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Characterizing structures of solid, liquid, and gas phases from simulations
• Solids are dense and have ordered structures. Molecules have vibrational and
maybe some rotational degrees of freedom in the lattice;
• Liquids are dense and have disordered structure. Molecules have vibrational
motion about their positions and hindered translational and rotational degrees of
freedom;
• Gases are dilute and disordered;
• We characterize the density in a phase by the number density, ρ = N/V 1
Solid Liquid Gas
ρsolid > ρliquid >> ρgas
2
Temperature / K
(state) E(K+U) /
kcal·mol-1
E(U) /
kcal·mol-1
ρbulk /
g·cm-3
d(N2∙∙∙N2) /
Å
20 (solid) -2.14 -2.24 0.431 3.39
100 (liquid) -0.77 -1.24 0.296 3.85
298 (gas) +1.51 -0.03 0.00141 22.9
Simulations of solid, liquid, and gas phases with molecular dynamics
Ala Nissila et al. Adv. Phys. 2002, 51, 949.
Energy and other characteristics of solid, liquid, and gas phase N2 from simulation
Schematic representation of the potential energy surface for solid from simulation
Potential
Energy
x y
r
Are other molecules
present
Microscopic structure and spatial correlations between molecules
Are there spatial correlations between the positions of the particles?
If there is a particle at some origin, what is the probability of observing a second
particle at a distance r? );0();( 2212 rPP rr
Are the placements of the two particles independent?
Spatial correlations extend to large distances in the solid phase
r1 r'2
r1 r'2
3
r''2
Yes
No
r''2
…
2r'2
4 4
r2 – r1
dr dV
Local density and the radial distribution function
bulkbulk
local dVdNg
/)()( 1212
122rrrr
rr
dV
dNlocal
)()( 12
12rr
rr
Local density at a r2 from an atom at r1. Written
in terms of the Cartesian volume element
Pair distribution function g(r2-r1) is the
ratio of local density to bulk density
Simplifying assumptions:
• In a homogenous fluid phase, all
molecules are equivalent, i.e., on
average, all molecules have the
same local environment;
• The distribution of molecules
depends only on the distance
between the two and not on their
orientation in the system;
• Distribution of molecules are
expressed in polar coordinates.
bulkbulk
local drrrdNrrg
24/)()(
Radial distribution function (RDF)
5 5
r1
r2 z
x 0
y
r
θ
1
2
bulkbulk
local dVdNg
/)()( 1212
122rrrr
rr
bulkbulk
local drrrdNrrg
24/)()(
Simplification of the radial distribution function
Simplifying assumptions:
• The absolute location of r1 is not important, but rather (r2 – r1) determines the
presence of the second molecule;
• For spherically symmetric systems, only r = |r2 – r1| is effects the distribution
• We integrate over the r1, θ, and .
Distribution of molecules in simulation
in terms of absolute coordinate system Distribution of molecules in simulation
relative to each others radial separation
6
ρi,1 → dN(ri,1)/dV
ρ i,2 → dN(ri,2)//dV
ρ i,3 → dN(ri,3) /dV
ρ j,1 → dN(rj,1) /dV
ρ j,2 → dN(rj,2) /dV
ρ j,3 → dN(rj,3) /dV
ρ k,1 → dN(rk,1) /dV
ρ k,2 → dN(rk,2) /dV
ρ k,3 → dN(rk,3) /dV
Average density at different distances at time t
• At each snapshot, we count neighboring molecules at all distances;
• Local density values are averages over all molecules in the system;
• Periodic boundary conditions must be considered. 6
i
j k
N
dVtrdNN
tr1
/,1
,
• For molecule i (1 to N) count the of
neighbors between r and r + Δr for a
snapshot at time t (binning of the
molecules)
Gather more data by averaging the radial distribution function over time
7 2Δt 3Δt 4Δt 5Δt 6Δt jΔt … Δt 0
RDF data saved along a trajectory: every 5 time steps
At time t
tN
t
rN
r1
),(1
)(
N
dVtrdNN
tr1
/,1
,
• Repeat the procedure every
nΔt time steps
•Accumulate statistics for all
molecules and all distances over
all times in the trajectory
At time 5Δt
8
Asymptotic behavior
• If r → 0 then g(r) → 0
• If r → then g(r) → 1
The radial distribution function for liquid Krypton at 80 K
ε = 1.3635 kJ/mol
= 3.827 Å
• The distance of the first peak in the RDF coincides with the distance of the
minimum in the intermolecular potential! (Lennard-Jones in this case);
• The next peaks in the RDF are not exactly at distances which are multiples of the
first peak distance (why?)
• Correlations become weaker with distance
• There are regions with less density than the bulk (why?)
