2.1. Modeling of Thermally Vibrated Amorphous SiO2 Membranes
We have used homemade FORTRAN source program for the present work. Intel
® Parallel Studio for Linux was employed as a compiler and the software was run on high performance computers with Intel
® Xeon core. The membrane material was amorphous silica that was composed of silicon and oxygen atoms. In order to replicate atomic distances and covalent bond angles, a combination of the modified Born–Mayer–Huggins pair potential and the Stillinger–Weber three-body interactions (BMH–SW) was employed. Details of the force-field functions and parameters can be found in the literature [
23,
24]. The melt-quench procedure was used to obtain amorphous silica structures with densities of 2.0, 2.1, and 2.2 g/mL. The unit cell size was around (2.9 × 2.9 × 2.9 nm
3).
Figure 1 shows snapshots of silica structures. The density was 2.2 g/mL. The β-cristobalite structure in
Figure 1a was melted at 8000 K and quickly quenched to obtain the amorphous silica structure shown in
Figure 1b.
Two different virtual silica models were prepared. One was a static model where all atoms were fixed to satisfy atomic pair distances and covalent bond angels of an equilibrated amorphous silica structure. Another was a dynamic model, where the thermal motion or kinetics of atoms was accounted for in addition to the static structure of amorphous silica.
The inorganic polymer compound in amorphous silica membranes is chemically stable. Those rigid silica membranes are generally assumed not to show dynamic behaviors such as micro-Brownian motion due to the thermal motion of polymer chains. Therefore, the molecular motion that should be simulated in this work is atomic thermal motion that originates from the atomic vibration that is addressed in solid-state physics. Since amorphous silica structures have a short-range order and a chemically stable regular tetrahedral structure, the thermal motion of the atoms on a membrane must be calculated in consideration of the equilibrated location of the atoms in a stable regular tetrahedral structure as base point positions. The thermal motion of atoms in a solid can be approximately described as a simple harmonic oscillation by employing a potential model that contains the natural frequency of an atom as a parameter. In this work, the simplest harmonic oscillation potential model was used. The potential equation is shown as Equation (1).
In Equation (1),
k is the force constant,
r is the inter-atomic distance of atom pairs such as Si–O and O–O in the case of silica,
r’ is the equilibrated distance between the two atoms. This inter-atomic potential model enables us to control the thermal motion that originates from the displacement of an equilibrated position by changing the value of parameter
k and also allows examination of the effect of the displacement on gas permeation properties. In order to prevent a change in the equilibrated silica network structures by the thermal motion, only the thermal motion of oxygen atoms was considered, and the silicon atoms were fixed at equilibrated positions. The silicon atoms were surrounded by a more electronegative electron cloud of oxygen atoms in an amorphous silica structure, and the interaction between permeation molecules and silicon atoms was assumed to be reasonably negligible [
25]. Therefore, fixing silicon atoms at their equilibrated positions on gas permeation properties would amount to a small effect.
The equilibrated distances between silicon and oxygen atoms, r’Si–O, and oxygen and oxygen atoms, r’O–O, were set to be the location of the first peak of the pair of radial distribution functions of the amorphous silica structure. The values of r’Si–O = 0.164 nm and r’O–O = 0.267 nm were employed. In order to simulate strict atomic motions using the oscillation vibration potential, in addition to the spreading motion of covalently bonded atoms (Si–O), several modes of motion caused by actual thermal molecular motion, such as the angle-changing motion of three-body atoms (O–Si–O, Si–O–Si) and torsion motion that originated from the rotation of atoms, should also be taken into account. However, so many numbers and complexed equations of motion increases computational cost, which makes it difficult to conduct sufficient calculations in order to obtain reliable permeation data for analysis. In the present study, to ease the handling of the thermal motion of the membrane constituent atoms, the vibration of the non-covalently bonded O–O atomic distance was considered to be an alternative motion for the O–Si–O angle movement. Since covalently bonded Si–O atom pairs are more stable than non-bonded O–O atom pairs, more force would be required to restore Si–O harmonic oscillation at an equilibrated location. The detailed differences between these two restoring forces remain unknown, and their optimum values are unclear. Therefore, the spring constant, kO-O, for O–O atom distance was assumed to be 1/10 of kSi–O for Si–O in this work.
