TIME DEPENDENCY OF CHLORIDE DIFFUSION COEFFICIENTS IN CONCRETE

Transcription

1 TIME DEPENDENCY OF CHLORIDE DIFFUSION COEFFICIENTS IN CONCRETE Visser, J.H.M. 1, Gaal, G.C.M. 2, and de Rooij, M.R. 1 1 TNO Building Construction and Research, the Netherlands 2 Delft University of Technology, the Netherlands Abstract Most of the currently used time-dependent models to predict chloride diffusion in concrete are based on short term exposure tests. Consequently, there is some serious doubt if these models can be used to model long term exposure tests (field data). An additional factor 1/(1-n) is required to model long term exposure tests accurately. In this paper, the difference and mathematical consequences between short term and long term exposure tests is explained. To obtain optimum values for the three variables (c s, D 0 and n) in either solution, the equations should be solved at the same time for chloride profiles determined at different ages. If the variables are determined by the correct approach, the values can be exchanged between the two models. 1. Introduction One of the main causes of damage of concrete structures is corrosion of reinforcement steel. The reinforcement can corrode due to external species migrating into concrete, e.g. chloride resulting in chloride induced corrosion. This process is assumed to begin when the chloride content in concrete at the position of the reinforcement reaches the chloride threshold value. To predict this moment in time diffusion based chloride ingress models are used. Thirty years ago, a first diffusion model was proposed by Collepardi [1]. Since then, many new models have been introduced including effects like the environmental (exposure) conditions [2] and concrete composition [3]. However, one of the most important factors influencing the chloride diffusion is the ageing of concrete. Due to ongoing hydration, the concrete becomes denser, resulting in a decreasing diffusion rate. 421

2 This paper focuses on the time dependency of the diffusion coefficient. It is shown that there are basically two ways of calculating the relationship: one by using long term penetration tests or field data (continuous exposure), the other by performing short term penetration or migration tests (discrete exposure). When the variables are determined by the correct approach, the values for the diffusion coefficient can be exchanged between the two methods. 2. Modeling chloride ingress 2.1. Apparent diffusion coefficient Collepardi [1] proposed a model to estimate chloride ingress based on Fick s second law of diffusion, assuming a constant diffusion coefficient and a constant surface chloride content. The analytical solution for this situation is: ccl ( x, t) = ccl; i + ( ccl; s ccl; i ) erfc x 4 Dapp t (1) wherein: c cl (x,t) chloride content at time t and depth x c cl;i initial chloride content in the concrete c cl;s surface chloride content x depth from exposure surface t exposure time D app apparent diffusion coefficient Based on this approach, the diffusion coefficient and the surface chloride content can be determined from a measured chloride profile, e.g. by means of least square approximation. However, by applying equation (1) to field data, it was found that the apparent diffusion coefficient changed with exposure time (see e.g. [4], [5]). As an example of such a timedependent behaviour of the apparent diffusion coefficient, Figure 1 shows the decrease in diffusion coefficient with exposure time for reinforced concrete specimens exposed in the splash zone at the UK Coast Time dependent diffusion coefficient A first conclusion that can be drawn from Figure 1 is that the apparent diffusion coefficient is not constant. This has a major impact on the prediction models for the onset of corrosion: a time dependency of the diffusion coefficient should be included in order to be able to correctly predict this time. A first concise attempt to include the effect of ageing of concrete on the diffusion coefficient in relation to the service life has been made within the Brite/Euram project DuraCrete [6]. 422

3 apparent diffusion coefficient (m 2 /s) 1E-11 PC 8E-12 PC/PFA 6E-12 PC/BFS 4E-12 2E exposure time (year) Figure 1 Decrease in apparent diffusion coefficient with exposure time (data from [4]), symbols: apparent diffusion coefficients determined from chloride content profiles (equation 1); solid lines: best-fit aging model (equation 2), concrete with PC = Portland cement, PC/PFA =Portland cement and 30 % fly ash, PC/BFS = Portland cement and 70 % blast furnace slag Deducted from data like presented in Figure 1, a reference diffusion coefficient was introduced that could be transformed to a time dependent diffusion coefficient by: n t0 D( t) = D0 t n 0 (2) wherein: D(t) time dependent diffusion coefficient D 0 reference diffusion coefficient at reference time t 0 t 0 reference time (usually 28 days) t time n age exponent This is not the only model used to include the time dependency of the diffusion coefficient (see e.g. [7]), but most models include a similar exponential time behavior, where the decrease in diffusion coefficient is introduced by the age exponent n. In principle, the age exponent n can have any value larger than 0. The larger the age exponent becomes, the faster the decrease in diffusion coefficient is, compared to a reference diffusion coefficient. The time t must be larger than t 0. The limitations for n and t stem from the fact that under normal conditions, it is not possible for the diffusion coefficient to increase with time, when the conditions are kept constant. 423

