Sorption Function

We now will focus on the issue of mathematically describing sorption as a function of time. In nearly all cases, we found that the water absorbed was proportional to t^1/2 over short times as described in Equation 1. For periods greater than a day, there was a clear trend away from the t^1/2 behavior as the rate of sorption decreased. Simple sorption theory based on tube models (see appendix) predicts an exponential decay in the rate of sorption due to gravitational effects. None of our samples exhibited a pure exponential decay. Indeed, the equilibrium height of the water should be tens of meters (see appendix). As a consequence of the large equilibrium height, finite size effects can have an important effect on capillary sorption studies concerning concrete. A detailed study of finite size effects will be presented in a future work. Barring finite size effects, the decrease in sorption rate may be due to several factors. First, as mentioned above, as the water invades the pore space it encounters smaller pores hence slowing the rate of sorption. Second, even if the capillary pores form a strongly connected network through the specimen, such as through the interfacial zone [20] around the aggregates where capillary pores may be larger, the ingress of water may still be slowed as the air/water interface relaxes to a stable or metastable configuration in the pore space (Figure 1). Then any further ingress of moisture would be controlled by capillary transport through the gel pores or moisture diffusion in the capillary and gel pores.

It has been recently suggested [14] that as water is absorbed in concrete, calcium hydroxide is dissolved into the pore solution producing a concentration gradient which diminishes the absorption rate. Our own preliminary studies, did not show a significant difference between the sorption of water or the sorption of calcium hydroxide saturated water.

Slow relaxation phenomena and diffusion in complex pore structures having a wide range of pore sizes is often described using the so-called stretched exponential function [21]. To model the transition from rapid to slow sorption rates we will use the following empirical function which incorporates the stretched exponential function to account for the crossover regime:

$$\frac{W}{A}=C(1-exp(-St^{1/2}/C))+ S_gt^{1/2}+S_o$$ (2)

where, as before, W is the volume of water absorbed, A is the cross-sectional area of the water exposed surface, and C is a constant which is related to the distance from the concrete surface over which capillary pores control the initial sorption (and may be most sensitive to finite size effects). The coefficient
S_g< < S describes the sorptivity in the smaller pores or the effects of moisture diffusion. We constructed this function so that at early stages, in the limit t^½ < C/S, expansion of the exponential gives C(1 − exp( − St^½ / C)) ≈ St^½ which is the same as Equation 1. At long times, the above equation is dominated by the S_g t^1/2 term.

Figures 7 and 8 show the fitting of data using Equation 2. Note the good agreement over four decades in time. Reasonably good fits were obtained for all the mix designs as well as for different initial saturations. Table 2 contains some representative values of coefficients for Equation 2 from fits to our data. Note, the effect of the S_o term is negligible when fitting data over very long periods hence we set S_o = 0.

One interesting point is that if, instead, the anomalous scaling form W / A = St ^α is assumed, an anomalous scaling exponent of α ≈ 0.25 is obtained for the data in Figure 7. However, it appears that this exponent is really due to an initial uptake due to surface effects and the crossover from large to small pore dominated sorption.

FIG. 8. Fit of equation 13 to HPC data.

At this point it is appropriate to mention a few caveats. First, Equation 2 cannot describe sorption in its entirety in that the additional t^1/2 term implies the specimen would never stop absorbing water. However, given the slowness of the sorption and the fact that capillary rise in small tubes can reach tens of meters, it is possible that Equation 2 may be usable over very long time periods, up to even years. Second, we were unable to accurately determine all the coefficients of Equation 2 over very short periods of measurement (of order a few hours) because the power law term will obviously have a greater impact on predicted sorption at later times. If the effect of the smallest pores (or moisture diffusion) is ignored (i.e. setting S_g = 0), simple fits of Equation 2, based on data acquired from the first day, give predictions of sorption within 30% of total sorption values recorded over a period of about one year. To closely predict sorption for long periods (i.e. one year) of exposure, we needed data from ten to twenty days of measurements in order to account for the crossover regime.

Appealing aspects of Equation 2 include its simplicity and ease of physical interpretation. However, since we did not derive the stretched exponential functional form from basic physical principles, we cannot rule out other "simple" functions that may work as well to describe our data. In addition, it should be emphasized that we have adopted the stretched exponential form to describe the crossover regime in contrast to its typical usage of describing very long time behavior. Further study is needed to determine the accuracy and utility of such a functional form.