The initial oscillated displacement from the equilibrated position of atom Δ
r was given according to Equation (2) so that the harmonic oscillation potential satisfied the Maxwell–Boltzmann energy distribution shown in Equation (2).
In Equation (2),
kB is the Boltzmann constant,
T is the absolute temperature, and
k is the spring constant. The three-dimensional harmonic oscillation is composed of three independent oscillation motions of an atom in the
x-,
y-, and
z-directions. Therefore, the harmonic oscillation potential was calculated for each direction of
x,
y, and
z in this work. The potential parameter,
k, in Equation (2) is defined as the strength of the chemical bond of atoms. In the case of atomic oscillation due to covalent bonding between the two atoms, the value of k depends on the atomic weight,
m (kg), and the oscillation frequency,
υ (s
−1). The frequency is given by Equation (3).
The frequency,
ν (s
−1), is described by the product of the wavenumber,
ω (1/m), and the propagation speed (speed of light),
c (m/s). The wavenumber is a unique value that depends on the material that is used, and it can be obtained by spectroscopic measurement. The wavenumber for the covalent bond of Si–O in silica glass has been measured and reported to be
ω = 700–1100 cm
−1 [
26,
27]. In this case, Equation (3) gives
k = 300–700 J m
−2. These values of
k, however, reflect only the stretching oscillation of Si–O, and it might not always be useful for simulating the overall motion of atoms. We employed different values for
k, such as
kSi-O = 100, 200, and 300 (
kO–O =
kSi–O/10), in the same range and order as the above values for trial simulations and empirically decided upon a suitable value of
k for simulating the performance of actual silica membranes. The thermal motion of an oxygen atom was characterized by calculating the mean squared displacement (MSD) for validation of the harmonic oscillation potential parameter.
2.2. Non-Equilibrium Gas Permetion Molecular Dynamics Simulation
Gas permeation simulations were carried out using the dual control plane non-equilibrium molecular dynamics (DCP-NEMD) method [
24] under a constant pressure difference and at a constant temperature. Two virtual boundary planes (control plane, CP) of the gas phase were settled at positions in the upstream and downstream sides sufficiently distant from a membrane surface, where permeating molecules were either generated or deleted to control the pressure difference across the membrane. Virtual helium- and hydrogen-like LJ particles, with size parameters of 0.26 and 0.289 nm, respectively, were employed as permeating molecules. For interactions among, and between, permeating molecules and oxygen atoms on the membrane, the well-known Lennard-Jones potential (Equation (4)) was used [
7,
28,
29]. Lennard-Jones potential parameters are summarized in
Table 1, where
kB is the Boltzmann constant. Interaction between an oxygen atom of silica and a permeating molecule was calculated using Lorentz-Berthelot mixing rules given by Equations (5) and (6).
In the gas permeation simulations of static models, the temperature of the system was controlled via the velocity scaling of guest gas molecules, and the membrane constituent atoms were fixed at equilibrated positions. On the other hand, in the case of dynamic models, the temperature was controlled via the velocity scaling of membrane constituent atoms and no artificial change was added to the velocity of the guest molecules. The time step of MD calculation for the static model was 1 fs, and 0.5 fs was used for the dynamic model. The fifth-order Gear algorithm was used to solve the Newtonian equation of motion for sequential computation of the location and velocity of each atom. The upstream and downstream side pressures were 1 MPa and 0, respectively. The gas permeability,
P [mol m m
−2 s
−1 Pa
−1], was calculated by counting the number of permeated, and then removed, molecules at the position of a control plane during a steady permeation state [
24]. Temperatures of 400–800 K were set to investigate the temperature dependency of gas permeability.