MOLECULAR MODELING OF MATTER :FROM REALISTIC HAMILTONIANS TO SIMPLE MODELS AND THEIR APPLICATIONS

Ivo NEZBEDAE. Hala Lab. of Thermodynamics, Acad. Sci., Prague, Czech Rep.andDept. of Theoret. Physics, Charles University, Prague, Czech Rep.

COLLABORATORS:J. Kolafa, M. Lisal, M. Predota, L. Vlcek; Acad. Sci., PragueA. A. Chialvo; Oak Ridge Natl. Lab., Oak RidgeP. T. Cummings; Vanderbilt Univ., NashvilleM. Kettler; Univ. of Leipzig, Leipzig

SUPPORT:Grant Agency of the Czech RepublicGrant Agency of the Academy of Sciences

Roughly speaking, there are

TWO WAYS OF MOLECULAR MODELING OF MATTER:

SPECULATIVE: MATHEMATICAL:

to model INTUITIVELY certain features(e.g. to capture the essence of intermolecularinteractions) or a specific property (e.g. freeenergy models)

starting from a certain mathematicaldescription of the problem (e.g. froma realistic Hamiltonian), well-definedapproximations are introduced in orderto facilitate finding a solution.

Although both methods may end up with the same model,there is a substantial difference between them.

WAYS OF MODELING…contd.

SPECULATIVE: MATHEMATICAL:

EXAMPLE 1: van der Waals EOS

Molecules are not immaterial points butOBJECTS with their own impenetrablevolume

molecules may be viewed as HARD SPHERES

EXAMPLE 1: Theory of simple fluids

The simple fluid is defined by a REALISTICpotential, u (r). Too complex for theory.

1. The structure is determined by short-range REPULSIVE interactions.

u(r)=u(r)rep + Δu(r)

and use a perturbation expansion2. To solve the problem for the reference, u(r)rep, is still too difficult

properties of the u(r)rep fluid are mapped onto those of hard spheres: Xrep Xhard sphere

zPP sphere hard)(

zPzPP sphere hardrep )()(

WAYS OF MODELING…contd.

SPECULATIVE: MATHEMATICAL:

PROBLEMS:(1) How to determine Δz?(2) How to refine/improve the approach?(3) How this is related to reality?

Hard sphere model is a part of the well-defined schemeThe correction term iswell-defined

There are evident ways to improve performance of the method

HOW TO REACH THE GOAL?Start from the best realistic potential models and use a perturbation expansion.

SIMPLE (THEORETICALLY FOOTED) MODELS ARE AN INDISPENSABLE PART OF THIS SCHEME.

ULTIMATE GOAL OF THE PROJECT:Using a molecular-based theory, to develop workable (and reliable) expressions for the thermodynamic properties of fluids

PERTURBATION EXPANSION – general considerations Given an intermolecular pair potential u, the perturbation expansion methodproceeds as follows:

(1) u is first decomposed into a reference part, uref, and a perturbation part, upert:

u = uref + upert

The decomposition is not unique and is dictated by both physical and mathematical considerations. This is the crucial step of the method that determines convergence (physical considerations) and feasibility (mathematical considerations) of the expansion.

(2) The properties of the reference system must be estimated accurately and relatively simply so that the evaluation of the perturbation terms is feasible.

(3) Finally, property X of the original system is then estimated as

X = Xref + X where X denotes the contribution that has its origin in the perturbation potential upert.

STEP 1: Separation of the total u into a reference part and a perturbation part,

u = uref + upert

THIS PROBLEM SEEMS TO HAVE BEEN SOLVED DURING THE LAST DECADE AND THE RESULTS MAY BE SUMMARIZED AS FOLLOWS:

 Regardless of temperature and density, the effect of the long-range forces on the spatial arrangement of the molecules of PURE fluids

is very small. Specifically: (1) The structure of both polar and associating realistic fluids and their short- range counterparts, described by the set of the site-site correlation functions, is very similar (nearly identical). (2) The thermodynamic properties of realistic fluids are very well estimated by those of suitable short-range models;

(3) The long-range forces affect only details of the orientational correlations.

