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Mesoscopic nonequilibrium
thermoydnamics
Application to interfacial
phenomena
Miguel Rubi
Dynamics of Complex Fluid-Fluid
Interfaces
Leiden, 2011
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

shear
bound

free
f 0
f 0
f 0


Activation

shear
F 0
stick
F 0

Examples:
Chemical reactions, adsorption,
evaporation, condensation,
thermionic emmision, fuel cells….
slip
Activation: to proceed the system has to surmount a potential barrier; nonlinear
NET: provides linear relationships between fluxes and forces
Nonequilibrium thermodynamics
• Global description of nonequilibrium processes (k0;
ω0)
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 reactions
1
   JA
T
L
L
J   A   (  2  1 )
T
T
Law of mass action
linearization
2
1
A

 kT

L
kT
kT
J   D  e  e    D(1  e )   A
T


Conclusion: NET only accounts for the linear
regime.
Activation
Unstable
substance
Final product
Naked-eye: Sudden
jump
Watching closely
Progressive molecular changes
Diffusion
Translocation of ions (through a protein channel)
Biological membrane
short time scale: local equilibrium along
the coordinate
biological pumps,
chemical and biochemical reactions
Local, linear Global, non-linear
Arrhenius, Butler-Volmer,
Law of mass action
Protein folding
Intermediate configurations, same as for
chemical reactions
Molecular motors
Energy transduction,
Molecular motors
Activated process
viewed as a diffusion process along a reaction
coordinate



L 
kL  kT  kT

 ( )  J  
 e
e   De kT
e kT
T 
P


From local to global:

2
1
d Je

kT
 kT
  D  d
e
1

2
   d  ...
2
1
 kT
J   D  e  e kT


   D( z2  z1 )

What can we learn from kinetic
theory?
Boltzmann equation
f A
  E AS  RA
t
S
A B
CD
Chapman-Enskog
2 (2)
f A  f A(0) 1   (1)


 A  .. 
A
LMA
J. Ross, P. Mazur, JCP (1961)
1
   JA
T
Thermodynamics and stochasticity
J.M. Vilar, J.M. Rubi,
PNAS (2001)
Probability conservation:
J x J v
P


t
x
v
Entropy production:


  Jx
 Jv
0
x
v
Fokker-Planck
P

 v D  
  vP    2  P
t
x
v    v 
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
J.M. Rubi, A. Perez, Physica A 264 (1999) 492
 v   Pudu
hydrodynamic
i) t   1
ii) t 
1
P  p1  kT  1
dv
    P    v
dt
Fick
Maxwell-Cattaneo
J   D  
1
dJ
dt

  D(k )k 2 
t
D(k )  D(1  D 1k 2 )
Burnett
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
( )
1
1
Physisorbed

2
0
2

Chemisorbed
MNET of adsorption
Langmuir equation
I. Pagonabarraga, J.M. Rubi, Physica A, 188, 553 (1992)
Evaporation and condensation
D. Bedeaux, S. Kjelstrup, J.M. Rubi, J. Chem.
Phys., 119, 9163 (2003)
Condensation coefficient

Stick-slip transition

shear
F 0
stick
F 0

f
b
J  l ( e kT  e kT )
slip
 f  kT ln c f   0  
b  kT ln cb   0
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….
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