Non-equilibrium thermodynamic approach

Part 1
Jean-eric.wegrowe@polytechnique.edu

•1) Short introduction to equilibrium thermodynamics

•2) Examples of simples transport equations

•3) Cross-effect example 1: The thermoelectric effect

•4) Continuity equations = conservation laws

• 5) Derivation of the transport equations

• 6) Onsager reciprocity relations

• 7) Drift-diffusion-reaction processes

• 8) Cross-effect example 2: The Hall effect 31 Slides

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 Short introduction to
thermodynamics
- Describes the role of heat in physical processes.
- Introduction of a new observable: the entropy
- Few variables define univokely the state of the system (« state variables »)
(Case of simple fluid: Entropy S, Volume V, Number of particles N).
Sadi Carnot
« Réflexions sur la puissance 1796 – 1932
Sadi Carnot 1824 : Fundations motrice du feu et sur les Ecole Polytechnique
machines propres à
First introduction of the entropy as physical concept (quantity Heat/T) développer cette puissance »

Rudolph Clausus1850
Coined the name the entropy. Formulation of the second law of thermodynamics

Lord Kelvin and Max planck (1903) reformulation of the second law

Constantin
Constantin Carathéodory 1909: axiomatic theory: "Investigations on the Foundations
Carathéodory
of Thermodynamics"
Theoretical expression of the second law of thermodynamics 1873 - 1950

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 First law
of thermodynamics
First law (principle):
For all systems , there is a state function, scalar, extensive, conserved,
associated with the homogeneity of time, called energy E.

Furthermore: or

Note :
- the decomposition of the the transfer of energy between the system and the environment in « work », « heat »
and « mater » is not univocal. It is often difficult to distinguish, for instance, between transfer of mass and
tranfer of heat, or transfer by radiation or tranfer of heat, etc.
However, the decomposition is rendered univocal by the choice of the set of state variables (see below).

- The concept of flux of heat is related to a lack of knowledge about the microscopic mechanisms responsible
for the transfer of energy (the action of the microscopic variables disappeared during the averaging process).
The decomposition between work and heat is related to this lack of information in the macroscopic description
of the system.
- The energy is defined close to an arbitrary affine transformation: E’ = a(E + E 0) is also valide.

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 Second law
of thermodynamics

Second law (principe):

For all systems , there is a state function S, scalar, extensive called the entropy,
that obeys to the following properties:
(a) Principle of evolution:
If the system is adiabatilly closed (no heat exchanded), noted 0, then:

Production I of entropy =
irreversibility = dissipation
(b) Principle equilibrium (thermostatics):
If the system is insulated, noted 0, the entropy of the system tends to a maximum,
compatible with the constraints.
« Thermal death »
space of the states

In nature, the case dS/dt = 0 is an idealization.

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 Theory of non-equilibrium thermodynamics
(applied to solide state physics)
Some milstones
De Groot-Mazur 1962 Onsager Nobel 1968 Prigogine Nobel 1977

E. C. Stueckelberg (1974)
Axiomatic approach

On the model of C. Caratheodory 1909.

and the Erlangen program (group theory)

free at (french):
https://www.epflpress.org/produit/659/9782889142248/thermodynamique-statistique

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 Justification about the
phenomenological approach

Josiah Willard Gibbs (1839 - 1903):

« One of the principal objects of theoretical research in any department of
knowledge is to find the point of view from which the subject appears in
its greatest simplicity. »
Rumford Medal address, January 12, 1881

Goal: description of transport processes

including of cross- effects

A microscopic approach is available through the linear response theory (and other non-equilibrium statistical methods).
The necessity of adding cross-effects renders this microscopic statistical approach difficult in practice.