THE REFERENCE MODEL IS A SHORT-RANGE FLUID:

uref = ushort-range model

STEP 2: Estimate the properties of the short-range reference accurately (and relatively simply) in a CLOSED form

HOW TO ACCOMPLISH THIS STEP ?

HINT: Recall theories of simple fluids:

uLJ = usoft spheres + Δu (decomposition into ‘ref’ and ‘pert’ parts)

XLJ = Xsoft spheres + ΔX

XHARD SPHERES + ΔX

SOLUTION: Find a simple model (called primitive model) that (i) approximates reasonably well the STRUCTURE

of the short-range reference, and (ii) is amenable to theoretical treatment

Re SUBSTEP (1): Early (intuitive/empirical) attempts(for associating fluids)Ben-Naim, 1971; M-B model of water (2D)Dahl, Andersen, 1983; double SW model of waterBol, 1982; 4-site model of waterSmith, Nezbeda, 1984; 2-site model of associated fluidsNezbeda, et al., 1987, 1991, 1997; models of water,

methanol, ammoniaKolafa, Nezbeda, 1995; hard tetrahedron model of waterNezbeda, Slovak, 1997; extended primitive models of water

PROBLEM:These models capture QUALITATIVELY the main featuresof real associating fluids, BUTthey are not linked to any realistic interaction potential model.

SUBSTEPS OF STEP 2:(1) construct a primitive model(2) apply (develop) theory to get its properties

GOAL 1: Given a short-range REALISTIC (parent) site-site potential model,

develop a methodology to construct from ‘FIRST PRINCIPLES’ a simple (primitive) model which reproduces the structural properties of the parent model.

IDEA: Use the geometry (arrangement of the interaction sites) of the parent model, and mimic short-range REPULSIONS by a HARD-SPHERE interaction,

ji

ijij ruu,

rangeshort )()2,1( ,

Example:carbondioxide

and short-range ATTRACTIONS by a SQUARE-WELL interaction.

OO

1.163Å

CCO O

1.163 A

rH-M

uH-M

0rH-H rM-M

uH-H

0R

H

H

O

M eeMe

M

rM-M

rH-M

hard-sphere repulsion

square-wellattraction

uM-M(hydrogen bonding)

M

M

O

H eeMe

H ee

H

H

O

M

M

PROBLEM:PROBLEM: We need to specify the parameters of interaction 1. HARD CORES (size of the molecule) 2. STRENGTH AND RANGE OF ATTRACTION

SOLUTION: Use the reference molecular fluid defined by the average site-site Boltzmann factors,

and apply then the hybrid Barker-Henderson theory (i.e. WCA+HB) to get effective HARD CORES (diameters dij):

constr

ijijij

ijBij

ij

ddure

reTkru

)2()1(]exp[)(

)(ln)(

RAM

r

uij/kT

repulsive part: uijrep

attractive part: uijatr

full angle averaged potential: uijRAM

rmin

0

rrudr

repijij dexp1

min

0

1-1. HOW (to set hard cores): ???FACTS: Because of strong cooperativity, site-site interactions cannot be treated

independently.HINT: Recall successful perturbation theories of molecular fluids (e.g. RAM) that use sphericalized effective site-site potentials and which are known to produce quite accurate site-site correlation functions.

O

HH 109 .47°

1.0Å

O

X

HH

X109 .47°

OO

1.163Å

C CO O

1.163 A

EXAMPLES:

SPC water

OPLS methanol

carbon dioxide

1-2. HOW (to set the strength and range of attractive interaction): ???HINT: Make use of (i) various constraints, e.g. that no hydrogen site can form no more than one hydrogen bond.

This is purely geometrical problem. For instance, for OPLS methanol we get for the upper limit of the range, λ, the relation:

The upper limit is used for all models.

(ii) the known facts on dimer, e.g. that for carbon dioxide the stable configuration is T-shaped.

