1 Introduction

In the context of concrete and mortar, carbonation describes the process of atmospheric CO\(_2\) reacting with hydration products such as calcium hydroxide (Ca(OH)\(_2\)) to form carbonates such as calcium carbonate (CaCO\(_3\)). Carbonation is a reactive transport process [1], meaning that the physiochemical process of carbonation is considered to have two main phases: the diffusion of CO\(_2\) through the concrete pore structure, and the chemical reaction within pores in the presence of pore water [2]. The diffused CO\(_2\) dissolves in pore water to form carbonic acid, which then reacts with calcium compounds [3]. The assumption is made that the rate of the carbonation reaction is much faster than the diffusion of CO\(_2\) through the concrete [4]. As such, the overall carbonation rate is diffusion controlled and can be described using diffusion equations derived from Fick’s first law [5].

Carbonation can alter the pore structure of concretes and mortars by affecting both the pore size and volume of pores [6]. Due to precipitation of carbonates on pore walls, carbonation typically reduces the total porosity of material in Ordinary Portland Cement (OPC) concretes [7], though the opposite effect has also been observed for blended cements [6]. The porosity of the carbonated cementitious layer impedes the diffusion of CO\(_2\) to the carbonation reaction front [8], and therefore, influences the carbonation rate. The location of the carbonation front, called the carbonation depth, varies approximately linearly with the square root of the time of CO\(_2\) exposure. It follows that any carbonation which occurs in the initial period of exposure of the specimen has the greatest carbonation rate, and so capturing the behaviour at this stage is of significance.

Carbonation leads to deterioration of reinforced concrete structures because the reaction reduces the alkalinity (pH) of the concrete [9], leading to depassivation of the steel reinforcement. Delayed onset or altogether prevention of carbonation-induced corrosion is a key factor when specifying performance requirements of concretes [10].

Carbonation behaviour is measured through destructive testing, in which a specimen of concrete is split open to reveal an internal surface. The depth of carbonation perpendicular to this surface is revealed by phenolphthalein indication which is pink in non-carbonated material and colourless in carbonated material, as shown in Fig. 1. These tests are conducted in a variety of laboratory controlled and natural exposure conditions, and can be accelerated by increasing the concentration of CO\(_2\). Many standards for accelerated test conditions exist, including [11,12,13,14], conditions of which are summarised in [15]. Correctly interpreting results of tests such as these is of importance to performance-based specification of concrete mixes as they enable quick prediction of carbonation behaviour in real-world conditions to be generated [13, 14], which can then be used for probabilistic design against failure.

Fig. 1
figure 1

Carbonation depth indicated on a cube of concrete exposed to a 4% CO\(_2\) accelerated test environment. Pink material is non-carbonated, grey material is carbonated

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It is not unusual for some initial carbonation to have occurred in a concrete before it undergoes measured carbonation testing, due to prolonged exposure to ambient air during the curing or a preconditioning period. In this paper, two alternative approaches to modelling carbonation in concretes with non-zero initial carbonation depth are presented. The approaches are compared to data collated from literature sources to assess how well they fit experimental results. The impact of the approach adopted and the initial carbonation depth on the estimate of carbonation coefficient is then assessed and discussed.

2 Modelling carbonation

2.1 Diffusion model

The following derivation is adapted from Kropp and Hilsdorf [5] (also in [16]). The carbonation process is modelled by Fick’s first law of diffusion, Eq. 1. The flux, F, of the CO\(_2\) through the concrete (in mol/(mm\(^{2}\cdot\)day)) is given as:

$$\begin{aligned} F=-D \frac{\partial C}{\partial x}=D \frac{c}{x} \end{aligned}$$
(1)

where x is the depth from the external surface (mm), D is the diffusion coefficient of CO\(_2\) through concrete (mm\(^2\)/day), C is the concentration of CO\(_2\) (mol/mm\(^3\)), which is c at the free surface of the concrete (concentration at the carbonation front = 0 mol/mm\(^3\)).

