3. SOLUTION OF THE NONLINEAR POISSON-BOLTZMANN EQUATION
3.1. Introduction
As the service life requirements of cementitious applications become more demanding, the need for mechanistically based, long-term prediction of material performance grows ever more pressing. More accurate estimation of constituent release is especially important for predicting performance of cementitious materials in waste management applications, wherein the
cementitious matrix is relied upon to retain and/or impede migration of contaminants. Describing ionic transport in cementitious materials is complicated by the interplay of several factors
including constituent retention via ion-exchange and adsorption, precipitation and dissolution of solid phases, the formation of expansive products which may lead to cracking, and the movement of ions through an often restrictive and tortuous pore space. Additional complications are
apparent at the boundaries between cementitious materials and geologic media, where dramatic disparities in chemistry can induce large gradients in ionic composition (e.g., pH, ionic strength, etc.). Exacerbating this complexity is the fact that, even if relatively high ionic strength cement
1 Published in Cement and Concrete Research, 44 (2013)
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pore solution were considered in isolation, localized ion-ion and ion-solvent interactions might still play a significant role in determining the rate of ion transport [47, 83, 84].
A further confounding effect that seems to have been obscured by the aforementioned aspects of cement complexity is the much faster apparent diffusion of anions as compared to cations in steady-state through-diffusion experiments. In such experiments, two parallel faces of a
cementitious specimen are exposed to two different solutions, an “upstream” solution with high tracer concentration and a “downstream” solution with little to no tracer present. One-
dimensional diffusion subsequently occurs through the cementitious membrane due to the concentration difference Δ𝐶𝑖 of species 𝑖 between the downstream and upstream compartments.
Observed diffusivity 𝐷𝑖𝑜𝑏𝑠 of species 𝑖 is calculated from 𝐽𝑖 = −𝐷𝑖𝑜𝑏𝑠 Δ𝐶𝑖
Δ𝑥 (3.1)
where Δ𝑥 is the width of the cementitious membrane, and the flux of species 𝑖, that is, 𝐽𝑖, is calculated from the change in concentration of 𝑖 in the downstream compartment per time interval between measurements of 𝑖. Steady state is assumed when 𝐽𝑖 is constant after successive measurements. Atkinson et al. found that 𝐷𝑖𝑜𝑏𝑠 for cations and anions measured in diffusion cell experiments may differ by up to an order of magnitude with decreasing water/cement ratio[85].
Others have reported observed anion diffusivities approximately five to twenty times greater than cation diffusivities, as demonstrated in Table 3.1 [85-88]. Such differences may become very important when predicting the rate of ingress of corrosive species, such as chloride or sulfate, and the rate of egress of constituents of potential concern (e.g., Tc-99 in nuclear waste
management applications).
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Table 3.1: Apparent diffusivities reported in the literature from diffusion cell experiments on cementitious materials. Free diffusivities of Na+, K+, and Cl- in water are 1.334, 1.957, and 2.032 [10-9 m2/s], respectively [82].
𝐷𝑖𝑜𝑏𝑠 [10-12 m2/s]
Diffusivity Ratio Reference Salt Upstream Salt
Conc. [M]
Water/Cement [-]
Cation Anion Anion/Cation
[87] NaCl* 0.5 0.45 0.83 4.2 5.1
NaCl* 0.5 0.45 0.85 10.8 12.7
NaCl# 0.5 0.40 1.35 826 612
NaCl# 0.5 0.40 1.6 7.0 4.4
NaCl 0.5 0.35 1.32 6.8 5.2
[88] NaCl 1.0 0.4 0.19 3.2 16.8
NaCl 3.0 0.4 0.33 1.8 5.5
KCl 0.5 0.4 0.2 3.7 18.5
KCl 1.0 0.4 0.35 2.9 8.3
KCl 2.0 0.4 0.46 1.9 4.1
*duplicate experiments.
#duplicate experiments (first case apparently in error in context of other reported data)
Observations of disparate cation/anion diffusivities coupled with measurement of significantly high membrane potentials and zeta potentials [86, 89-91] in cementitious materials and their constituent phases led Chatterji and Kawamura [88, 92] to suggest the electric double layer (EDL) as the underlying cause of disparate ionic diffusivities and to propose a qualitative description of the process. The quantitative descriptions of EDL theory are rooted in the early 20th century [93-95], and since that time, numerous investigations have aimed at illuminating manifestations of the double layer in various materials. Of particular interest are studies of the EDL in clays wherein anions are repelled by Coulombic forces from the negatively charged clay surface [96-98]; the effect of this repulsion is an exclusion of ions from a fraction of the pore space which serves to limit the area available for anion diffusion and, hence, slow anion diffusion with respect to cation diffusion [36, 99-103]. Slower anion diffusion is precisely opposite the observation of ionic diffusion in cement diffusion-cell through-diffusion
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experiments despite the expectation that the surface charge of calcium-silicate-hydrate (C-S-H), the primary solid phase of cementitious materials, is expected to be negative also at the pH of the pore water (typically about 12.5 for ordinary Portland cements). As the surface charge of C-S-H is determined by the amphoteric silanol site, depression of the pore solution pH could lead to a reversal of the surface charge and cation exclusion, but this situation is unlikely given the buffering capacity of Portland cement systems.
Due to the complexity of cementitious materials in both composition and in pore space
morphology, a number of factors in addition to EDL effects may be fully or partially responsible for enhanced anion diffusion in cementitious materials. Unfortunately, a full mechanistic
description of disparate ionic diffusivities originating from EDL phenomena in cementitious materials has yet to be developed; a first step toward such a description is the modeling of the EDL. Typically, the effects of the EDL are addressed through a simplified version of the Poisson-Boltzmann equation (PBE), the Gouy-Chapman formulation, which is valid for low ionic strength symmetric systems with symmetric electrolyte composition, that is, every cation of the bulk solution with valence +𝑧 is accompanied by an anion of valence of −𝑧. Recently, Friedmann et al. [104] have demonstrated the effects of the EDL on transport in simple pore geometries using the analytical solution of the PBE and assuming a symmetric electrolyte composition of the pore solution, a condition that need not hold in real systems.
The objective of the research presented here is to develop and use a numerical solution of the full nonlinear PBE for the purposes of modeling the electric double layer for asymmetric non-dilute pore water compositions of cements. For the sake of conceptual and computational simplicity, pore boundaries in this work are envisaged as uniformly charged cylinders or parallel plates;
however, a primary advantage of the finite difference solution is that it is extensible to arbitrary pore geometries provided that the pore boundary is sufficiently smooth. The pore water
composition and pore size distribution of a blended cementitious mortar were measured in order to demonstrate the possible effects of the EDL on diffusion in a model cementitious material.
Solutions of the PBE are used, then, to ascertain 1) the degree of divergence of the linear and nonlinear solutions for expected conditions in cementitious materials and 2) whether the effect of the EDL on diffusion may be significant for the relevant range of pore sizes encountered in cementitious materials.
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3.2. Experimental Methods