Gas: no long- or mid-range order
Liquid: no long-range order
Solid: long-range order
Radial distribution function of phases of N2
9
Radial distribution function of liquid and solid phases can be experimentally
determined from the Fourier transform of the X-ray scattering intensity
10
FT
Calculating the coordination numbers for molecules: local environment in
solutions
0
24)( drrrgN bulk
11
Integration of the RDF over the entire range
gives the number o molecules in the system
(from the definition of g(r))
RDF plot for Lennard-Jones Fluid
-0.5
0
0.5
1
1.5
2
2.5
3
0.0 1.0 2.0 3.0 4.0
r (Angstrom)
g2
(r)
r / Å
g(r)
rmin
min0
2min 4)()( r
bulkcoord drrrgrN Integration of the RDN up to the first
minimum gives the coordinate number
We can gain information
about solution structure
RDF of NaCl in aqueous solution
Simulation of a NaCl solution
Red: oxygen
White: hydrogen
Light blue: chlorine
Dark blue: sodium 12
How can we characterize the structure of a NaCl solution?
BBAA VXVXsolutionV )(
Thermodynamic properties of an ideal
solution are determined by:
Information obtained on the solution includes:
• The microscopic nature of solvation;
• Number of solvent molecules in the solute
coordination sphere;
• Solute – solute interactions in the solution
• Volume of solvation;
• Energy, enthalpy, and free energy of solvation
(which gives the solubility);
• Solvation dynamics
• …
BBAA MXMXsolutionM )(
Radial distribution functions for Na+ and Cl- in aqueous solution
13
• How do ions interact with the water solvent?
• Are the cation and anion loosely associated
or do they form ion-pairs?
• Many RDF pairs can be defined:
-gNa-Na(rNa-Na) gNa-Cl(rNa-Cl)
gNa-O(rNa-OW) gNa-HW(rNa-HW)
-gOW-OW(rOW-OW) gOW-HW(rOW-HW)
gOW-Cl(rOW-Cl)
-gCl-Cl(rCl-Cl) gCl-HW(rCl-HW)
-gHW-HW(rHW-HW)
)OW(
4/)(
)OW(
)()(
2OWOWNa
OWNabulkbulk
drrrdNrrg
)Na(
4/)(
)Na(
)()(
2NaNa-OW
Na-OW
bulkbulk
drrrdNrrg
Number of water OW atoms
surrounding each Na+ ions
Number of Na+ ions around
each water OW atom
Bulk density with respect to the specific partner
Radial distribution functions for Na+ and Cl− in aqueous solution
Atom
pair σ / Å
O-O
Na-O
Cl-O
3.166
3.248
3.791
The solution has considerable structure: ions have more than one solvation shell
of water 14
54 NaCl formula units + 804 Water molecules
15
N
N
kTU
NNN dddQ
edddP
N
rrrrrrrrrrrr
......),...,,( 21
/),...,,(
2121
21
Statistical mechanics of phase structure
s
N
Nss
kTU
sss dddQ
dddedddP
N
rrrrrr
rrrrrr
rr,r
......
...),...,,( 21
21
/),...,(
2121
21
The spatial distribution function in a fluid:
The reduced distribution function in a fluid:
probability of finding molecule in volume element dr1 at r1, molecule β at
volume element dr2 at r2, …, molecule ν at volume element drN at rN is,
probability of finding molecule in volume element dr1 at r1, molecule β at
volume element dr2 at r2, …, molecule σ at volume element drs at rs,
regardless of where the other σ + 1, σ + 2, …, N molecules are.
molecule α
dr1 r1
molecule β
dr2 r2
dr3 r3
molecule γ
P3(r1,r2,r3) ),,()!3(
!),,( 32133213 rrrrrr P
N
N
“Specific” distribution function “Generic” distribution function
Characterizing spatial correlations in molecular systems
Probability of simultaneously
seeing molecule:
α at r1 + dr1
β at r2 + dr2
γ at r3 + dr3
Probability of simultaneously
seeing any three molecules at:
r1 + dr1;
r2 + dr2;
r3 + dr3 16
any molecule
dr1 r1
any molecule
dr2 r2
dr3 r3
any molecule
17
21
43
/),...,(
21212
...)1(),(
21
rrrrr
rrrr
rr,r
ddQ
dddeNNdd
N
N
kTU N
Nji
ijNN ruU1
21 )(),...,,( rrr
Statistical mechanics and the radial distribution function
Assumption of pairwise additivity is that the total potential energy can be broken
to a sum of two-atom interactions
2
Pairwise additivity should not be taken for
granted and must be validated with quantum
chemical calculations and by comparison of
computations with experimental results!
1
- Would the interaction of water 1 with an ion
affect its interaction with water 2?
18
0
2
23 4),,()(
2drrTrgru
kTNkT
E bulk
220
22
)(4),,()(6
bulkbulkbulk
bulk TBdrrTrgrurkTkT
p
Knowledge of the RDF allows us to calculate thermodynamic properties of non-
ideal fluids
Energy
Pressure
(equation of state;
virial expansion)
N
N
kTu
N
N
kTU
Q
dddeV
Q
dddeNN
N
Vg
ji ji
N
rrr
rrrrr
r,r
rr,r
...
...)1(),(
43
/)(
2
43
/),...,(
2
2
212
,
21
Statistical mechanics and the radial distribution function
]),(),(1[),,( 22
1/)( TrgTrgeTrg
bulkbulkkTru
bulk
The RDF can be expressed as a series expansion in terms of density
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