4 3. Mathematical consequences of time dependency 3.1. Equation for discrete exposure tests In the DuraCrete program, the time dependent diffusion coefficient as given in equation (2) was originally derived for the Rapid Chloride Migration (RCM) test. In this test concrete that has not been previously exposed to chloride is tested briefly (test-time one or two days) for its chloride migration behaviour. If identical concrete samples differing only in age are tested in the RCM-test, the obtained diffusion coefficients plotted versus time should look like the graph given in Figure 2. The exposure time is so brief compared to the age of the concrete, that a discrete diffusion coefficient is obtained. The solid line going through the three discrete diffusion coefficients is described by the relation presented in equation (2). For a reference time t 0 a reference diffusion coefficient D 0 is found (usually t 0 = 28 days). D(t) D =D(t ) discr 1 1 D =D(t ) discr 2 2 t 1 t 2 t 3 D =D(t ) discr 3 3 Figure 2 Schematized decrease in diffusion coefficient (solid line) and calculated diffusion coefficients at discrete exposure times Because the testing time in the RCM test is brief, the discrete diffusion coefficient can be considered a constant for the concrete samples of that specific age. Because the discrete diffusion coefficient can be considered as a constant, the discrete diffusion coefficient can be substituted in equation (1), leading to the following equation for the prediction of the chloride profile: c cl x, t) = c cl; i + ( c cl; s c cl; i ) erfc ( 0 n 1 (3) x t t D t In this equation t age is now the age of the concrete, and t is the exposition time, valid only for brief exposures. age n t 424

5 3.2. Equation for field data When chloride profiles from field data are analyzed a different approach is necessary. In the field, concrete samples or structures are exposed continuously to chloride since their reference time t 0. In the section on chloride modeling it has already been shown that the apparent diffusion coefficient that can be calculated from chloride profiles, is not constant but time dependent. In essence this means equation (1) cannot be used for evaluating chloride profiles from field data. A time dependent diffusion coefficient is required. It was then shown that as a first concise attempt for a time dependent diffusion coefficient equation (2) was proposed. Generally, as the next step, equation (2) is substituted in equation (1) to obtain a relationship for the time dependent prediction of the chloride profile. However, this substitution is mathematically inconsistent! Equation (2) can only be substituted in equation (1) if the time dependent diffusion coefficient can be considered constant and the exposure time is brief. The discrete coefficient as obtained from the RCM-test can be used as such. When a concrete sample or structure is exposed continuously from the beginning of its life time, the penetration of chloride over time depends on a continuously changing diffusion coefficient. This is illustrated in Figure 3. D(t) t1 D cont = 0 D(t)dt t2 D cont = 0 D(t)dt t3 D cont = 0 D(t)dt t 1 t 2 t 3 t Figure 3 Schematized decrease in diffusion coefficient (solid line) and calculated continuous diffusion coefficients at different exposure times The total history of the changing diffusion coefficient is only included when solving Fick s second law anew. It results in a time-history of the diffusion coefficient given by integrating equation (2): t 0 n D0 t0 D( t) dt = t n 0; n 1 (4) 1 n t 425

6 The factor 1/(1-n) is missing in most analyses, as noted earlier by Bentz and Feng [8]. Due to the integration it is now valid again to substitute equation (3) in equation (1) resulting in: c cl x ( x, t) = ccl; i + ( ccl; s ccl; i ) erfc 0 n <1 (5) n D0 t0 4 t 1 n t In this equation t is representing the exposure time. Notice that by the chosen integration boundaries, the equation is valid only for exposure times much larger than the age of the concrete, and that the age at which the exposure start should be small Optimization of parameters There are now two equations for the prediction of chloride profiles: one applicable to short term discrete diffusion approach (equation 3), and one applicable to field data using a long-term continuous diffusion approach (equation 5). In these equations there are three unknown variables that need to be optimized when solving the equations: the chloride surface content c s, the reference diffusion coefficient D 0 and the age exponent n. When solving this equation for various chloride profiles, it is not correct to determine at first the apparent diffusion coefficient by equation (1) and next the time dependent factor by equation (2) or (4) because then not the optimum values for the three variables are found, but only the optimum values for two sets of two variables (D(t) and c s and D 0 and n). It should be noted that equation (5) is indeterminate when only one chloride profile is available, i.e. there will be many solutions resulting in the same well-fitted chloride profile. Hence, when only one chloride profile is available, at best the time-dependent diffusion coefficient and the surface chloride content can be determined. Finally, it should be noted that the three unknown variables in the two equations (3 and 5) represent exactly the same variables. Failing to recognize this and thus using only one equation for the two different types of exposure tests will result in variables that are different. 4. Consequences for modeling field data 4.1. Used data Data from Thomas and Bamforth [4] have been used to illustrate the consequences when modeling field data based on the two different approaches, i.e. the old method determining first (D app and c s ) and next (D 0 and n), and the new approach by means of solving equation 5 for all profiles simultaneously. The data have been obtained from field experiments in which concrete specimens have been exposed for 8 years in the splash zone at the southeast coast of the UK, near 426