Defining a as the amount of CO\(_2\) required to carbonate a unit volume of the material (mol/mm\(^3\)), then the amount of CO\(_2\) required to carbonate an incremental volume, \(\mathrm{{d}}V\) (mm\(^3\)), is \(a\mathrm{{d}}V\) (mol). Assuming the incremental volume has area A (mm\(^2\)) such that \(\mathrm{{d}}V=A\mathrm{{d}}x\), this is equivalent to the amount of CO\(_2\) delivered by diffusion flux to the area in a given time increment, \(\mathrm{{d}}t\) (days), as in Eq. 2:

$$\begin{aligned} FA \mathrm{{d}}t=a A \mathrm{{d}}x \end{aligned}$$
(2)

combining Eqs. 1 and 2 leads to:

$$\begin{aligned} D\frac{c}{x} \, \mathrm{{d}}t=a \, \mathrm{{d}}x \end{aligned}$$
(3)

which may be integrated up to the carbonation depth at time t, \(x_t\) (mm), to give:

$$\begin{aligned} \int _{0}^t \mathrm{{d}}t&= \frac{a}{Dc} \int _{0}^{x_t} x \, \mathrm{{d}}x \end{aligned}$$
(4)
$$\begin{aligned} t&=\frac{a}{2Dc} x_t^2 \end{aligned}$$
(5)
$$\begin{aligned} x_t&=\sqrt{\frac{2Dc}{a}t} \end{aligned}$$
(6)

This is the familiar carbonation equation, in which the carbonation depth is proportional to the square root of time, which is often expressed as:

$$\begin{aligned} x_t=K\sqrt{t} \end{aligned}$$
(7)

where

$$\begin{aligned} K=\sqrt{\frac{2Dc}{a}} \end{aligned}$$
(8)

The constant K, known as the carbonation coefficient, captures the influence of many different material properties and environmental conditions, including but not limited to: CO\(_2\) concentration, relative humidity, temperature, curing, water/cement ratio, and clinker content. K is therefore unique to each concrete and its test conditions. Many analytical models for carbonation propose methods of estimating K. These are explored elsewhere by the authors [17].

2.2 Initial carbonation depth

2.2.1 Linear approach

To account for any initial carbonation depth in specimens prior to accelerated carbonation testing, a linear shift of the initial depth, \(x_0\), is often applied to Eq. 7, as in the European standard EN 12390-12:2020 [12] and the Swiss standard SIA 262/1 [14]. This results in the linear Eq. 9:

$$\begin{aligned} x_t=x_0+K\sqrt{t} \end{aligned}$$
(9)

where \(x_0\) represents the initial carbonation depth prior to accelerated carbonation exposure and t is the duration of exposure to accelerated carbonation conditions.

Equation 9 may be rearranged for an estimate of K where:

$$\begin{aligned} K=\frac{x_t-x_0}{\sqrt{t}} \end{aligned}$$
(10)

Equation 10 is herein referred to as the linear approach for estimating K from experimental carbonation depth data.

2.2.2 Non-linear approach

Alternatively, following the derivation of Eq. 7 for non-zero initial conditions results in a different, non-linear, equation for carbonation depth, as previously shown by Moreno [16, 18]:

$$\begin{aligned} \int _{0}^t \mathrm{{d}}t&= \frac{a}{Dc} \int _{x_0}^{x_t} x \, \mathrm{{d}}x \end{aligned}$$
(11)
$$\begin{aligned} t&=\frac{a}{2Dc} (x_t^2-x_0^2) \end{aligned}$$
(12)
$$\begin{aligned} x_t&=\sqrt{x_0^2+\frac{2Dc}{a}t} = \sqrt{x_0^2+K^2t} \end{aligned}$$
(13)

Equation 13 may be rearranged for K to give:

$$\begin{aligned} K=\sqrt{\frac{x_t^2-x_0^2}{t}} \end{aligned}$$
(14)

Equation 14 is herein referred to as the non-linear approach for estimating K from experimental carbonation depth data.

