 10 L to the modeled cement paste. At water addition greater than 10 L, the instantaneous and global equilibrium reactions of the modeled cement with the water predict unrealistic phase changes. Moreover, literature and experimental data made clear that kinetic considerations are crucial in terms of predicting the correct mineralogical sequences of precipitation products such as the layered formation of brucite and aragonite in sea water environments. Accordingly, the modeled secondary formation of ettringite is irrelevant for autogenous self-healing due to slow reaction kinetics. Therefore, it is desirable to extend future models with reaction kinetics and diffusion processes to predict timeand space-dependent mineralogical changes.
However, in terms of a qualitative assessment of the healing efficiency the models clearly predicted a higher increase of the volume of the original cement paste in seawater compared to tap water environment. This is probably due to the high magnesium concentration in seawater and consistent with a higher self-healing efficiency that was reported in the literature. Interestingly, experimental data showed that also in tap water environments the precipitation of Mg-containing phases is kinetically faster than CaCO3 precipitation. This needs further mineralogical consideration and should be examined in future studies regarding the influence on the self-healing efficiency.