7 Folkstone. Chloride profiles have been determined for 6 concrete compositions and three curing regimes after 6 months, 1, 2, 3, 6 and 8 years exposure. The data shown in this paper are for three compositions only and averaged over the three curing regimes. For full details of the experiments, see [4]. The concrete compositions are given in Table 1. Table 1 Concrete mix details (in kg/m 3 ) for the Folkstone specimens. Code PC PC/PFA PC/BFS Portland cement Fly ash Slag Water Wbr Gravel Sand Calculation by apparent diffusion approach Based on the Folkstone chloride profiles, the surface chloride contents c s and the apparent diffusion coefficients D app were recalculated using equation (1) (Table 2). Next, the reference diffusion coefficients D 0 (at 28 days) and the age exponent n were determined using equation (2) (Table 3, see also Figure 1). Table 2 Recalculated best fitted coefficients for the apparent diffusion approach using equation (1), surface chloride content c s in % by mass of concrete, apparent diffusion coefficient D app in m 2 /s. PC PC/PFA PC/BFS age (year) c s D app c s D app c s D app Table 3 Best fitted coefficients of the ageing formula in the apparent diffusion approach using equation (2) (standard deviations between brackets) Code PC PC/PFA PC/BFS c s (average, t% by mass of concrete) 0.45 (0.15) 0.52 (0.36) (0.09) D 0 (10-12 m 2 /s) n the outlier at 8 year has been omitted in the calculation of the average 427

8 In Table 3, it can be seen that the age exponent of PC is zero, i.e. the diffusion coefficient remains essentially constant during the exposure time. This is a consequence of the low level of variation in the apparent diffusion coefficients with age of this concrete (see Table 2). For the other two concrete mixtures, the apparent diffusion coefficient decreases very rapidly with age, hence high age exponents are found. The reference diffusion coefficient (at 28 days) for PC with a wbr of 0.66 is low compared to values normally found in laboratory experiments, whereas the reference diffusion coefficient for PC/BFS is unrealistic high. It should be noted that the BFS used in these experiments is relatively slow reacting (compared to Dutch slag cement). Note also that the surface chloride contents are averaged over the best-fit values for each chloride profiles, and not optimum values Calculation by continuous diffusion approach The chloride profiles from Folkstone have also been used in the calculation of the optimum values for c s, D 0 and n by means of the continuous diffusion approach (using equation 5). As an example, the results for PC are shown in Figure 4 and Table 4. chloride content (wt-% on concrete) 0,5 0,4 0,3 0,2 0,1 6 months 1 year 2 year 3 year 6 year 8 year distance from exposure surface (mm) Figure 4 Chloride profiles of PC at various times (symbols) and best-fitted profiles (lines) on the basis of the continuous diffusion approach (equation 5) assuming continuous exposure; optimized for all profiles together. Table 4 Best fitted coefficients in the continuous diffusion approach Code PC 1 PC/PFA PC/BFS 2 c s (average) / 0.49 D 0 (10-12 m 2 /s) / 2.1 n / the outlier at a depth of 5 mm after one year of exposure is omitted in the calculations 2 the bold values are calculated by omitting the chloride content at 5 mm after one year of exposure 428