The non-linear approach can also be derived by considering the simple case of a shift in the start time of measurement for a carbonation test. The carbonation profile of a test under constant conditions, started at \(t_1=0\) follows the expected relationship of Eq. 7, shown in Fig. 2a.

Fig. 2
figure 2

Carbonation profile for Fickian diffusion relative to a \(t_1\) and b \(t_2\)

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Considering the case where there is a delay in measuring the carbonation (but no delay in exposure) such that there is a shift in the recorded time axis: \(\Delta {t}=t_1-t_2\). The carbonation depth at \(t_2=0\) is equal to \(x_0\), shown in Fig. 2b. \(x_0\) is therefore defined:

$$\begin{aligned} x_0=K\sqrt{\Delta {t}} \end{aligned}$$
(15)

The behaviour in the new time axis, \(t_2\), in Fig. 2b is described by applying a time-shift to Eq. 7:

$$\begin{aligned} x_{t_2}=K\sqrt{t_2+\Delta {t}} \end{aligned}$$
(16)

substituting for \(\Delta {t}\) results in Eq. 17, which is equivalent to Eq. 13:

$$\begin{aligned} x_{t_2}=K\sqrt{t_2+\frac{x_0^2}{K^2}}=\sqrt{x_0^2+K^2t_2} \end{aligned}$$
(17)

2.2.3 Effect of the discrepancy

The estimation of K obtained from experimental data by fitting the linear Eq. 10 will give a different result to fitting the non-linear Eq. 14. The relative difference in estimate of K from Eq. 10 compared to Eq. 14 is defined:

$$\begin{aligned} e_K=\frac{K_{linear}-K_{non-linear}}{K_{non-linear}}=\frac{x_t-x_0}{\sqrt{x_t^2-x_0^2}}-1 \end{aligned}$$
(18)

and is plotted as a function of the ratio \(\frac{x_0}{x}\), shown in Fig. 3.

Fig. 3
figure 3

Theoretical error in estimate of K from linear approach (Eq. 10) relative to K from non-linear approach (Eq. 14)

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Figure 3 suggests that use of the linear Eq. 10 results in an underestimate of the carbonation coefficient of the material compared to the non-linear Eq. 14 estimate. The error is insignificant when \(\frac{x_0}{x_t}<<1\), i.e. for small \(x_0\) and large final carbonation depth.

However, when \(x_0\) is significant compared to the final carbonation depth, the choice of either model can result in a significant difference in the calculated carbonation coefficient K. This highlights the importance of correctly accounting for the initial carbonation depth to prevent underestimating carbonation coefficient and therefore overestimating the carbonation resistance of a concrete from an accelerated test.

2.3 Effect of CO\(_2\) concentration

A higher environmental concentration of CO\(_2\) results in a faster rate of carbonation, since the diffusion process is faster in the presence of a greater concentration gradient between the atmosphere surrounding the concrete and the carbonation front within the material. A sudden change in concentration occurs for specimens which have undergone some natural carbonation prior to accelerated carbonation testing. The initial natural exposure results in the initial carbonation depth \(x_0\).

By Eq. 8, the carbonation coefficient, K, for any given test is theoretically proportional to the square root of the CO\(_2\) concentration, c. Therefore, the carbonation coefficient for the natural carbonation, \(K_\mathrm{{NC}}\), is lower than the coefficient for the accelerated carbonation exposure, \(K_\mathrm{{AC}}\).

Assuming that the characteristics of carbonated material formed at both concentrations are similar, the non-linear approach for including initial carbonation depth presented in Sect. 2.2 may be extended to the case where there is a change in concentration, and therefore a change in carbonation coefficient, at the start of an accelerated carbonation test. This is plotted in Fig. 4.

