Model reduction for molecular diusion in nanoporous media Gast on A. Gonz alez1Ruben A. Fritz1Yamil J. Col on2and Felipe Herrera1 3 1Department of Physics Universidad de Santiago de Chile Av. Victor Jara 3493 SantiagoChile

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Model reduction for molecular diffusion in nanoporous media
Gast´on A. Gonz´alez,1Ruben A. Fritz,1Yamil J. Col´on,2and Felipe Herrera1, 3
1Department of Physics, Universidad de Santiago de Chile, Av. Victor Jara 3493, Santiago,Chile
2Department of Chemical and Biomolecular Engineering, University of Notre Dame, IN, USA
3Millennium Institute for Research in Optics, Chile∗
(Dated: October 27, 2022)
Porous materials are widely used for applications in gas storage and separation. The diffusive
properties of a variety of gases in porous media can be modeled using molecular dynamics simulations
that can be computationally demanding depending on the pore geometry, complexity and amount
of gas adsorbed. We explore a dimensionality reduction approach for estimating the self-diffusion
coefficient of gases in simple pores using Langevin dynamics, such that the three-dimensional (3D)
atomistic interactions that determine the diffusion properties of realistic systems can be reduced
to an effective one-dimensional (1D) diffusion problem along the pore axis. We demonstrate the
approach by modeling the transport of nitrogen molecules in single-walled carbon nanotubes of
different radii, showing that 1D Langevin models can be parametrized with a few single-particle 3D
atomistic simulations. The reduced 1D model predicts accurate diffusion coefficients over a broad
range of temperatures and gas densities. Our work paves the way for studying the diffusion process
of more general porous materials as zeolites or metal-organics frameworks with effective models of
reduced complexity.
I. INTRODUCTION
The simulation of gas diffusion in nanoporous solid-
state materials is important for applications such as gas
filtering, separation and storage [1–5]. The self-diffusion
coefficient of a gas in a porous medium is an essential
physical quantity that characterizes mass transfer and
is a relevant parameter for designing industrial separa-
tion processes [6], diffusion of gas mixtures [4], and the
selectivity of gas separation techniques [3,7–10]. The dif-
fusive properties of gases in porous media is ultimately
related to the short and long-range interaction potentials
between gas particles and between gas molecules and the
condensed-phase environment [11].
The growing interest in estimating the diffusive prop-
erties of target gases in porous materials reported in
public databases [5] has stimulated the search for meth-
ods to accelerate large scale screening efforts based on
fully-atomistic simulations, which in general are compu-
tationally demanding [8,12,13]. Acceleration strategies
based on machine learning are promising because training
sets with acceptable predictive power can be constructed
with a smaller number of calculations than an exhaus-
tive database search [14,15]. An alternative acceleration
strategy would be to develop generalizable physics-based
models that are sufficiently accurate for ranking materi-
als based on their transport properties, but at a much
lower cost than atomistic simulations.
In this context, we study the dimensionality reduction
capabilities of one-dimensional (1D) Langevin dynam-
ics for modeling gas diffusion inside carbon nanotubes
at different temperatures. The predictions of the re-
duced model are compared to the three-dimensional (3D)
∗Electronic address: felipe.herrera.u@usach.cl
molecular dynamics simulations. For concreteness, we
consider the transport of molecular nitrogen in single-
walled carbon nanotubes (CNT) and obtain self-diffusion
coefficients with 1D Langevin dynamics for different nan-
otube radii, temperatures and gas densities. We show
that it is possible to construct effective 1D pore poten-
tials and model parameters that can reproduce the dif-
fusive 3D transport behavior over a broad range of con-
ditions. The proposed parametrization scheme could be
extended to other porous materials such as zeolites and
metal-organic frameworks.
The rest of the article is organized as follows: Sec-
tion II describes the theoretical methodology and the set-
tings for the atomistic molecular dynamics simulations.
In Sec. III we discuss the results obtained for the diffu-
sion constant of nitrogen in carbon nanotubes, comparing
the predictions of the reduced 1D Langevin model, 3D
molecular dynamics simulations, and the Lifson-Jackson
formula from Brownian motion theory. In Sec. IV, we
suggest possible applications and generalization strate-
gies.
II. METHODS