9 From Figure 4, it can be seen that individual chloride profiles are sometimes not fitted well at all, contrary to the single fits in the apparent diffusion approach. This is a consequence both of the requirement that the surface chloride content remains constant with time, as well as the increase of the chloride profiles over time. In the continuous diffusion approach all profiles are fitted at the same time to get the best optimization of the three values. Unfortunately, it is very difficult to judge the value of the first point in the profiles, when the first point is low compared to the entire profile. Furthermore, an a- priori assumption is made on a constant surface chloride content. However, the used data is obtained in the splash zone and a constant surface chloride content may not be present [2]. On the other hand, there is no trend in the variation of the surface chlorides (see Table 2), so that the observed variations may well be within the normal variation within the splash zone. The calculation of the optimum values in the continuous diffusion approach is very stable. However, for age factors close to 1, the calculated reference diffusion coefficient D 0 is very sensitive to the data in the calculation (since n=1 is a singularity point). This is illustrated in Table 4 by the second set of results of PC-BFS. Omitting just a single point in the calculation results in a reference diffusion coefficient that is 4 times as high. Note that the surface chloride content and the age exponent are much less sensitive Comparison between apparent and continuous diffusion approach Comparing the results of the continuous diffusion approach with those of the apparent diffusion approach, it can be seen that the two approaches give almost similar results for PC/BFS, with the exception of the reference diffusion coefficient (which differ about a factor 1/(1-n)), while for PC/PFA and PC the results of the two approaches differ. In Figure 5, the implication of these differences can be seen. For the PC concrete, the age exponent is 0 for the apparent approach, whereas for the continuous approach the age exponent is 0.12 so that the chloride ingress is faster for the apparent diffusion solution even though the reference diffusion coefficient is lower and the surface content is higher. For the PC/PFA concrete, the age exponent is however lower in the continuous approach than in the apparent diffusion approach so that the chloride ingress is slower for the apparent diffusion solution. For the PC/BFS concrete, an age exponent of 1 was found in the apparent diffusion approach. As can be seen as well in the figure, this implies an instantaneous build-up of chloride penetration limited to a surface layer after which the diffusion stops, i.e. there is no further chloride penetration. Although this seems unlikely in practice, as an approach for long term chloride penetration, it may be valid in case the concrete becomes dense so quickly that the chloride penetration can be neglected after initial ingress. For the continuous approach, the age exponent is slightly smaller than 1 so that this initial build-up of a chloride profile takes a slightly longer time. The progress with time is however negligible as well. 429

10 chloride content (% on concrete) 0,5 0,4 0,3 0,2 0,1 PC, app PC, cont PC/PFA, app PC/PFA, cont PC/BFS, app PC/BFS, cont 0, exposure time (year) Figure 5 Chloride content at a depth of 25 mm 5. Conclusion It is found that different solutions exist of Fick s second law of diffusion for different experiments differing by the length of exposure times. Two solutions have been derived, one for short term exposure tests, in which the diffusion coefficient during the test can be considered constant and one for long term exposure tests in which the diffusion coefficient is time dependent during the exposure. Failure to recognize the existence of two solutions for the two types of tests can result in a wrongful prediction of future chloride profiles. On the other hand, when the variables are determined by the correct approach, the values can be exchanged between the two methods. To obtain optimum values for the three variables (c s, D 0 and n) in either solution, the equations should be solved at the same time for chloride profiles determined at different ages. When only chloride profiles of one age are available the time-dependency of the chloride diffusion coefficient cannot be determined; just an apparent diffusion coefficient over that particular exposure time can be obtained. 430

11 References 1. Collepardi, M., Marcialis, A., Turriziani, R.: Penetration of Chloride Ions into Cement Pastes and Concretes, J.Am.Cer.Soc., October 1972, pp Costa, A. and Appleton, J., Chloride penetration into concrete in marine environment Part 1: Main parameters affecting chloride penetration, Mat.&Struct. 32: , Johannesson, B., Transport and sorption phenomena in concrete and other porous media. Lund University Report TVBM-1019, Thomas, M.D.A. and Bamforth, P.B.: Modelling chloride diffusion in concrete. Effect of fly ash and slag. Cem.&Concr.Res. 29: , Sandberg, P., Tang, L. and Andersen, A., Recurrent studies of chloride ingress in uncracked marine concrete at various exposure times and elevations. Cem.&Concr.Res. 29(10): , DuraCrete, Performance based design for durability of concrete. Modelling of degradation (Task 2 Summary Report), CUR, Gouda, Dhir, R.K., Jones, M.R. and Ng, S.L.D.: Prediction of total chloride content profile and concentration/time-dependent diffusion coefficients for concrete, Mag.Concr.Res. 50(1): 37-48, Bentz, D.P. and Feng, X., Time-dependent diffusivities: possible misinterpretation due to spatial dependency. In: Testing and modeling the chloride ingress into concrete (eds. C. Andrade and J. Kropp), RILEM Proc.19, RILEM Publ.,