Fig. 4
figure 4

Carbonation profile over whole life of concrete exposed to natural carbonation followed by accelerated carbonation

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3 Validation

3.1 Method

Experimental data of carbonation depth development over time for a total of 361 data series with an initial carbonation depth greater than zero is collated for validation of the two approaches. The data is both the authors’ own and collected from literature sources. Specimens include both mortars and concretes. The key features of the data used are summarised in table 1. AC refers to accelerated carbonation testing, NC denotes natural carbonation testing.

Table 1 Summary of experimental validation dataset
Full size table

At high concentrations it is believed that the nature of carbonation reactions may be altered, preventing the application of a Fickian diffusion equation to results. For this reason, it is common for AC test standards to recommend a concentration of 4% or below. However, results obtained at concentrations of 10%, 20% and even 100% are included in the dataset to increase the total number of data series and to assess whether the same model discrepancy is observed across an exhaustive range of concentrations.

In the case of the data from Leemann and Moro [21], the values of t were given as approximately 1, 2 and 5 years for each measurement for the specimens, which were subject to sheltered and unsheltered outdoor exposure. Since higher accuracy was not available, these were used as the t values for the curve fitting. The calculated values of K gave reasonably good agreement with the authors values computed with true t values using this assumption, so this was considered a valid approximation.

Vanoutrive et al. [15] observed a statistically significant difference between carbonation of different faces depending on casting orientation. This was also observed in [19], when comparing trowelled (top) and non-trowelled (bottom) surfaces. Therefore, where possible, results from different faces are kept separate when curve fitting, as their values of K are expected to be different. Results for different curing conditions, curing duration, and preconditioning environment are also kept distinct, as these have been shown to affect the value of K [22,23,24]. Each curve fitting is referred to as a separate data series in further analysis. In all cases, the values used in curve fitting are the measured mean carbonation depths from each surface exposed to CO\(_2\), revealed using an indicator solution—typically phenolphthalein.

The validity of the linear and non-linear approaches is assessed by curve fitting Eqs. 9 and 13 to the carbonation front data for each sample. The curve fit is conducted using the curve fitting toolbox in Matlab, using linear (in the case of Eq. 9) and non-linear (in the case of Eq. 13) least squares fitting methods for the relationship between x and \(\surd t\) for each data series. The initial carbonation depth, \(x_0\), is fixed as the mean value measured at the start of the test as per recommendations in EN12390-12-2020 [12], and the regression line is forced through \(x_0\). Figure 5 shows an example of the curve fitting to generate the carbonation coefficients using either the linear or non-linear approach for a given sample (this data series relates to concrete mix 2-C in [19] exposed to 4% CO\(_2\)). The \(R^2\) statistic indicates how well the respective version of the diffusion model fits the observations.

Fig. 5
figure 5

Curve fitting of linear (red) and non-linear (blue) fickian diffusion equations to exemplar data from concrete mix 2-C in [19]

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The full carbonation coefficient results for the dataset introduced in Table 1 are given in [25].

Fig. 6
figure 6

Comparison of goodness of fit statistics for two versions of the model. R\(^2\) values below 0.5 (6 data points) and RMSE above 4 mm (2 data points) not shown

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Figure 6 shows the \(R^2\) statistics and Root Mean Square Error (RMSE) for all of the data compared. By either model, the \(R^2\) value is generally very high, and the RMSE low, indicating that both models may be considered a good fit to carbonation depth data over a wide variety of curing and preconditioning conditions, carbonation concentrations, exposure types, and different concrete or mortar mixes. The median \(R^2\) of the linear and non-linear approaches are found to be 0.965 and 0.959 respectively. The difference between the distributions of \(R^2\) from fitting either approach is not statistically significant. There is a strong correlation between the \(R^2\) values from each model, indicating that where the quality of the original data is high, both models fit the results well. When the quality is low, both models give a lower \(R^2\) statistic and higher RMSE.

The European Standard EN 12390-12 [12], which uses the linear model approach to account for initial carbonation depth, states that an \(R^2\ge 0.95\) should be expected, otherwise reasons for a poorer fit should be investigated. The same standard dictates that measurements with a deviation of under 4 mm may be used in calculation of the mean carbonation depth, therefore, a RMSE of up to 4 mm may be expected.