2OHOOOHOO 3 ldld

SELECTED RESULTS (OPLS methanol):

3 4 5 6 7 8

gOO

0

1

2

3

4

5

r[Å]4 6 8 10

gMeMe

0

1

2

3

r[Å]

3 4 5 6 7 8

gOMe

0

1

2

filled circles: OPLS methanolsolid line: primitive model

number of H-bonds per molecule

0 1 2 3 4

pe

rce

nta

ge

dis

trib

utio

n

0

20

40

60

80

Average bonding angles θ and φ: θ φ

prim. model 147 114OPLS model 156 113

O

HH 109 .47°

1.0Å

106.4° HHH

N

1.01Å

waterTIP5P

waterTIP4P

waterSPC/E

ammoniaOPLS

O

HH 104.52°

0.957

2Å

MM

0.7Å

109.47°

N

XH

H

XX

H

Y

O

X

HH

X

O

X

HH

XM

OX

HH

X

Y Y

MM

ethanolOPLS

C H2

H

O

108.5°

C H3

108°

1.43Å

1.53

Å

O

HH 104.52°

0.957

2Å

M

C 1

H

O

X

C 2

3 5 7 9

gNN

0

1

2

3

2 4 6 8

gNH

0.0

0.5

1.0

1.5

r[Å]

2 4 6 8

gHH

0.0

0.5

1.0

AMMONIA

circles ….. realitylines ……. prim. model

3 4 5 6 7

gOO

0

2

4

6

1 2 3 4 5 6 7

gOH

0

2

4

6

8

10

r[Å]

2 3 4 5 6 7

gHH

0

2

4

6

r[Å]

3 4 5 6 7 8 9

gOC2

0.0

0.5

1.0

1.5

3 4 5 6 7 8 9

gOC1

0.0

0.5

1.0

1.5

2.0

r[Å]

ETHANOL

3 4 5 6 7

gOO

0

1

2

3

4

5

1 2 3 4 5 6 7

gOH

0

1

2

3

4

r[Å]

2 3 4 5 6 7

gHH

0

1

2

SPC/E WATER

circles ….. realitylines ……. prim. model

POLAR FLUIDSAlthough they do not form hydrogen bonds, the same methodology can be applied also to them.

OO

1.163Å

CCO O

1.163Å

92 .0°

1.34Å

HH

S

H H

X X

S

92 .0°

C

M eM e

O

XCM eM e

121.35°

O

1.2 2

3Å

1.572Å

ACETONEsite-site correlation functions

ACETONEsite-site correlation functions

APPLICATIONS (of primitive models):

1. As a reference in perturbed equations for the thermodynamic properties of

REAL fluids [STEP 3 of the above scheme: X = Xref + X ].

Example: equation of state for water [Nezbeda & Weingerl, 2001]

Projects under way: equations of state for METHANOL, ETHANOL, AMMONIA, CARBON DIOXIDE

2. Used in molecular simulations to understand basic mechanism governing the behavior of fluids. Examples: (i) Hydration of inerts and lower alkanes; entropy/enthalpy driven changes [Predota & Nezbeda, 1999, 2002; Vlcek & Nezbeda, 2002] (ii) Solvation of the interaction sites of water [Predota, Ben-Naim & Nezbeda, 2003] (iii) Mixtures: water-alcohols, water-carbon dioxide,… (iv) Preferential solvation in mixed (e.g. water-methanol) solvents (v) Aqueous solutions of polymers (with EXPLICIT solvent) (vi) Clustering/Condensation (i.e. nucleation) (vii) Water at interface

xMeOH

0.0 0.2 0.4 0.6 0.8 1.0

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

vEx

[cm3mol-1]

xMeOH

0.0 0.2 0.4 0.6 0.8 1.0

hEx

-1000

-800

-600

-400

-200

0

-1000

-800

-600

-400

-200

0

Excess thermo-properties

WATER-METHANOLat ambient conditions

circles….. exptl. datasquares… prim. model

Excess thermo-properties

CO_2 - WATERat supercritical CO_2 conditions

circles….. exptl. datasquares… prim. model

xCO2

0.0 0.2 0.4 0.6 0.8 1.0

hEx, gEx

[J.mol-1]

0

500

1000

1500

2000

0

500

1000

1500

2000

xCO2

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

0

2

4

6

8

vEx

[cm3mol-1]

THANK YOUfor your attention