Intensive Course on Theoretical Chemistry and Computational Modelling

Universities: Perugia, Autonoma de Madrid, Paul Sabatier and Porto

Liquid State and Phase Transitions

Fernando M.S. Silva Fernandes

Department of Chemistry and Biochemistry

Centre for Molecular Sciences and Materials

Faculty of Sciences, University of Lisbon, Portugal

“Tamed, but not entirely domesticated, analytically intractable yet not to be coerced by approximation, liquid state physics has for a long time been the enfant terrible

of the phase diagram”

(in Introduction to Liquid State Physics, Clive Croxton, J.Wiley

& Sons, 1975)

After 35 years, multifarious difficulties still persist...

Liquids, contrary to solids and gases, do not have an ideal reference model. As such, a molecular theory for liquid sate or phase transitions usually starts from rigorous correlation and/or partition functions which, sooner or later, have to be tamed by analytical approximations and computer simulation techniques.

Real Liquids Model LiquidsSet up Models

Perform Experiments

Carry OutSimulations

Statistical MechanicalTheories

ExperimentalResults

Results forModels(Exact)

TheoreticalPredictions

Compare Compare

ModelsValidation

TheoriesValidation

Mechanical,dielectric and transport properties; Free Energies and chemicalpotentials; Phase transitions; Structure; Motion at molecular level; Predictionof properties not easily observed in laboratory.

Phase Diagram and Projections The Liquid State Pocket

Pressure – density projection

Kinetic and Potential Energies

Gibbs Energy, Entropy and Pressure

p T

G GS VT p

Max Born – Tisza square

First-order and Continuous Phase Transitions

Fluid - Liquid Transitions

12

12

L V

c

L Vc

c

TBT

TAT

; ,

1

XX

TT

SC T X V pT

p

Cp for butane

Ferromagnetic spontaneous magnetization

; ,C XX

TT

SM T T C T X H MT

MH

Critical Exponents for fluid and magnetic systems

C Ct T T / T

Universality

Reduced Units

* 3

*

*

3*

1/ 2*

2

*

Density

Temperature

Energy

Pressure

Time

Force

Bk TT

EE

pp

t tm

ff

The sui generis critical point!

Continuous (order-disorder) phase transitions

NH4

Cl TC

= 243 K beta-brass TC

= 733 K

Summary

•

A phase transition is signaled by singularities in the free energies (Helmholtz or Gibbs) or in their derivatives.

•

If the free energies are continuous but first derivatives have finite discontinuities the transition is termed first-

order.

•

If the first derivatives are continuous but second derivatives are discontinuous or divergent (infinite) the transition is called higher order, continuous or critical.

Statistical MechanicsTheory

Fundamentals

• In an equilibrium state the macroscopic properties are invariant on time.

•

During its time evolution the system goes through very many microstates, that is, it is subjected to thermal fluctuations.

•

There are very many microstates consistent with a given thermodynamic state (macrostate) identified by constraint variables.

•

The complete specification of a thermodynamic state leaves the microstates undefined.

Trajectory in state space (each box represents a different microstate)

1 2 1 2, ,..., , , ,..., , ,N N N N N NN N d dp r r r p p p r p r r p

Classical microstates (phase space). Solutions of classical equations of motion:

;N t r

Quantum microstates (Hilbert’s space). Solutions of quantum equations of motion (Schrödinger, Heisenberg or Dirac):

Dynamical (or mechanical) microscopic properties are defined for

each microstate: instantaneous energy, temperature, pressure, etc.

Statistical mechanics aims to establish a bridge between the macroscopic observable properties and the underlying microscopic

properties, by averaging the dynamical properties over the microstates consistent with preset thermodynamic constraints. Consider time and ensemble averages:

0

1 M

obs tmG G GM

2

1 2

NNi

insti i

E t Um

p r

( )obsnG G ergodic hypothesisM

obs enG P G G

Typical Ensembles

Ensemble Constraints

Microcanonical E, V, N Canonical T, V, N Isothermal - Isobaric T, p, N Grand - Canonical T, μ, V (μ is the chemical potential)

Transformation between Ensembles Legendre and Laplace Transforms

Trajectory in state space with each box representing a different state

13, , ! ,N N N N NE V N N h E H d d r p r p

, , , , /E V N E V N E

13, , ! ,N N N N NE V N N h E H d d r p r p

Constraints : E, V, N

(Phase space Volume)

(derivative of unit step function)

(Phase space density)

Generalized Boltzmann’s Equation

n Sn Xn dn Zn Yn

1 S1 (E,V,N) E drN dpN E

2 S2 (H,p,N) H - pV drN dpN dV H

3 S3 (L,V,) L + N drN dpN L

4 S4 (R,p, ) R - pV + N drN dpN dV R

S kn n/ ln n n n nZ X d H

n n nY / n n n nZ X d H

13N h N!