There are cases where either model performs more strongly than the other, with respect to the R\(^2\) statistic. These appear to be randomly distributed, indicating that neither model is exclusively a better fit to the data than the other. There is a slightly higher RMSE produced by the non-linear model, which is still within a reasonable range for variability of the carbonation depth data.

3.2 Results

The values of carbonation coefficients, K, calculated by either model are compared in Fig. 7. The estimate of K from the non-linear approach is always equal to or higher than the estimate of K from the linear approach. This confirms the hypothesis in Sect. 2.2. For the data assessed here, the estimates of K from the non-linear approach are, on median, 10% greater than the estimates of K using the linear approach. This is a small but non-negligible discrepancy.

Fig. 7
figure 7

Comparison of values of K calculated using both linear and non-linear approaches (3 data points with \(K>5\) not shown)

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Figure 8 plots the relative error between the estimates of K calculated using the non-linear and linear approaches, where \({x_{fin}}\) is the carbonation coefficient calculated by the non-linear approach model for the end of the test, and \(x_0\) is the initial carbonation depth from the data. These values are included in the full results table [25]. The results closely follow the relationship determined in Eq. 18, that the discrepancy between the two estimates of K are a function of the ratio of initial and final carbonation depths. The greater the ratio \(x_0/x_{fin}\), the greater the difference between estimates of K. Despite being very close, the results do not always perfectly fit the theoretical relationship because neither model perfectly captures all variability in the original data (\(R^2<1\)).

Fig. 8
figure 8

Numerical error in K from using linear approach of Fickian diffusion model with initial carbonation depth, relative to non-linear approach

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Figure 9 demonstrates the distribution of the relative difference in the estimate of K calculated with either a linear or non-linear approach for the initial carbonation depth for various concentrations of exposure. The results show that the highest discrepancy is observed in the experimental tests conducted at natural concentrations (0.032–0.04%), where the carbonation rate is low (less than 0.5 mm/\(\surd\)day). In these cases, the ratio of initial carbonation depth to final measured depth is largest, which generally corresponds to specimens with a low maximum measured carbonation depth. This suggests that the effect of initial carbonation depth is most significant in the natural exposure carbonation tests assessed here, but the effect could be reduced by increasing the overall duration of such tests and therefore the maximum measured depth.

Fig. 9
figure 9

Relationship between error from linear approach and K estimated using the non-linear approach, for various concentrations indicated by colour

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4 Discussion

4.1 Modelling carbonation

The results suggest that both linear and non-linear approaches to including initial carbonation depth into the Fickian diffusion equation for carbonation of concrete are a good fit to experimental data. Using either approach results in different predictions of carbonation coefficient, demonstrated in Fig. 7. The linear version of the model gives a consistently lower prediction of carbonation coefficient in the presence of non-zero initial carbonation depth, which shows that it is a non-conservative assumption. Both versions of the model are identical in the case where carbonation depth at the start of measurement is zero. Overall, the results suggest that it is important not only to measure the carbonation depth at the start of a carbonation test, but also to understand how this is being considered in the subsequent analysis.

In this work, \(x_0\) is fixed as the value measured at the start of the test. This aligns with the recommendations of EN12390-12-2020 [12] and maintains close relation to the physical meaning of \(x_0\) as the initial carbonation depth. Nonetheless, it is possible that there is some uncertainty in measurement of this value which may affect the estimated K. However, alternatively allowing \(x_0\) to vary as part of the regression fitting gave similar results, and generated the same overall conclusions. Therefore, a fixed \(x_0\) approach was used, albeit with a caveat about uncertainty.

4.1.1 Benefits of linear approach

The linear approach as in Eq. 10 may be preferred as it is less computationally complex to carry out linear regression analysis than non-linear analysis. This is the approach which has been historically adopted by standards. For cases where the values of K produced by either model are very similar, the benefit of higher accuracy in K may be outweighed by the complexity of using the non-linear equation.

