Conductivity

Calculated electrolyte conductivity _calc can be expressed as a weighted sum of the equivalent conductivity _i of each ionic species [11]:

$$\sigma_{\mbox{\small calc}} = \sum_i \ z_i\, c_i\, \lambda_i$$ (2)

The quantities z_i and c_i are the species valence and molar concentration, respectively. At low concentrations (c0.01 mol/L), the equivalent conductivity is practically constant, and the solution conductivity is proportional to concentration. At higher concentrations, the equivalent conductivity decreases noticeably with increasing concentration. The OH$^-$ concentration in pore solution is typically in the range 0.1 mol/L to 1.0 mol/L [12]. Therefore, accurately estimating pore solution conductivity requires accurately estimating the equivalent conductance concentration dependence.

While a number of highly accurate equations containing numerous coefficients exist for estimating the equivalent conductivity [8], a new single-parameter model is proposed for its simplicity, with the objective that the equation should be accurate to within 10 % for typical pore solutions. Previous work [6] indicates that the uncertainty in estimating the bulk conductivity _b can be less than a few percent. From Eqn. 1, an uncertainty of 10 % in pore solution conductivity $\sigma_p$_p would translate into a similar uncertainty in the calculated formation factor $\Upsilon$. Such a level of uncertainty would be difficult to improve upon using existing diffusion cell experiments.

The concentration dependence of the individual equivalent conductivities at 25 ºC is approximated using the following single-parameter model that characterizes low concentration data well, and remains reasonably accurate at concentrations near 1 mol/L:

$$\lambda_i = \frac{\lambda^\circ_i}{1 + G_i\, I_M^{1/2}}$$ (3)

The quantity $\lambda ^\circ$º is the equivalent conductivity of an ionic species at infinite dilution, and is only a function of temperature; the values of $\lambda ^\circ$º for Na$^+$, K$^+$, OH$^-$, Ca$^{2+}$, Cl$^-$, and SO$_4^{2-}$ at 25 ºC can be found in the literature[8], and are shown in Table 1. The quantity I_M is the ionic strength (molar basis), and has the following definition [11]:

$$I_M = \frac{1}{2}\, \sum_i z_i^2 c_i$$ (4)

The empirical coefficients G_i are chosen to best agree with published data for the electrical conductivity of solutions. In principle, the coefficient G_i will also depend upon temperature.

The algebraic form of Eqn. 3 is based on previous work on the conductivity of electrolytes. It is known that the leading term in the correction should be proportional to c^½ [13]. At higher concentrations, however, this is an overcorrection. Onsager and Fuoss (OF) [14] gave additional terms that are proportional to c log c and c. Although rigorous, using the OF equation would require multiple coefficients for each species, which violates the objective of simplicity desired here. As a compromise, Eqn. 3 is a modification of a relationship (for binary salts) by Walden [15] that is a function of the salt concentration and requires an empirical coefficient for each salt. The extension to electrolytes containing many ionic species was achieved by changing the salt concentration to the molar ionic strength I_M . This change is motivated by similar relationships for estimating the activity of ionic species in concentrated electrolytes [8].

Based on Eqn. 2, the most significant contributor to the pore solution conductivity of a cementitious system is the OH$^-$ ion; its equivalent conductivity is a factor of two greater than that for sodium or potassium (see Table 1), and it is present at the highest concentration. Because the equivalent conductivity of the remaining ionic species in the pore solution of a well hydrated specimen are all of the same magnitude, the Na$^+$ and the K$^+$ should be secondary contributors due to their relatively high concentrations after 1 d [12].

Two other species to consider are calcium and sulfate. Due to high alkalinity, the equilibrium calcium concentration in pore solution is typically on the order of 0.001 mol/L [10]. The corresponding calcium contribution to the overall conductivity (assuming I_M 1.0 mol/L and $\sigma_p$_p = 20 S/m) is on the order of 0.003 S/m, and so can be neglected. Using the pore solution speciation model by Taylor [9], the concentration of sulfate can be roughly approximated by the potassium and sodium concentrations:

$$c_{\mbox{\tiny SO$_4^{2-}$}} \approx \alpha \, \left( c_{\mbox{\tiny K$^+$}} + c_{\mbox{\tiny Na$^+$}} \right)^2$$ (5)

($\alpha$ = 0.06 L/mol) Using this approximation, sulfate will make the greatest relative contribution when the sum of the potassium concentration and the sodium concentration approaches 1 mol/L (it is unlikely they will be significantly greater). The corresponding sulfate contribution to the pore solution conductivity is approximately 0.25 S/m, or less than 2 % of the anticipated total conductivity.

Therefore, the electrical conductivity of most pore solutions of well-hydrated cement-based materials could be accurately estimated from the contribution of the Na$^+$, K$^+$, and OH$^-$ ions alone. In those cases where other species are present at significant concentrations, additional coefficients are provided in Table 1, but are not part of the validation experiment.

Table 1: Equivalent conductivity at infinite dilution $\lambda ^\circ$º and conductivity coefficients G at 25 º C. Reference data typically report the product z$z\lambda ^\circ$ (z: species valence).

Species	z$z\, \lambda^\circ$º (cm2 S/mol)	G (mol/l)-½
OH$^-$	198.0	0.353
K$^+$	73.5	0.548
Na$^+$	50.1	0.733
Cl$^-$	76.4	0.548
Ca$^{2+}$	59.0	0.771
SO$_4^{2-}$	79.0	0.877