Materials for Energy PHY563 Fundamentals of energy ...

Materials for EnergyPHY563

Fundamentals of energy conversion and transport

06/01/2021

Jean-François Guillemoles,

Nathanaëlle Schneider

Outline

PHY579 – JF Guillemoles 2

• Thermodynamics bases

o Physical grounds

o Potentials

o Stability

• Irreversible thermodynamics

o Flux and Forces

o Entropy production

o Linear theory

• Thermoelectricity

• Endoreversible thermodynamics

o Thermal

o Chemical

THERMODYNAMICS


Thermodynamic System

5

• A thermodynamic system is clasified according to the nature of hisexchanges with other systems: matter, energy, et entropy/heat.

• It is bounded by permeable or impermeable membranes (cf Callen)

• His state is defined by extensive (V,m,U,…) & intensive (T,P,µ,ϕ,…) variables

(P, T,…)

(V, m, U…)

Q1

Q2

W

Extensive (V,m,U,…) & intensive (T,P,µ,ϕ,…) variables are conjugate

Thermodynamics 101

• Large number of degree of freedom

o Microscopic level => 6N, huge phase space (h3N per state)

o Statistical Properties at macroscopic scale (local macroscopic variables, in volumes wherefluctuations are small dN/N<<1)

o Macroscopic Quantities define a thermodynamic state (slow variables)

• Equilibrium

o Mecanical (P)

o Chemical (µi)

o Thermal (T)

• Gradient of T, P, µ, … => non-equilibrium/flux unbalance => dynamics to return to equilibrium

• Energy, matter, charge : additive & conserved

• Entropie : additive & non-conserved

7

Entropy

Thermodynamics :

2nd principle : Work cannot be extracted from a single temperaturereservoir.

Is an extensive state function S.

dS=dQ

T

æ

èç

ö

ø÷

rev

Statistical physics :

Ensemble of microstate Ω, with occupation probability Pi.

Boltzmann : equiprobable microstates (isolated system at equilibrium)

• Shannon Entropy

• Boltzmann Entropy

Jaynes’ Information Interpretation: Entropy represents what is not known (equiprobability=hypothesis from ignorance) Ss = - Pi logPi

i

å

Sb = kB logW

Other extensive Variables

-Polarisation

- Magnetic Moment

-Displacement (in a solid)

-Surface

-Charge (test charge)

-Mass

Coupled intensive Variables ?

Variables couplées

- Electric field

- Magnetic Field

- Stress

- Surface Tension

- Electric Potential

- Gravitationnal field

-Polarisation

- Magnetic Moment

-Displacement (in a solid)

-Surface

-Charge (test charge)

-Mass

10

Thermodynamic laws

Isolated System :

Evolutions Equations

The Laws of Thermodynamics

1. You Can’t Win

you can’t get more energy out of the system than you put into it.

2. You Can’t Break Even

any transfer of energy will result in some waste of energy (unless a

temperature of absolute zero can be achieved)

3. You Can’t Get Out of the Game

you cannot achieve absolute zero (in a finite number of steps)


C.P. Snow, 1975

dU=dQ+dW

dS= dQ/T > 0

dS 0 when T 0

Callen

• Evolutions when contact between unbalanced systems

- Some allowed flux (conductors)

- Some forbiden flux (insulators)

•Approximation: ideally insulating/conducting materials

•NB: both are necessary for a controled energy conversion (while limitingentropy production)

FLUX AND FORCES: NON EQUILIBRIUM THERMODYNAMICS


Gradients as source of work

• Gradient of pressure (Mechanical energy: momentum tranfer)

o Wind, waves, electrokinetic effect…

• Temperature gradients (thermal energy: internal energy transfer)

o Thermoelectricity, convection, ..

• Chemical gradients (chemical potential energy)

o Diffusion

• Potential gradients (i.e. force)

o Hydropower (gravitation)


Non equilibrium and gradient

• Force gradients

• Flux of conserved quantities

o Stationnary & et homogeneous: constant gradient

o flux ~ DT/L

• Dissipative Flux

o lead to return to equilibrium with dS>0


L

Non equilibrium Systems

• Empirical relationships

Electronic transport

Potential gradient → flux !

