EDL modeling

3. SOLUTION OF THE NONLINEAR POISSON-BOLTZMANN EQUATION

3.3. EDL modeling

Adsorption of ions onto solid surfaces induces a gradient in electrical potential 𝜓 [V] near the solid-pore liquid interface of charged porous media. The excess surface charge due to adsorbed species must be balanced by the presence of counter-ions in a “diffuse swarm” in the near- surface region, and these two regions are collectively referred to as the electric double layer (EDL). The interdependence of the diffuse layer charge density 𝜌_𝐷 [C/m³] and 𝜓 gives rise to the elliptic partial differential equation termed the Poisson-Boltzmann equation:

∇²𝜓=− 𝜌_𝐷

ϵ0ϵ (3.2)

where ϵ0 is the permittivity of free space [8.854 x 10^-12 C/V/m], ϵ is the dielectric constant of the solution [-], and ∇² is the Laplace operator. Concentration of a particular species at a distance 𝑥 [m] normal to a solid surface depends upon the energy state of that species and is therefore given

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by the Boltzmann distribution. Thus, 𝜌_𝐷 is simply the difference in positive and negative charge distributions for all species 𝑖, such that

𝜌𝐷 =𝑒NA� 𝑧𝑖𝐶𝑏,𝑖exp�−𝑧𝑖𝑒𝜓 kB𝑇 �,

𝑖

(3.3) where 𝐶_𝑏,𝑖 is the concentration in the bulk pore solution [mol/m³], 𝑒 is the elementary

electrostatic charge [C/particle], N_A is Avogadro’s number [particles/mol], and k_B is the Boltmann constant [J/K] [110]. For small distances from a charged surface and for very large values of 𝜓, the Boltzmann distribution runs into difficulties because it may predict impossibly high values of concentration of ions given their finite sizes , however, this problem is alleviated by limiting the number of near surface charges to a maximum determined by the number of available “adsorption” sites within the near-surface region of finite thickness termed the “Stern layer” [111]. For the purposes of this work, the effects of the Stern layer are not taken into account explicitly; rather, it is assumed that the thickness of the Stern layer is negligible as compared to the width of the pores in question and that balancing of the structural charge by the Stern layer is negligible such that the effective potential boundary condition to be solved, 𝜓𝑏𝑐, is equivalent to the potential of the outer Helmholtz plane 𝜓_𝛿 [112]. Moreover, the systems

considered are assumed to be open such that the bulk concentrations 𝐶_𝑏,𝑖 are constant values which implies that masses of ions in the EDL are negligible compared to the bulk concentrations.

Combining Eqs. (3.2) and (3.3) yields the general form of the full nonlinear PBE:

∇²𝜓= −𝑒N_A

ϵ0ϵ � 𝑧^𝑖𝐶𝑏,𝑖exp�−𝑧_𝑖𝑒𝜓 kB𝑇 �.

𝑖

(3.4) Eq. (3.4) may be linearized if 𝑧_𝑖𝑒𝜓 ≪k_B𝑇, that is, where the electrical potential energy of an ion (𝑧𝑖𝑒𝜓) is small compared to its thermal energy (kB𝑇), such that

∇²𝜓= 𝜅²𝜓 (3.5)

where 𝜅²is the Debye parameter, given as [94]

𝜅² = 𝑒²NA

ϵ₀ϵk_B𝑇 � 𝐶^𝑏,𝑖𝑧_𝑖²

𝑖

. (3.6)

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Eq. (3.4) is by far the most commonly employed version of the PBE; however, electrophoretic measurements of hydrated cement and its constituent solid phases indicate that the magnitude of the electrokinetic (zeta) potential (which we assume approximates the potential of the Helmholtz plane, 𝜓𝛿) for cement is typically on the order of tens of millivolts, in which case the exponential terms may not be neglected [89, 92]. In systems for which linearization of the PBE is not

appropriate, simplification of Eq. (3.4) is still possible with the assumption that the system is comprised of a single symmetric electrolyte, that is, for every cation of valence +𝜁 there exists an anion of valence –𝜁. Assuming symmetry, the exponentials of Eq. (3.4) may be combined to yield

∇²𝜓= −2𝑒N_A

𝜖₀𝜖 𝜁𝐶^𝑖sinh�𝜁𝑒𝜓

k_B𝑇� (3.7)

for which analytical solutions are possible; however, experimental evidence indicates that constituents of real cementitious pore solutions are unlikely to be symmetric or of a single valence [113].