A. 1D stochastic Langevin dynamics
The stochastic motion of Brownian particles can be
described by a Langevin equation [16], which for a 1D
system of Nparticles with trajectories z(α)(t) can be
written as
˙p(α)(t) = −∂V (zN(t))
∂z(α)−γ(α)p(α)(t) + ξ(α)(t)
α=1,2,.,N
(1)
arXiv:2210.14663v1 [physics.chem-ph] 26 Oct 2022
2
where αis the particle index, pis momentum, Vis the
total potential, and z(α)the position of the α-th parti-
cle. The interaction of particle αwith a large ensem-
ble of bath particles is effectively taken into account by
introducing the momentum loss (dissipation) term pro-
portional to the damping parameter γand a random mo-
mentum kick given by the random process ξ(t), which in-
duces energy fluctuation. These terms together take into
account the multiples collisions of the system (Brownian)
particle with the reservoir [1,16]. The random momen-
tum kick has zero bias, i.e., hξ(α)i= 0 and its autocorre-
lation function is given by
hξ(α)(0)ξ(β)(τ)i= 2δ(τ)δαβ m(α)γ(α)kBT, (2)
where mis the particle mass, kBis the Boltzmann con-
stant, Tis temperature, δ(t) is the Dirac delta function
and δαβ is a Kronecker delta. In other words, momentum
fluctuations are Markovian in time and proportional to
the thermal energy kBT.
We solve Eq. (1) numerically for a system of Nparti-
cles using the impulsive Langevin leap-flog algorithm [1],
which is a modification of the classical Verlet algorithm
that involves an intermediate velocity correction at each
time step of the form
∆v(α)= ˙v(α)h−γ(α)v(α)(t)h+q2kBT γh/m(α)ξ(3)
where v(α)and ˙v(α)are the velocity and acceleration of
the α-th particle, and his the time step of the simulation.
For a free Brownian particle at thermal equilibrium, the
damping coefficient γcan be obtained from the Einstein
relation [1]
D0=kBT
mγ (4)
where D0is the free-particle diffusion coefficient. In this
work, the damping parameter γencodes the interaction
of gas molecules with the carbon nanotube walls.
B. Diffusion from mean squared displacements
We calculate the self-diffusion coefficient Dsusing the
mean squared displacement (MSD) method from the sim-
ulated particle trajectories. For a trajectory composed of
cartesian vectors ~ri= (xi, yi, zi) at times ti, the MSD can
be calculated as [17]
MSD(τ=nh) = 1
M−n
M−n
X
i=1
(~ri+n−~ri)2(5)
which uses all available offsets τof a given duration nh
with nthe offset step. The advantage of this definition is
that the number of such displacements is M−nand there-
fore large for small n, resulting in well-averaged MSD
values. MSD is related to the self-diffusion coefficient by
FIG. 1: (a) Radial viewpoint of the nanotube (11,0) with
molecular nitrogen molecule (N2) its pore volume. (b) Axial
viewpoint of the nanotube (11,0).
the expression [18]
MSD = 2aDsτ(6)
where ais the system dimensionality (a= 1 for 1D, a= 3
for 3D). By solving Eq. (1) for all the particles in the
system at fixed temperature and density, we calculate
MSD from Eq. (5) and obtain Dsfrom the slope of a
linear fit plot of Eq. (6) using the least-squares method.
For short simulation times, particle transport is dom-
inated by the initial condition and the absence of inter-
molecular interactions (ballistic regime). After equilibra-
tion is reached though multiple collisions, the linear scal-
ing of MSD with time is established (diffusive regime).
Several methods have been proposed to analyze trajecto-
ries with coexisting transport regimes [19]. In our work,
the diffusive regime is established when a log-log plot
MSD vs τ, averaged over particles and simulation repli-
cas has unit slope.
C. Lifson-Jackson model for 1D diffusion
The Lifson-Jackson formula is an analytical expression,
first derived in Ref.[20], for the diffusion coefficient of a
periodic 1D potential in terms of the potential depth.
The periodic nature of a pristine carbon nanotube po-
tential along its axis allows us to use this theory directly
at different temperatures. For a periodic potential V(x)
with period L, the Lifson-Jackson diffusion coefficient can
be written as [20–22]
D0
0(T) = D0(T)L2
hRL/2
−L/2e−V(x)
kBTdxihRL/2
−L/2e
V(x)
kBTdxi(7)
摘要：

Modelreductionformoleculardi usioninnanoporousmediaGastonA.Gonzalez,1RubenA.Fritz,1YamilJ.Colon,2andFelipeHerrera1,31DepartmentofPhysics,UniversidaddeSantiagodeChile,Av.VictorJara3493,Santiago,Chile2DepartmentofChemicalandBiomolecularEngineering,UniversityofNotreDame,IN,USA3MillenniumInstitutefor...

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Model reduction for molecular diusion in nanoporous media Gast on A. Gonz alez1Ruben A. Fritz1Yamil J. Col on2and Felipe Herrera1 3 1Department of Physics Universidad de Santiago de Chile Av. Victor Jara 3493 SantiagoChile.pdf

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