Drift

Electrical potential= bias

Diffusion

Chemical potential= concentration

18

Einstein Relationship

q

TkD B

2.6 Notion de transport dans les SCs

Transport equations

Drift Diffusion

Semiconductors - transport

)( ee

nEej nn n

nenDneDj nnn

en

c

nnkT

enD

N

nenDj

ln

ee

enn eenj

ee

ee

n ee

ej

)(

Chemical potential in a force field

• Charge and particle number are proportional => electric and chemicalparts of µ can’t be measured independently

• Chemical potential needs to be generalized to include the work of the forces (electrochemical potential)


Non equilibrium

• Relaxation towards equilibrium

Non equilibrium

• Relaxation towards equilibrium with entropy production:

Entropy creation

• DZ acts as a driving force

• dX (or dX/dt) acts as an evolution coordinate, or a flux


𝑑𝑖𝑆𝑡𝑜𝑡𝑎𝑙 =dS1+dS2= (Z1-Z2) dX →

𝜕𝑆

𝜕𝑡= σ𝑗(𝑍𝑗,1−𝑍𝑗,2)

𝜕𝑥𝑗

𝜕𝑡

Entropy creation: example I

• True for each part of the subsystem


𝑑𝑆 = σ𝑑𝑄

𝑇= σ𝑒𝑥𝑡 dQ/T +𝑑𝑄𝑖𝑛𝑡(

1

𝑇1

−1

𝑇2

)

𝑑𝑆𝑖𝑛𝑡 = 𝑑𝑄𝑖𝑛𝑡(1

𝑇1

−1

𝑇2

)>0 => 𝑑

𝑑𝑡𝑆𝑖𝑛𝑡 =

𝑑

𝑑𝑡𝑄𝑖𝑛𝑡(

1

𝑇1

−1

𝑇2

) >0

CONVERSION SYSTEMS

Coupled Fluxes

• New variables / équilibrium: flux

• Additional relationships flux/force are needed

• Limited expansion near equilibrium (Empirical):


Continu

Partitionné

NB: Onsager relationships

Coupled fluxes and forces


• A gradient can produce a work to set (or upset) another gradient

• thereby ensuring dS>0 while work is “spontaneously” produced

o Thermal electrical : thermoelectricity (Peltier, Seebeck, Thomson)

o Concentration heat : thermodiffusion (Soret, Dufour)

o Chemical chemical : coupled reactions

o Concentration electrical : batteries

From J.M. Rax

Thermodynamic Flux

Thermodynamic Affinity

Coupled variables

Conversion systems

• Coupling of fluxes enables conversion

• Direct energy conversion: when electrical power is produced directly (electric current being one of the fluxes)


Dissipative Transport

• Transport is dissipative

o e.g. RI²


Coupled Transport


• Coupled transport enables conversion (with dissipation)

Electric Generators

• Find Voc, Isc

• How much is max power as a function of Voc & Isc?

• Impedance adaptation of generators

• Is linearity between fluxes and forces optimal?

o Could non linear relationship between fluxes and forces results in a higher output power? Do you know an exemple?


Electric Generators

• How much is max power as a function of Voc & Isc?

o ¼ of Voc.Isc

• Impedance adaptation of generators

o Rload =V/I equal to internal resistance of the generator = dV/dI at maximum power

• Is linearity between fluxes and forces optimal?

o No, Diodes can do better


Thermal enginesCarnot engines

44

T1

T2<T1

Q1

Q2

W

2 temperature sources, T1>T2.

Reversible operation for optimal efficiency

Transformation as cycles.

Thermal engines I

45

Entropy production Sources :

(a) T drops (necessary for conduction),

(b) Heat leaks (conduction or radiation),

(c) Dissipation (friction, viscosity, resistivity,…).