13N h N!

13

0 N h NN !

13

0 N h NN !

High Dimensional Geometry (sphere and spherical shell volumes)

ln , , ln , ,

nn

V rV A r nV r

S k E V N k E V N

In an imaginary world of high dimensionality there would be an automatic and perpetual potato famine, for the skin of a potato would occupy essentially its entire volume!

(H.B. Callen

in Thermodynamics and An Introduction to Thermostatistics)

Canonical Ensemble. Probability and Partition Functions

13, , ! exp ,N N N N NQ T V N N h H d d r p r p

exp ,, ,

, ,

N N N NH d dP T V N

Q T V N

r p r p

, , exp ii

Q T V N E

exp

, ,exp

i

ii

EP T V N

E

Thermodynamic Properties from Partition Function

, ,ln , ,

A T V NQ T V N

kT

,

ln , ,T N

p kT Q T V NV

,

ln , ,V N

S kT Q T V NT

,

ln , ,T V

kT Q T V NN

Structure and Correlation

24n rVg r

N r r

r g r

Radial distribution function (pair correlation function; pair distribution function)

Solid and Liquid RDF’s

Pair Correlation Function and Thermodynamics

2

0

3

0

3 42 2

46

NE NkT u r g r r dr

du rNkT Np g r r drV V dr

Potential of mean force

1 2

3 1,

1 3

1 21

... / exp...

... exp

ln , ln

NNfixed

N

d d dU d Ud Ud d d U

dkT g w r kT g rd

w r potential of mean force

r r

r r rr

r r r

r rr

From Introduction to Modern Statistical Mechanics, D.Chandler

Hard Spheres

Theories for g(r)

0lim 0

1

w r

u r

g r e

w r u r w r

w r

g r e O

For higher densities we have to deal with the deviations of ∆w(r)

from zero. In the most successful approaches ∆w(r)

is estimated in terms of ρg(r) and u(r),

yelding integral equations for g(r)

that are essentially mean field theories.

Van der Waals Theory. A mean field theory

Van der Waals Theory

The theory predicts an order parameter scaling factor β=0.5

WCA theory, Science, 220(1983)787

WCA theory

WCA theory

Weeks, Chandler and Andersen Theory (WCA)

Ferromagnetic spontaneous magnetization

; ,C XX

TT

SM T T C T X H MT

MH

Ising

Model

1 2

1 2

1

1

1 1

1

; 1

, , ... exp

, ... exp ; 0, /

N

N

N

i j iij i

N

ii

N

i j is s s ij i

i js s s ij

Hamiltonian J s s H s s

M s

Q N H J s s H s

Q K N K s s H K J kT

Renormalization Group Theory

/ 2

1/ 2 1/8

1

1/ 4

0

0

, , / 2 ( )

3 ln cosh 4 /8

ln 2 cosh 2 cosh 4

ln1 cosh exp 8 / 34

1 1 ln 2exp 2 / 3 cosh 4 / 32 2

N

K K

K

Q K N f K Q K N Kadanoff transformation

K K K J kT

g K g K K K

Q Ng K

K K

g K g K K K

0

3 ln cosh 48

0.50698 ( ) 0.44069

C

K

K K K

C C

c C

K K

K RG K exact

Renormalization Group Theory Ising Model : T >=1.22TC

Adapted

from Statistical Mechanics of Phase Transitions, J.M.Yeomans

Renormalization Group Theory Ising Model : Critical Temperature

Adapted

from Statistical Mechanics of Phase Transitions, J.M.Yeomans

Scale invariance

Renormalization Group Theory Ising Model : T <=0.99TC

Adapted

from Statistical Mechanics of Phase Transitions, J.M.Yeomans

Computer Simulation

Potential Energy and Force Curves

Force Fields and Total Intermolecular Potential Energy

12 6

6 80

:

4

:

exp4

:

ijij ij

i j ij ijij ij ij

ij ij ij

N

iji j

Lennard Jones Potential

u rr r

Born Mayer Huggins Potentialq q c d

u r b B rr r r

Total Intermolecular Potential Energy

U u r

Polyatomics

Dynamics of a system of polyatomic molecules:•

Intramolecular: bond stretching, angle bending, torsion alongbonds and non-bonding interactions.

