Diffusion coefficients and radial distribution functions of NaCl and HMT-PMBI(Cl - ) determined using molecular dynamics simulations with periodic boundary conditions Colton Lohn Department of Physics, Simon Fraser University, Burnaby BC Canada Abstract The relationship between system size and diffusivity in a periodic cubic simulation cell was studied. The diffusion coefficients of Na + and Cl - in TIP4P/2005 water show significant system-size dependence. No such system-size effects were found for either the diffusion coefficient of Cl - or the radial distribu- tion functions for chloride–oxygen, chloride–chloride, chloride–nitrogen, and chloride–carbon in 87.5% methylated HMT-PMBI at hydration level λ = 16 using TIP4P/2005 water. Introduction Molecular dynamics simulations have emerged as an indispensable tool for studying the nanoscopic, meso- scopic, and macroscopic properties of matter. These simulations frequently make use of a periodic boundary simulation cell. In a periodic cell, particles can pass through the cell boundary and re-emerge on the opposite side. This allows a finite system to act as if it is infinite, and therefore bulk properties can be investigated without the use of large computationally demanding systems. However, in a periodic system, it is possible for a particle to interact with itself through the periodic boundary. This becomes particularly problematic with long-range interactions, such as the Coulombic r -1 interaction, as well as hydrodynamic interactions which also exhibit r -1 decay. 1 This self-interaction introduces a system-size dependence which can result in the underestimation of diffusivity. 2 We first perform molecular dynamics simulations of NaCl aqueous solution using various water models. Diffusion coefficients and radial distribution functions are subsequently calculated. We then compare our results to those from Cheatham 3 and from experiment. Next, we investigate the relationship between system size and diffusivity. Finally, we look at whether this relationship exist in systems of poly-(hexamethyl-p- terphenylbenzimidazolium), referred to as HMT-PMBI. 1
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Diffusion coefficients and radial distribution functions of NaCl and
HMT-PMBI(Cl−) determined using molecular dynamics
simulations with periodic boundary conditions
Colton Lohn
Department of Physics, Simon Fraser University, Burnaby BC Canada
Abstract
The relationship between system size and diffusivity in a periodic cubic simulation cell was studied.
The diffusion coefficients of Na+ and Cl− in TIP4P/2005 water show significant system-size dependence.
No such system-size effects were found for either the diffusion coefficient of Cl− or the radial distribu-
tion functions for chloride–oxygen, chloride–chloride, chloride–nitrogen, and chloride–carbon in 87.5%
methylated HMT-PMBI at hydration level λ = 16 using TIP4P/2005 water.
Introduction
Molecular dynamics simulations have emerged as an indispensable tool for studying the nanoscopic, meso-
scopic, and macroscopic properties of matter. These simulations frequently make use of a periodic boundary
simulation cell. In a periodic cell, particles can pass through the cell boundary and re-emerge on the opposite
side. This allows a finite system to act as if it is infinite, and therefore bulk properties can be investigated
without the use of large computationally demanding systems. However, in a periodic system, it is possible
for a particle to interact with itself through the periodic boundary. This becomes particularly problematic
with long-range interactions, such as the Coulombic r−1 interaction, as well as hydrodynamic interactions
which also exhibit r−1 decay.1 This self-interaction introduces a system-size dependence which can result in
the underestimation of diffusivity.2
We first perform molecular dynamics simulations of NaCl aqueous solution using various water models.
Diffusion coefficients and radial distribution functions are subsequently calculated. We then compare our
results to those from Cheatham3 and from experiment. Next, we investigate the relationship between system
size and diffusivity. Finally, we look at whether this relationship exist in systems of poly-(hexamethyl-p-
terphenylbenzimidazolium), referred to as HMT-PMBI.
1
Methods
Molecular dynamics simulations were performed in LAMMPS,4 an open source classical molecular dynamics
code. The OPLS force field was used to model interatomic interactions. The functional form of this force
field consists of harmonic bond stretching and angle bending terms, a Fourier series for dihedral energetics,
and Coulomb and Lennard-Jones terms for nonbonded interactions.
Ebonds = kr,i(ri − r0,i)2, (1)
Eangles = kθ,i(θi − θ0,i)2, (2)
Edihedrals =V1,i2
(1 + cos(φi)) +V2,i2
(1− cos(2φi))+
V3,i2
(1 + cos(3φi)) +V4,i2
(1− cos(4φi)),
(3)
Enonbonded =qiqje
2
4πε0rij+ 4εij
[(σijrij
)12
−(σijrij
)6]. (4)
Here, kr,i and kθ,i are force constants, V1,i, V2,i, V3,i, and V4,i are Fourier coefficients, qi and qj are the partial
atomic charges, εij is the well depth for the Lennard-Jones potential, and σij is the finite distance where
the Lennard-Jones potential is zero. The parameters εij and σij were obtained using geometric combination
rules εij = (εiεj)1/2 and σij = (σiσj)
1/2.