1

W

Q Q

Thermal engines II

46

Entropy production Sources :

a- T drops (necessary for conduction)

= > endoreversible analysis

b- Heat leaks (conduction or radiation)

=> Effective input heat reduced

c- Dissipation (friction, viscosity, resistivity,…)

=> Coupled fluxes description of work generation

1Q

W

Thermoelectric converter

49

Direct thermal energy conversion → electrical current :

• 2 sources with different temperatures

• 2 (semiconductors, is best) doped N/PFaster creation and migration of pairs on hot side

→ potential difference & flux of charges

Materials: good electrical/poor thermal conduction

Thermoelectric effect

• We consider an isotherm material driven by an electric field. Link the Onsager’s coefficients to well known coefficients.

We can demonstrate that a linear response of the system can be written using the Onsager’scoefficients with the following expressions:

jQ = LQQÑ1

T

æ

èç

ö

ø÷+ LQN

1

TÑ -m( )

jN = LNQÑ1

T

æ

èç

ö

ø÷+ LNN

1

TÑ -m( )

jQ = LQQÑ1

T

æ

èç

ö

ø÷+ LQN

E®

T

jN = LNQÑ1

T

æ

èç

ö

ø÷+ LNN

E

T

QNNQ LL

• We consider a material with a temperature gradient in the open circuit condition. Link the Onsager’s coefficients to well known coefficients.

jQ = LQN

E®

T=

LQN

LNN

j®

N = p j®

N

jN = LNN

E

T=s E

Loi d’Ohm

Effet Peltier

jQ =1

T2

LNQ

2

LNN

- LQQ

æ

èç

ö

ø÷ÑT = -lÑT

E =LNQ

TLNN

Ñ T( ) =aÑ T( ) Effet Seebeck

Loi de Fourier

52

E = electro-chemical field


dTdV .

Seebeck effect (1821)

53

A thermal gradient applied at the ends of an open circuit induces a finite voltage difference

- Qualitative picture

Higher carrier density on the cold sideLower carrier density on the hot side an electric field is established

Source: Chaikin, An introduction to thermopower for thosewho might want to use it in « Organic superconductors », 1990

- Actual observation: junction between two materials

α = Seebeck coefficient (sometimes noted S) or thermopower in μV.K-1

Intrinsic material property : α > 0 (p-type) or α < 0 (n-type)α (metal) ~ few μV.K-1 α (SC) ~ 100 μV.K-1

Machines endoreversibles

•1st principle- energy conservation

•Second principle - entropy conservation

•Definition of Q and efficiency

Carnot efficiency

T1

T2

W

Q1

Q2

0 EWQ

0/ TQS

1Q

W

1

21T

T

Cas reversible!

The Carnot Engine

T1

T2

W

Q1

Q2

T4

T3

g1

g2

•1st principle- energy conservation

•Second principle - entropy conservation

•Definition of Q (Fourier) and efficiency

Carnot efficiency

0 EWQ

0/ TQS

1Q

W

4

3

1Carnot

T

T

)( 3111 TTgQ

What is the total efficiency ?

g : thermal conductance

CONDUCTION

if reversible !

The Curzon-Ahlborn Engine

1. Discuss the approximation for Q1=g.(T1-T3)

2. Deduce from 1st and 2nd law equations the relation Q1(T1,T2, ). What is the

value when the efficiency is zero ? What is the efficiency when Q1=0 ?

3. Express W. What is the efficiency at Wmax ? W max ?

4. Determine a relation S(T1,T2, ).

The Curzon-Ahlborn Engine (2)

A. De Vos, Thermodynamics of solar conversion, Ed. Wiley

W max

1

21T

TC

1

2& 1

T

TAC

Irreversibility => lower efficiency

True for other engines (Joule, Diesel…)NB: for radiation such as solar radiation, a different analysis applies

Independent of g!

Efficiency at max. Power for the C-A Engine

Optimum

• Maximal efficiency vs maximal powero Reversibility : max efficiency at zero power!o Max power

• An economic optimumo Amortization vs fuel costs

• Exemple:

Dissipation

• Entropy production is a positive bilinear form

Definitions of adimensional parameters

Converting fluxes (merit factor in transfer efficiency)

Normalisation factor

Operating point


• Et écrire:

Operation

• What work is flux I2 doing?