R. Kubo, M. Toda, N. Hashitsume : « Statistical physics II. Nonequilibrium Statistical Mechanics »

Springer Series in Solide-State Sciences no 31 (1985)
Robert Zwanzig : «Nonequilibrium statistical mechanics  » Oxford University Press, Ney-York (Ed. 2001) Reference Textbooks:
Hamiltonian approaches
Dieter Forster : « Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions »,
for response function
Taylor & Francis (1975) and stochastic equations
E. Fick and G. Sauermann « The Quantum Statistics of Dynamical Processes »,
Springer Series in Solide-State Sciences no 86 (1990)

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 Conjugate variables in thermodynamics
and statistical physics
A system is defined by the set of state variables (including S necessarily). Simple fluids: (S,V,N)
Energie U = Product of extensive by intensive state variables:
Xi extensive (S,V, N, …)

Yi intensive variable, thermodynamically

conjugate to Xi:
Justification from statistical physics :
m= microstate

Extensive random variable Parameter Yi is describing a réservoir:

defined by the distribution - Temperature: reservoir of energy (thermostat),
Mean value: - Pressure: reservoir of Volume (piston),
-Chemical potential  : reservoir of particles N,
Energy
(random variable) -Magnetic field: reservoir of magnetization,

Equilibrium = Boltzmann distribution:

System in contact to a thermostat at temperature T :

General case (many reservoirs Yi): Partition function:

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 Thermodynamic potential: internal energy
case of simple fluids

1er law: there is a state function, scalar, extensive, conserved for an insulated system:the energy U

Thermodynamics: case of a simple fluid. Model with three extensive state variables (or «observables»)

Extensive means homogenous of degree 1:

Simple Fluids
State funcion: three kinds of power
related to the variables
Three conjugate intensive variables: (S, N, P) (T, , P):
heat, chemical, mechanical:
Chemical
Temperature: Pressure:
potential:

In fact, we know that we have the expression of U:

Q+ ch + W
Why not S, N and V?

Answer (math oriented): The Euler throrem:

if f is differentiable en and homogeneous of degree 1, i.e. then:

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 Non-equilibrium thermodynamic approach
Part 1

•Short introduction to equilibrium thermodynamics

•Examples of simples transport equations

•Cross-effect, example 1: The thermoelectric effect

•Continuity equations = conservation laws

• Derivation of the transport equations

• Onsager reciprocity relations

• Drift-diffusion-reaction processes

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 Some simple transport equations
=> relation between flux and force

Two reservoirs in contacted through the system generation of a flux JX

discrete system continuous system

flux force
The electric field generates an electric current.
Ohm’s law:  is the electric conductivity
= scalar transport coefficient

flux force Thermoelectric

Fourier’s law: The temperature difference generates a heat current
 is the thermal conductivity

flux force Thermodiffusion

 is the chemical potential.
Fick’s law for diffusion: The gradient of concentration
generates a flux of particles
« entropy force »
Common Diffusion constant:
expression:
(same as in for diffusion equation)

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 Two supplementary simple transport equations

Hook’s law for elasticity: The stress  generates a strain 

Compliance where Y is the Young modulus

flux force
The flux – force relation is

Kinetics of a chemical reaction : Extent of the A : Chemical

relaxation of internal degrees of freedom reaction or relaxation affinity
flux? forces?

Chemical Stoichiometric
coefficients
reaction :
reactant product
Reaction rate Flux
Variation of the number of molecules C: or Relaxation rate

Knowing nc(0), nc(t) is defined univocaly by the extent of the reaction (t)

State variable: 
Chemical affinity:
conjugate to  chemical potentials
Generalized force
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 Non-equilibrium thermodynamic approach
Part 1

•Short introduction to equilibrium thermodynamics

•Examples of simples transport equations

•Cross-effect, example 1: The thermoelectric effect

•Continuity equations = conservation laws

• Derivation of the transport equations

• Onsager reciprocity relations

• Drift-diffusion-reaction processes

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 Wiedemann-Franz law
(for noble metals)
Lorenz number

If same carriers q and Wiedeman-Franz Law (metal)

same dissipation mechanisms 

Consequences:
(1) Moving elecric charges with temperature gradient. Lorenz number
(2) Moving heat carriers with electric field.