NaCl
The diffusion coefficients and radial distribution functions of Na+ and Cl− were investigated using the three-
site water model, SPC/E,5 and the four-site water models, TIP4P-Ew6 and TIP4P/2005.7 Three-site water
models describe three points of interaction, each assigned a charge and Lennard-Jones parameters. It is,
however, common to treat hydrogen as a point particle with no Lennard-Jones potential. Four-site water
models shift the oxygen charge to a new location along the bisector of the H–O–H angle, denoted M in Fig.
1. This is done to improve the charge distribution of the molecule. The geometry for each water model is
summarized in Table 1.
���
ZZZ
O
H H
���
ZZZ
O
MH H
Figure 1: Three-site water model (left) compared to four-site water model (right).
2
SPC/E5 TIP4P-Ew6 TIP4P/20057
O–H distance (A) 1.0 0.9572 0.9572
H–O–H angle 109.47◦ 104.52◦ 104.52◦
O–M distance (A) 0.1250 0.1546
Table 1: Water molecule geometries.
All of these water models are rigid, meaning the O–H distance and H–O–H angle are fixed. Therefore,
all interactions are described by the nonbonded term of the OPLS force field (Eq. 4). The water model
parameters were previously determined to imitate the various properties of water,5–7 while the ion parameters
were optimized to reproduce their experimental hydration free energies and other quantities.8,9 Both water
and ion parameters are summarized in Table 2.
SPC/E5,8 TIP4P-Ew6,8 TIP4P/20057,9
qi
(e)
εi
(kcal/mol)
σi
(A)
qi
(e)
εi
(kcal/mol)
σi
(A)
qi
(e)
εi
(kcal/mol)
σi
(A)
O −0.8476 0.1553 3.166 −1.0484 0.1628 3.1644 −1.1128 0.1852 3.1589
H 0.4238 0 0 0.5242 0 0 0.5564 0 0
Na+ 1 0.3526 2.160 1 0.1684 2.185 1 0.0005 4.07
Cl− −1 0.0128 4.831 −1 0.0117 4.918 −1 0.71 4.02
Table 2: Charges and Lennard-Jones parameters for water and ions.
The simulation cells were initialized using the PACKMOL software package.10 This package returns
the coordinates for the desired atoms and molecules which have been randomly distributed within the cell,
separated by at least the specified distance tolerance. Each of the simulation cells consisted of a cubic periodic
box of side length 40 A containing 1500 water molecules and 27 ion pairs. This gives a concentration of
approximately 1 mol/kg.
In LAMMPS, the direct nonbonded interaction cutoff was set to 9 A. This means that the nonbonded
energy between water molecules and ions was only evaluated when they were separated by distances less than
the cutoff. At distances greater than the cutoff, electrostatic interactions were calculated using the particle-
particle particle-mesh solver with accuracy 5 × 10−4. The SHAKE algorithm11 was used to constrain the
water molecule geometries with a tolerance of 10−5 A. A simulation timestep of 2 fs was used. The target
temperature was set to 298 K, while the target pressure was set to 0.9869 atm. Temperature and pressure
time constants were 200 fs and 2000 fs, respectively. The system energies were minimized using the conjugate
gradient method for 1000 iterations. The systems were subsequently equilibrated at constant volume (NVT)
for 40 ps and then at constant pressure (NPT) for another 40 ps. Finally, simulations were performed at
3
constant volume (NVT) for 10 ns. A snapshot of the simulation cell is shown in Fig. 2a. A performance of
approximately 39 ns/day was acheived with 16 CPU cores on the Cedar cluster at Simon Fraser University.
For a 3 dimensional system, the diffusion coefficient is related to the mean squared displacement by,
D = limt→∞
∂
∂t
〈|r(t)− r0|2〉6
. (5)
Therefore, the diffusion coefficient can be determined using least squares fitting to estimate the slope of
the mean squared displacement over time. For a single particle, very long simulation times are required to
obtain accurate results; however, averaging over an ensemble of identical particles significantly improves the
statistics allowing for shorter run times.12
The systems were analyzed using VMD,13 a visualization program for large molecular systems. The
diffusion coefficient tool14 was used to calculate the mean squared displacements of Na+ and Cl− for a 100
ps lag time with a 1 ps timestep, shown in Fig. 3. The associated instantaneous diffusion coefficients are
displayed in Fig. 4. To reproduce the methods used by Cheatham,3 a straight-line fixed at the origin was
fit by least squares to the plots of mean squared displacement for a 10 ps lag time. The fits for TIP4P/2005
are shown in Fig. 5. The diffusion coefficients were subsequently calculated and the results are summarized