Mass Transport of Multicomponents Solute in Bentonite Clay
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(a) Original image
(b) Enlarged height image
Figure 1. Confocal scanning laser photomicrographs of compacted bentonite (ρdry=1.6).
Since the crystalline structure of montmorillonite determines the fundamental properties of the montmorillonite hydrate, we need to analyze its molecular behavior. Thus by applying a molecular dynamics (MD) simulation we inquire into the physicochemical properties of the montmorillonite hydrate such as diffusivity of chemical species. For extending the microscopic characteristics of constituent materials to a macroscopic diffusion behavior of the micro-inhomogeneous material we apply a homogenization analysis (HA; Sanchez-Palencia 1980). Note that before applying HA we summarize the existing diffusion models for soil. Then we present our model, which can treat the micro-inhomogeneous properties with adsorption behavior on the surface of clay minerals. A CLASSICAL DIFFUSION THEORY OF MULTICOMPONENT SOLUTION IN POROUS MEDIA First we treat a classical diffusion theory of chemical species in a dilute solution and the diffusion in a porous media saturated with solution, because it gives the fundamental idea why we need to introduce the micro/macro analysis based on the molecular simulation and the homogenization method. Diffusion of multicomponent chemical species in solution Let us think a dilute solution with N-components, in which (1,2,…,N-1)-th components are solutes (i.e., diffusing chemical species), and the N-th component is solvent (i.e., water in our case). We introduce the mass-percent concentration of the α-th species, cα , by cα = mα/m where mα is the mass of the α-th species, and m the total mass. Let ρα = mα/V (V the volume) be the mass
density of the α-th species, and ρ = m/V = ΣN ρα the average mass density. Due to the mass conservation law of the α-th species we have ∂ ( ρ cα ) ∂ d α ρα & + fα = + ρ cα viα + f&α = 0 dt ∂t ∂xi
(
)
(1)
where vαi is the velocity of the α-th component particles, d α φ / dt = ∂φ / ∂t + ∂ (φ viα ) / ∂xi implies the material time derivative of a function φ with respect to the α-th component, and f&α gives the source term per volume per unit time because of chemical reactions, etc. The average velocity v is defined by v=
1
N
N
∑ ρ vα = ∑ cα vα
ρ α =1
(2)
α =1
A diffusing mass flux of the α-th species, jα, is introduced as jα = ρα ( vα − v) = ρ cα ( vα − v) , and by using the Fick’s first law jα = − ρ ∑ β Dαβ ∇cβ , Eqn(1) can be written as ∂cβ & * ∂cα ∂c ∂ N + vi α − ∑ Dαβ + fα = 0 ∂t ∂xi ∂xi β =1 ∂xi
(3)
where we set as f&α* = f&α / ρ . The source term f&α is determined by an inter-species relation either at equilibrium or on reaction process. Here we treat only on the equilibrium state. Let us consider a stoichiometric (reversible) process given by
ν
f f A1 + ν A2 + ... 1 2
ν
b b B1 + ν B2 + ... 1 2
or
να Mα ∑ α
=0
(4)
where ν αf and ν αb are the forward and reverse stoichiometric coefficients of α-th species, respectively. Note that the stoichiometric coefficients, ν α , of Eqn(4)2 corresponding to the le
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