•

Intermolecular: translation and rotation.•

Translation as in monoatomics

(motion of mass centre).

•

Rotation as rigid rotors (Euler angles or equivalent).•

Translation and rotation: total of 6 degrees of freedom.

•

Intramolecular: as a first approximation (molecularmechanics) treated classically with force fields similar to

the

intermolecular ones

A Typical Force Field Energy

2

2

,0 ,0

12 6

1 1 0

1 cos2 2 2

44

N i i ni i i i

bonds angles torsions

N Nij ij i j

iji j i ij ij ij

k a vU l l n

q qr r r

r

Common Force Fields

Molecular Dynamics Method

Starting from a set of initial positions and velocities for the N molecules in the model, and taking into account the forces between molecules, the motion equations for each molecule are numerically integrated to produce a trajectory of the system in phase space.

2

2

'

ii i

d tt m

d tNewton s equation for molecule i

r

F

MD or MC box and fcc unit cell

Boundary Conditions

Boundary Conditions, Minimum Image Convention and Truncation

Thermal Properties

V3VTNk

p

:essurePrNk3

mT

:eTemperatur

rum21E

:EnergyTotal

N

jiijij

B

B

N

1i

2ii

N

1i

N

jiij

2ii

rF

v

v

Argon Equation of State (dens=0.6; 1977)

Phase Diagram LJ-Argon (1969)

Heat Capacity, Free Energy and Structure

22

2

1 0 0

2

3 ( )2

ln exp / ( )

ln /

( )4

v

exc test

U UC k heat capacity

kT

kT u kT excess chemical potential

A A A kT U kT

A Helmohltz free energy

n rVg r radial distribution functionN r r

RDF’s for Lennard-Jonesium

Dynamic and Spectroscopic Properties

0

2

0

0 ( )

1 ( )3

0 ( )

lim ( )6

1 . 0 ( )

i i

i i

t

x x

z t t velocity autocorrelation function

D z t dt self diffusion cofficient

msd t t mean squaredisplacement

msd tD self diffusion coefficient

t

I t I dt electrical conductivitykTV

v v

r r

1

0

( )

1 . 0 cos

; ( ; )( )

Nx

x i ii

I t q v t elecric current

I f t f t dt

f dipole moment polarizability tensor IR Ramanf nuclear angular momentum NMR

Mean Square Displacement, KCl clusters (before and after freezing)

0

2

4

6

8

10 m

ean

squa

re d

ispl

acem

nt /

(ang

st.2

)

0 1 2 3 4 time / ps

liquid773 K

solid769 K

Velocity autocorrelation functions

Velocity acf’s and Fourier spectra

Simulated velocity acf and power spectra

Phonon spectra

Vibratory and diffusive modes in liquids

Power Spectra of KCl clusters

Phase Transitions – KCl Clusters

-680

-670

-660

-650

-640

-630

-620

-610

Con

fig. E

nerg

y / k

J.m

ol-1

0 200 400 600 800 1000 1200 Temperature / k

heating

cooling

IonicCluster64 ions

0.4

0.5

0.6

Rad

ius

of G

yrat

ion

/ box

uni

ts

0 200 400 600 800 1000 1200 Temperature / K

heating

cooling

IonicCluster 64 ions

Radial Density Functions, KCl clusters

0

20

40

60

80

100

rdf (

arbi

trary

uni

ts)

0 2 4 6 8 10 r / A

- +

+ +

heatingT=851Kbeforemelting

0

10

20

30

40

50

60

70

rdf (

arbi

trary

uni

ts)

0 2 4 6 8 10 r / A

- +

+ +

heatingT=861 Kaftermelting

Melting, Freezing and Glass Transitions, KCl clusters

-700

-680

-660

-640

-620

-600

-580

Tota

l Ene

rgy

/kJ.

mol

-1

0 200 400 600 800 1000 1200 temperature / K

slow heating slow cooling fast cooling

512 IonsCluster

Glasses, Solids and Liquids

Glass transition

Monte Carlo

Simulation

Basics

2

1

, exp ,

exp ,

2

( ) :

exp32 exp

:

N N N N N N

N N N N

NNi

i

N N N

N N

idea

H H d dE

H d d

H U Hamiltonian of the systemm

Integrating out analytically the kinetic ideal part

U U dNkTEU d

In general

G G

p r p r p r

p r p r

p r

r r r

r r

exp

exp

N N N

l excess ideal N N

G G dG G

G d

r r r

r r

Canonical Configurational Average

... ( )exp ( )( )

... exp ( )

N N NN

N N

G U dG

U d

r r rr

r r

exp( )

... exp

NN

N N

U

U d

rr

r r

( ) ...N N N NG G d r r r r

Discretized Canonical Configurational Average

1

exp ( )( )

exp ( )M

U

U

1

1

( ) exp ( )( )

exp ( )

M

NM

G UG

U

r

1

( )M

NG G

r

Metropolis at canonical Monte Carlo

•

Choose M configurations with probability proportional to exp[-βU(rN)]

•

Weight them evenly:

1

1( ) ( )M

NG GM

r

How to choose?

Choosing configurations

•

N molecules enclosed in a cubic box of side L at any initial configuration

•

There exists a process (to be defined ahead) to move the molecules, generating M configurations (A,B,C,...) consistent with the available configurational space.

•

Transitions from state A to any other possible state, and vice-versa.

•

[A]=nA

/M; [B]=nB

/M (nA

,,

nB

–

number of states A and B) can be interpreted as “generalized concentrations of species A and B or the probabilities

of states A and B”:

1AB

BA

A B reversible process of st order

Choosing configurations

•

After many transitions a dynamic equilibrium should be reached:

AB BAeq eqA B

principleof microscopic reversibility

expexp

exp exp

AB

BAeq

BA

U BB BA A U A

U B U A U

General Scheme

underlying (trial) matrix; acceptance matrix1

/

1

min 1, /

' : ( ) 1/

B A

R

A B A B acc A Bacc

A B A B if B A

A B B A if B A

A A A B

acc A B B A

Metropolis first choice symmetric N

MC trial move and acceptance or rejection decision

Phenol adsorption on gold electrodes

R.S.Neves;A.J.Motheo;F.M.S.S.Fernandes;R.P.S.Fartaria,J.Braz.Chem.

Soc., 15(2004)224-231.

1-decanethiol adsorption on gold electrodes

R.P.S.Fartaria;F.F.M.Freitas;F.M.S.S.Fernandes,J.Electroan.Chem.,574(2005)321-331.

Monte Carlo steps by Gibbs Ensemble method: particle displacement, volume rearrengement

and particle interchange

Gibbs Esemble: particle displacement and volume rearrangement Panagiotopoulos et al, Mol.Phys.,63(1988)527-545

Gibbs Ensemble: particle interchange

Vapor-Liquid Equilibrium of Argon. Simulation with Nonadditive Potentials

S.P.J. Rodrigues;F.M.S.S.Fernandes, J.Phys.Chem., 98(1994)3917

Gibbs-Ensemble on Polyatomics

F.F.M.Freitas;B.J.C.Cabral;F.M.S.S.Fernandes

Gibbs-Duhem integration method

( )

ln ( )

, 1

1 ; 12 2

cl

L V L Vc

c c

d hd vdp Gibbs Duhem equation

d p h Clapeyron equationd p v

S r SSP r u r

T TB AT T

D.A.Kofke, J.Chem.Phys., 98 (1993) 4149-4162

C60 (buckminsterfullerene)

Potentials for C60

The subtlety of C60

F.M.S.S.Fernandes;F.F.M.Freitas;R.P.S.Fartaria, J.Phys.Chem.B, 2004, 108, 9251-9255

Thermodynamic Integration

2

1

2

1

2 1

2 1

2 1

1 1

, ,( )

, ,( )

, ; ,

B B B

T

B B BT

A T A T p d isothermNk T Nk T k T

A T A T E T dT isochoreNk T Nk T Nk T T

A T A T reference states of known free energies

Reference free-energy for solids: Einstein crystal method

0 0

2

0,1

0

1

, 1

, 1

'( )

N N N NS S S S

N

i ii

S Ein

B B B

Ein

U U U U

A T UA dNk T Nk T Nk T

A freeenergyof Einstein scrystal analytically knownseeFrenkel and Smit

r r r r

r r

D.Costa et al., J. Chem. Phys., 118 (2003) 304-310

KCl phase diagram using two potential models

P.C.R.Rodrigues;F.M.S.S.Fernandes, J.Chem.Phys, 126 (2007) 024503