The full nonlinear PBE only assumes that the configuration of charges in proximity to a charge surface follows a Boltzmann distribution and may be solved numerically as follows. The left hand side of Eq. (3.4) is replaced by a second order central finite difference, and the exponential on the right hand side is expanded as the infinite Taylor series:

exp(𝜉) = �𝜉^𝑛 𝑛!

∞

𝑛=0

= 1 +𝜉+�𝜉^𝑛 𝑛!

∞

𝑛=2

(3.8) with 𝜉=−𝑧_𝑖𝑒𝜓⁄k_B𝑇. Thus, the exponential function may be subdivided into a linear part, 𝜉, and a nonlinear part, 1 +∑^∞_𝑛=2^𝜉_𝑛!^𝑛. Solution of Eq. (3.4) then follows closely the method proposed by Nicholls and Honig [114] excepting that electrolyte symmetry was not assumed.

The value of potential at given node, 𝜓0, is expressed as a function of the values of 𝜓 at the nearest neighboring nodes, the constant term, and the power series of 𝜉

𝜓0 = �� 𝜓𝑘 𝑎𝑙𝑙 𝑘

+�𝑒NA

ϵ0ϵ � 𝑧^𝑖𝐶𝑏,𝑖 �1 +�𝜉^𝑛 𝑛!

∞

𝑛=2

�

𝑎𝑙𝑙 𝑖

� ℎ²� �� 1

𝑎𝑙𝑙 𝑘

+𝜅²ℎ²�

� , (3.9)

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where 𝑘 represent the indices of the six nearest neighbors [(𝑥 −1,𝑦,𝑧), (𝑥+ 1,𝑦,𝑧), … ] and ℎ is the grid spacing [m]. For a discretization of 𝐿³ nodes, Eq. (3.9) may be written in vector notation as

𝚿^𝑡+1= T𝚿^𝑡+𝐐 (3.10)

where 𝚿^𝑡+1 is the 𝐿× 1 vector of 𝜓 values computed at the next iteration 𝑡+ 1, 𝚿^𝜏 are the 𝜓 values at the current iteration 𝑡, T is an 𝐿×𝐿 matrix of coefficient values, and 𝐐 is the vector of nonlinear terms. Successive “under relaxation” is applied to Eq. (3.10), and the nonlinearity 𝐐 is treated as a perturbation of the solution [114]. To maintain stability, 𝐐 is scaled by a factor 𝜔_𝑸 that grows with each iteration; for example:

𝜔𝑸 =�0.5𝑡 for 𝑡 ≤20

1 for 𝑡> 20 . (3.11)

The final iteration algorithm is written as

𝚿^𝑡+1 =𝜔_ΨT𝚿^𝑡+ (1− 𝜔_Ψ)𝚿^𝑡+𝜔_𝑸𝐐 (3.12) where 𝜔Ψ is the relaxation parameter. Dirichlet boundary conditions (𝜓𝑏𝑐 = constant) are imposed at the pore wall corresponding to the potential just beyond the Stern layer, assumed herein to be approximated by the zeta potential. Periodic boundary conditions are imposed at the

“ends” of a pore, and values of 𝜓 are then computed from Eq. (3.12) until the maximum change in 𝜓 is less than a specified tolerance, here 10^-4 mV. Once satisfactory convergence of the solution has been obtained, the concentration of any particular ion may be computed from the Boltzmann distribution

𝐶𝑖 =𝐶𝑏,𝑖exp�−𝑧_𝑖𝑒𝜓

kB𝑇 �. (3.13)

The solution scheme was implemented in the commercial computational software package MATLAB (MATLAB 7.10.0, The MathWorks Inc., Natick, MA, 2000).

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3.4. Results and Discussion