Liquid State and Phase Transitions

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

Set up Models

Perform Experiments

Carry Out Simulations

Experimental Results

Results for Models(Exact)

Model Liquids

Statistical Mechanical Theories Theoretical Predictions

Compare

Compare

Models Validation

Theories Validation

Mechanical,dielectric and transport properties; Free Energies and chemical potentials; Phase transitions; Structure; Motion at molecular level; Prediction of 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  G     S T   p

 G    V p   T

Max Born – Tisza square

First-order and Continuous Phase Transitions

Fluid - Liquid Transitions



 T   B 1   2  Tc   T   L  V  c  A 1   2  Tc 

 L  V

S  CX  T   ; X V, p  T  X

T 

1      p T

Cp for butane

Ferromagnetic spontaneous magnetization

  S  M  T  TC ; C X  T   X  H,M T   X  M  T     H T

Critical Exponents for fluid and magnetic systems

t   T  TC  / TC

Universality

Reduced Units Density

 *   3

Temperature

T 

k BT

Energy

E 

E

Pressure

p 

*

*

*



 p 3

 1/ 2

Time Force

   t*  t  2   m  f f* 



The sui generis critical point!

Continuous (order-disorder) phase transitions

NH4Cl

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

Statistical Mechanics Theory

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)

Classical microstates (phase space). Solutions of classical equations of motion:    r1 , r2 ,..., rN , p1 , p 2 ,..., p N    r N , p N    r N  dr N , p N  dp N 

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

    r N ; t  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: E inst  t  

Gobs

G obs 



N

 i 1

1  M

n G M

p i2  U r N 2 mi

M

G   0



  G  tm

( ergodic hypothesis )

Gobs   P G  G en 

Typical Ensembles

Ensemble Microcanonical Canonical Isothermal - Isobaric Grand - Canonical

Constraints E, V, N T, V, N T, p, N 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

Constraints : E, V, N  E,V , N    N !h3N 

1

 E  H r

N

, pN   drN dpN

(Phase space Volume)

  E ,V , N      E ,V , N  /  E

 E,V, N   N!h3N 

1

(derivative of unit step function)

 E  H r ,p  dr dp N

N

N

N

(Phase space density)

Generalized Boltzmann’s Equation  n  Zn   X n  H d n

Sn / k  ln n

n

Sn

Xn

dn

1

S1(E,V,N)

E

drN dpN

1 N ! h 3N

E

2

S2(H,p,N)

H - pV

drN dpN dV

1 N ! h 3N

H

3

S3(L,V,)

L + N

drN dpN

1 3N N  0 N !h

4

S4(R,p, )

R - pV + N

drN dpN dV

1 3N N  0 N !h

n  n / Yn

Zn





Yn

L





R

 n  Zn   X n  H d n

High Dimensional Geometry (sphere and spherical shell volumes) V r n V r S  k ln   E , V , N   k ln   E , V , N 

V  An r 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

P T ,V , N  

Q T , V , N    N !h

exp    H  r N , p N   dr N dp N Q T , V , N 

3N

  exp  H  r 1

N

, p N   dr N dp N

exp    Ei  P T , V , N    exp   Ei  i

Q T , V , N    exp    Ei  i

Thermodynamic Properties from Partition Function

A T , V , N    ln Q T , V , N  kT

   p  kT ln Q T , V , N   V T , N    S   kT ln Q T , V , N   T V , N        kT ln Q T , V , N   N T ,V

Structure and Correlation

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

V n  r  g r     2 N 4 r r

 r    g r 

Solid and Liquid RDF’s

Pair Correlation Function and Thermodynamics



N 3 2 E  NkT  u r g r r dr 4       2 2 0  NkT N  du  r  3 p g r 4  r dr     V 6V 0 dr

Potential of mean force



  dr3 ...drN  dU / dr1  exp    U  d U  r N  r1 ,r2 fixed   ... dr1 d r ... d r exp U     3 N 

 kT

d ln g  r1 , r2   w  r    kT ln g  r  dr1

w  r   potential of mean force

From Introduction to Modern Statistical Mechanics, D.Chandler

Hard Spheres

Theories for g(r) g r   e

  w r 

w  r   u  r   w  r  lim w  r   0  0

g  r   e   u  r  1  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

  S  M  T  TC ; C X  T   X  H,M T   X  M  T     H T

Ising Model N

Hamiltonian   J  si s j   H  si ; s  1 ij 

i 1 N

 M     si i 1

N   Q   , N , H   ...  exp   J  si s j    H  si  s1 s2 sN 1 i 1  ij  

  Q  K , N   ...  exp  K  si s j ; s1 s2 sN 1  ij  

H  0, K  J / kT

Renormalization Group Theory

Q  K , N    f  K   3 K   ln cosh  4 K  8

N /2

Q  K  , N / 2  ( Kadanoff transformation) K  J / kT



g  K    g  K   ln 2  cosh  2 K  

1/ 2

 cosh  4 K  

1/8



ln Q  Ng  K  1 cosh 1  exp  8K  / 3  4 1/ 4 1 1 g  K   g  K    ln 2 exp  2 K  / 3 cosh  4 K  / 3  2 2   

K



K 0

K 

  

K 0

K 

    

K 0

KC

K 

3 K C  ln cosh  4 K C  8 K c  RG   0.50698 K C (exact )  0.44069



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