Mesoscopic nonequilibrium thermoydnamics

Application to interfacial phenomena

Dynamics of Complex Fluid-Fluid Interfaces Leiden, 2011

Miguel Rubi

Interfaces• The interface is a thermodynamic system;

excess properties; Local equilibrium holds.• Transport and activated processes take place• The state of the surface can be described by

means of an internal coordinate

bound free

shear

000

fff

0F 0F

stick slip

shear

Activation

Examples:

Chemical reactions, adsorption, evaporation, condensation, thermionic emmision, fuel cells….

Activation: to proceed the system has to surmount a potential barrier; nonlinearNET: provides linear relationships between fluxes and forces

Nonequilibrium thermodynamics• Global description of nonequilibrium processes (k0; ω0) Shorter scales: memory kernels (Ex. generalyzed

hydrodynamics, non-Markovian)

• Description in terms of average values; absence of fluctuations

Fluctuations can be incorporated through random fluxes (fluctuating hydrodynamics)

• Linear domain of fluxes and thermodynamic forces

Chemical reactions1 JAT

2 1( )L LJ AT T

Law of mass action

2 1

(1 )A

kT kT kT LJ D e e D e AT

Conclusion: NET only accounts for the linear regime.

linearization

Unstable substance

Final product

Naked-eye: Sudden jump

Progressive molecular changes

Activation

DiffusionWatching closely

Translocation of ions (through a protein channel)

short time scale: local equilibrium alongthe coordinate

biological pumps,chemical and biochemical reactions

Arrhenius, Butler-Volmer,Law of mass action

Local, linear Global, non-linear

Biological membrane

Protein folding

Intermediate configurations, same as for chemical reactions

Molecular motors

Energy transduction,Molecular motors

( ) kT kT kT kTL kLJ e e De eT P

2 2

1 1kT kTd Je D d e

2 1

2 1( )kT kTJ D e e D z z

Activated process

viewed as a diffusion process along a reaction coordinate

From local to global: ...d

What can we learn from kinetic theory?

J. Ross, P. Mazur, JCP (1961)

A B C D A

AS AS

f E Rt

(0) (1) 2 (2)1 ..A A A Af f

Boltzmann equation

LMA 1 JAT

Chapman-Enskog

Probability conservation:

x vJ JPt x v

Entropy production:

0x vJ Jx v

2

P v DvP Pt x v v

Fokker-Planck

Thermodynamics and stochasticity

J.M. Vilar, J.M. Rubi, PNAS (2001)

Molecular changes: diffusion through a mesoscopic coordinate

:( , ) :mesoscopic coordinate

P t probability

Second law D. Reguera, J.M. Rubi and J.M. Vilar, J. Phys. Chem. B (2005); Feature Article

Meso-scale entropy production

Relaxation equations

v Pudu

hydrodynamic

dv P vdt

1 1P p kT

1)i t   Fick

1)ii t    Maxwell-Cattaneo

1 dJJ Ddt

2( )D k kt

1 2( ) (1 )D k D D k

Burnett

J.M. Rubi, A. Perez, Physica A 264 (1999) 492

References

• A. Perez, J.M. Rubi, P. Mazur, Physica A (1994)• J.M. Vilar and J.M. Rubi, PNAS (2001)• D. Reguera, J.M. Rubi and J.M. Vilar, J. Phys.

Chem. B (2005); Feature Article• J.M. Rubi, Scientific American, November, 40

(2008)

Adsorption

Physisorbed Chemisorbed

( ) 1 2

1 0 2

MNET of adsorption

Langmuir equation

I. Pagonabarraga, J.M. Rubi, Physica A, 188, 553 (1992)

Evaporation and condensationD. Bedeaux, S. Kjelstrup, J.M. Rubi, J. Chem. Phys., 119, 9163 (2003)

Condensation coefficient

0F 0F

stick slip

shear

Stick-slip transition

0

0

( )

ln

ln

f bkT kT

f f

b b

J l e e

kT c

kT c

C. Cheikh, G. Koper, PRL, 2003

Conclusions

• MNET offers a unified and systematic scheme to analyze dissipative interfacial phenomena.

• The different states of the surface are characterized by a reaction coordinate.

• Chemical reactions, adsorption, evaporation, condensation, thermionic emmision, fuel cells….