General case (demonstration Lecture VIII):

Coupled transport equations = cross-coefficients = thermoelectric effects:

Onsager Transport coefficients:

Onsager reciprocity relation: (described below)

Electric field: reservoir of charge carriers

Electric current: flux of charged particles

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 Seebeck and Peltier effects: old observations
Conjugate variables:
Energy E and 1/T => Energy current JE and force
Number of particles N and chemical potential /T
=> Particule current JN and force Forces: justified below(general theory)

Thermocouple and Seebeck electric generator:

Peltier cooler

Seebeck coeffcient 
Peltier coeffcient 

Peltier Generator
with a Kerosene
lampe: 2 Watts

T. J. Seebeck A. F. Ioffe J.-Ch. Peltier

1770 - 1831 «Semiconductors Thermoelements and 1785 - 1845
Thermoelectric cooling», London, 1957
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 Seebeck and Peltier coefficients: measured coefficients
General Onsager reciprocity relation:
Mathematical
expression L12 = L21
For linear processes
(see discussion below)
Ohm’s law:
First line of the matrix: Peltier coefficient:
Redefinition of the Relation between the two fluxes
phenomenological at zero temperature gradient:
transport coefficients
Definition: at
Seebeck coefficient: Second line of the matrix:
Relation between electric field
and temperature gradient (for an open circuit)
Definition (measure) : First line of the matrix:
at
First line of the matrix:

L12 = L21

Confirmed experimentally: reciprocal effects

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 Thermoelectric equations

Fourrier law: Definition ? at (short-cicuit) or at (open circuit) ?

Second line of
the matrix:

Same effect but different coefficients

Or? depending on the experimental
configurations (constraints)

The transport coefficients define the «effects »:

Warning:
Usually you find Ohm Seebeck effect
the definition ( => -):

Heat
Because the heat flux conduction
goes from hot to cold.
Peltier effect Or:

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 The dimentionless figure of merit zT
for a Thermoelectric generator
Question: how to optimize the materials in order to have the better conversion from heat flux to
Electric voltage? Looking for a compromise…
Thermal conductivity at zero electric field (short-circuit)
=> maximizes the heat flux but minimizes the output voltage.

Thermal conductivity at zero electric current (open-circuit)

=> maximizes the voltage but minimizes the flux:

Hypothesis: The good strategy for a thermoelectric

generator is to maximize the ratio:

Let us define the zT coefficient,

which will be shown to be the « figure of merit »

The TE properties are summerized in the expression of zT

First proposed by Ioffe in a famous work :
A. F. Ioffe « semiconductors Thermoelements and
Thermoelectric cooling », infosearch Limited, London, 1957
A. Ioffé
1880 - 1960 17
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 Thermoelectric generator: maximal efficiency

The efficiency of a thermoelectric generator (TEG) is given by η, defined as :

Figure of Merit: z is « intrinsic »

Strong limitation today (materials):

Limitation related to the Wiedemann-Franz law

  Maximum efficiency:
Demonstration as exercise (difficult: correction on Moodle)


Max carnot cycle efficiency Irreversibility
(« reversible » S = 0)
(dissipation)

M0 is the optimum ratio M = Rload/R for the TEG

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 Non-equilibrium thermodynamic approach
Part 1

•Short introduction to equilibrium thermodynamics

•Examples of simples transport equations

•Cross-effect, example 1: The thermoelectric effect

•Continuity equations = conservation laws

• Derivation of the transport equations

• Onsager reciprocity relations

• Drift-diffusion-reaction processes

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 Conservation laws (continuity equations)
« Conserved » extensive variable Ni
Variation inside a volume = «  everything which is comming in » minus « everything which is flowing out ».

leak

Leak term (relaxation) for the « particles » Ni

«Particles » i are relaxing to « particles j »: relaxation channel = chemical reaction

Relaxation = chemical reaction. leak i
« Internal degrees of freedom »
Two « species » of particles i, j ...
r relaxation channels: source i

source j

Chemical affinity: leak j

the force that drives the leak
De Dooder 1924 i.e. the relaxation channels from j to k

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