Page:4SIGHT manual- a computer program for modelling degradation of underground low level waste concrete vaults (IA 4sightmanualcomp5612snyd).pdf/55

$D_{o}$ for cement paste can also be related to $\phi$ using the universal relation for the formation factor due to hydration, $\vartheta$, [10]:

$\vartheta ={\frac {D_{o}}{D_{Cl-}^{f}}}=0.001+.07\phi ^{2}+1.8(\phi -.18)^{2}H(\phi -.18)$

(27)

where $D_{o}$ is the concrete chloride diffusivity, $D_{f}$ is the free ion diffusivity of chloride ions, and $H(x)$ is the Heaviside function. The quantity $\vartheta$ is a constant for all ions.

The above equations relate the diffusivity of cement paste to water:cement ratio or porosity. A relationship is now needed between cement paste diffusivity and concrete diffusivity. Experimental results of Luping and Nilsson[11] for cement paste and mortar suggest that their diffusivities are approximately equal. Additionally, results from numerical experiments by Garboczi, Schwartz, and Bentz[12] investigating the effects of aggregate-paste interfacial zone diffusivity upon bulk diffusivity suggest that concrete diffusivity is approximately equal to the paste diffusivity.

Given ${\frac {w}{c}}$, the permeability can be approximated from the data in Hearn, et al.[13]:

$k=10^{5.0{\frac {w}{c}}}\times 10^{-21}m^{2}$

(28)

for ${\frac {w}{c}}$ in the range (0.35,0.80).

10.2Leaching

Once $D$ and $k$ have been established, changes due to leaching can be calculated from $\vartheta (\phi _{o})$, where $\phi _{o}$ is the initial porosity due to hydration. As the $Ca(OH)_{2}$ is leached, the porosity increases. Let the value of porosity after leaching be $\phi ^{\prime }$, and the diffusivity be $D^{\prime }$. Unfortunately, the ratio $D^{\prime }/D^{f}$ is not simply $\vartheta (\phi ^{\prime })$ because as calcium hydroxide is leached from the paste the ratio $D^{\prime }/D^{f}$ does not retrace eqn. 27. Rather, the ratio $D^{\prime }/D^{f}$ is greater than $\vartheta (\phi ^{\prime })$, as demonstrated by the NIST microstructural model.

The NIST cement microstructural model[14] was used to determine the relation for $D^{\prime }/D_{o}$ upon leaching. Results for a ${\frac {w}{c}}=0.35$ paste are shown in Figure 2. The formation factor decreases with decreasing porosity due to hydration, denoted by circles. Upon leaching of the calcium hydroxide, the formation factor follows the curve denoted by squares. Given the following definitions:

$\vartheta _{o}=\vartheta (\phi _{o})$$\vartheta ^{\prime }=\vartheta (\phi ^{\prime })$

(29)

An empirical relation was developed to relate the leached pore structure to the undamaged pore structure.

$\xi =\vartheta _{o}+2.0(\vartheta ^{\prime }-\vartheta _{o})$

(30)

and is shown by the solid curve in Figure 2. Therefore, the ratio of the leached value of diffusivity, $D^{\prime }$, to the initial value $D_{o}$ is

${\frac {D^{\prime }}{D_{o}}}={\frac {\xi }{\vartheta _{o}}}$

(31)

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