The phonon theory of liquid matter

Dima Bolmatov

Queen Mary University of London

Barcelona, 2012

Kostya Trachenko

Dima Bolmatov

Vadim Brazhkin

Plan

• Introduction

• The phonon theory of liquids

• Results: theory vs experiment

• Current and future work

States of Matter: gases, solids, liquids

• Solids: P. Debye's phonon theory, small displacements E=E_0+E_harm+E_anharm

• Gases: energy of ideal monatomic gas, small interactions E=3/2NT+P, k_B=1

• Liquids: neither

Lev Landau: liquids have no “small parameter” (no expansion is possible). Landau&Lifshitz, Statistical Physics: “Interactions in a liquid are both strong and system-specific… Liquid energy can not be calculated in general!

Energy and heat capacity of matter

A. Granato, J. Non-Cryst. Sol. 307-310, 376 (2002)

Gases: E_gas=3/2NT, k_B=1 Solids: E_solid=3NT, k_B=1, in classical limit Dulong–Petit law

Liquids: E_liquid=?

Liquid heat capacity?

“can not be done in general form” (L. D. Landau and E. M. Lifshitz, Statistical Physics,1964)

Over the last 30 years, liquid heat capacity is not mentioned in popular books about liquids:

J. M. Ziman, Models of Disorder, Cambridge University Press, Cambridge, 1979.

J. P. Boon and S. Yip, Molecular Hydrodynamics, Dover, New York, 1980.

N. H. March, Liquid Metals, Cambridge University Press, Cambridge, 1990.

R. Zwanzig, Non-Equlibrium Statistical Mechanics, Oxford University Press, 2001.

J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, Elsevier, New York, 2007.

Facts

D Wallace, Phys. Rev. E (1998)

meltingpoint

The same behaviour is seen in complex liquids

Two approaches to liquid energy

From the gas phase, by switching on interactions in a gas (L&L argument): E=3NT/2+U(r)

The exact results exist only for small densities and high temperatures that describe interacting gases, but not real liquids. CVan Der Vaals=CIdeal gas

Virial expansions do not work for real dense liquids: vN/V≈1 (Allen&Tildesley)

For simple models (hard spheres, LJ systems etc), analytical results agree with MD simulations, but still depend on parameters and correlation functions

Require knowledge of multiple correlation functions and interatomic potentials, calculations are difficult and approximations are hard to control

Correlation functions (especially higher-order) and interatomic potentials are not available beyond very simple liquids (Ar, Xe and LJ-type liquids)

RESULT:Not easy to see how CV=3N at low temperature and CV=2N at high

temperature

From the solid phase:

Take strong interactions into account from the outset: E=3NT + (?)

Brillouin, classical case:

Esolid=3NT=NT/2+NT/2+2(NT/2+NT/2). Eliquid=NT+2(NT/2)=2NT, in disagreement with experiments. Led to assume the existence of small crystalline domains with easy cleavage directions

J Frenkel: introduced relaxation time τ.

Two approaches to liquid energy

Local relaxation events

Dynamics in viscous liquids: “rattling” motion inside cages plus large atomic jumps with cage relaxation.

These thermally excited jumps give liquid flow

Liquid is a solid for times smaller than τ !

Frenkel proposed that liquids support transverse phonons with frequency ω>1/τ

High-frequency transverse modes widely observed in liquids experimentally

We took up this approach and accounted for the phonon energy

The phonon theory of liquids

or

Debye function

Virial theorem:

Heat capacity (cv=dE/dT ): theory vs experiment

Heat capacity (cv=dE/dT ): theory vs experiment

Heat capacity (cv=dE/dT ): theory vs experiment

Heat capacity (cv=dE/dT ): theory vs experiment

Quantum liquids: Helium, P=45 MPa

Quantum liquids: Hydrogen, P=50 MPa

Quantum liquids: Parahydrogen, P=120 MPa

Complex liquids

Chloropentafluoroethane (R115)

Complex liquids

Trifluoromethane (R23)

Universal behaviour of liquids near critical point

Universal behaviour of liquids near critical point

Universal behaviour of liquids near critical point

Current and future work

• Calculation liquid energies and heat capacities of quantum

exotic liquids: Hydrogen, Parahydrogen and Helium; organics: Octafluorocyclobutane, Octafluoropropane, Hexafluoroethane, Chloropentafluoroethane and etc. • Analytical expression of c_V for 2->3/2 regime.

• Connection the dynamics (phonon states) and structure (gas phase approach) of liquids: numerical calculations, MD simulations. • Description of liquids at microscopic level: spontaneously breaking symmetry, Goldstone excitations and etc.

• Liquids in confined and restricted geometries.

Thank you!