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