As demonstrated by Fig. 8, the difference in estimate of K from either approach is less than 10% when the initial carbonation depth is less than 10% of the final measured depth. In real terms, for a concrete with \(K_{\text {NC, non-linear}}=0.3\) mm/\(\surd\)day and \(K_{\text {NC, linear}}=0.27\) mm/\(\surd\)day, this equates to an additional 4 mm of carbonation across a 50 year design life predicted by the value obtained using the non-linear approach. This is low and is unlikely to be significant when compared to quality control tolerances of the cover thickness.

Non-uniformity of carbonation fronts leads to large uncertainty in measured carbonation depths, meaning that variability of the carbonation coefficient from experimental results can also be large, even if taken from the same specimens. The stability of the coefficient has also been found to depend on the duration of the test [15]. In these instances, the variability in K due to choice of linear or non-linear approach may be insignificant when compared to the expected variability in the value.

4.1.2 Benefits of non-linear approach

The non-linear approach demonstrated by Eq. 14 is more consistent with the accepted mathematical model for carbonation as a diffusion driven process, as demonstrated by integration in [16], which supports it’s adoption. It is seen in Fig. 7 that the calculated value of K is consistently larger than that estimated using the linear approach. This makes it a safer estimate, which will more accurately predict the future performance of specimens.

Perhaps the most useful feature of the non-linear approach is that no information is required about the prior exposure of a specimen prior to initial measurement to determine the carbonation coefficient. If we correctly account for the initial carbonation depth then it will have no impact on the subsequent results. A variation of Eq. 14 is obtained where the Fickian diffusion equation is integrated between two points in time, \(t_A\) and \(t_B\):

$$\begin{aligned} K=\sqrt{\frac{x_{B}^2-x_{A}^2}{t_B-t_A}} \end{aligned}$$
(19)

This can be used to obtain the carbonation coefficient for any period of known exposure conditions.

4.2 Assessment of existing structures

In real structures constructed from unknown concrete or where age of the structure is uncertain, the behaviour of the material may be desired to predict future performance. During routine maintenance, the carbonation depth may be measured from bored specimens. If this is conducted at separate time intervals, or the bored specimen is removed and then exposed to accelerated carbonation conditions for testing, the carbonation coefficient may be calculated using Eq. 19. In these cases, it is likely that the initial carbonation depth, \(x_A\), is significant relative to the later measurement, \(x_B\), and therefore the error induced by using the linear model approach is also significant.

4.3 Performance based design

Performance-based specification is a growing area of research within concrete durability design, including design for resistance to carbonation [26,27,28,29,30]. This enables greater potential for use of novel materials and new concrete technologies which can subsequently improve material efficiency and performance. To satisfy carbonation performance requirements, concrete carbonation resistance must be demonstrated through controlled tests such as those assessed in this analysis. The carbonation coefficient obtained in such tests is often converted to an estimate of the natural carbonation coefficient for determination of the expected performance of real structures [13, 14]. Therefore, it is important that the measurements are interpreted coherently. The results presented in this work show the potential for different estimates of carbonation resistance to arise depending on the method of consideration of the initial carbonation depth; highlighting the importance of understanding which approach has been adopted when interpreting the carbonation coefficients achieved for further analysis.

5 Conclusions

To account for concrete carbonation that occurs prior to carbonation testing, a linear or non-linear approach for the initial carbonation depth is incorporated into the Fickian diffusion equation to calculate the carbonation coefficient, K. This paper evaluates the two approaches to model the initial carbonation depth across 361 data series for various exposure conditions. It is found that both fit experimental data well, with a median \(R^2\) of 0.965 for the linear approach and 0.959 for the non-linear approach.

The linear approach inherently produces an underestimate of K as compared to the non-linear approach, which represents the theoretical value derived by integration of Fickian diffusion. The difference in the calculated K values from the linear and non-linear approaches is not significant when the initial carbonation depth is small relative to the final carbonation depth measured, but is shown to be more significant at greater values of initial carbonation.