• What is the dissipation associated to I1 flux?

• Efficiency? Efficiency at max power?

• Plot efficiency versus Power, and comment

Operation

• flux I2 yields work:

• flux I1 yields dissipation

Exemple : thermal engine


• Machine operating between 2 temperatures & 2 electric potentials

• Thermodynamic efficiency

• Efficiency of flux conversion :

=normalized flux ratio

W

Q

Efficiency vs power

Merit factor: q²/(1-q²)*

Efficient systems have efficiency very dependent on operating point

=

Realistic efficiencies

• If q=1, carnot engine internally

• If q < 1, coupled fluxed analysis

• Corresponds to linear transport

o Close to equilibrium

o Symetric Onsager matrixes

• Half Carnot often found in systems close to equilibrium


Summary

• Coupled fluxes

o A fraction is dissipated (price for stability)

o A fraction converted

o Dissipation increases with fluxes

• Classification

o Driven by force or flux. NB: often mixed

o transport parameter of the fluxes independent or not (coupled)

o Linear transport or not. NB: can depend on regimes


Device Drive Coupling Linear

Thermoelectric, Electrokinetic Force Y Y

Photovoltaic Flux N N

Nernst Generator, MHD Flux Y Y

Fuel Cell Force N N

Hydraulic Force N Y

Thermoelectrics Outline

• TE effect• Seebeck, Peltier and Thomson effects• Use of TE effect for energy applications• Thermodynamics• Figure of merits

• TE materials• Criteria / Design of a TE material• Bulk materials• Low-dimensional systems• Organic TE

76


dTdV .

Seebeck effect (1821)

77

A thermal gradient applied at the ends of an open circuit induces a finite voltage difference

- Qualitative picture

Higher carrier density on the cold sideLower carrier density on the hot side an electric field is established

Source: Chaikin, An introduction to thermopower for thosewho might want to use it in « Organic superconductors », 1990

- Actual observation: junction between two materials

α = Seebeck coefficient (sometimes noted S) or thermopower in μV.K-1

Intrinsic material property : α > 0 (p-type) or α < 0 (n-type)α (metal) ~ few μV.K-1 α (SC) ~ 100 μV.K-1


Peltier effect (1834)

Heat production at the junction of two conductors in which a current is circulated.Reversible: heating or cooling as orientation of current is reversed

2nd Kelvin relation (Onsager)

q= p. j

Q= pa -pb( ) I

p =a.T

Q heating rate

π = Peltier coefficient in V Intrinsic material property

Q.

= -t. j.ÑT

TT

τ = Thomson coefficient

Q heat production rate.

78


80

Two basic applications of the Peltier and Seebeck effects : generators and coolers

Cooling module (Peltier)Carrier transport

N: Electrons move against currentP: Holes move along currentBOTH leave cold end to reach hot end

Both processes correspond to lowering of entropyof cold end

Power generation module (Seebeck)


Let’s take a homogeneous and isotrope material. We work at P and V constant. Let’s note ν theelectrical potential and T the temperature. u, s, ne are the volumic density of internal energy,entropy, and free carriers (charge q and chemical potential μ).

Fondamental relations

• What are the extensive variables ?

• What is the general expression of the differential entropy? Deduce the expression of the entropy current as a function of the extensive variables

dQdNTdSdU dQT

dNT

dUT

dS

1

jS =1

TjU -

m

TjN -

n

TjQ

81

U, N,Q,S

U = internal energyN = number of particlesQ = charge (not heat)S = entropy


• The electric driven force comes from an electric potential (extensive variable). Introduce the electrochemical potential that links the electrical and the chemical ones.

• Deduce a simplified expression of js. Express the electric and heat current density

82

dQ= qdN mdN +ndQ= mdN +qndN = mdN

m = m +nqdQdNTdSdU

q = elementary charge

NUS jT

jT

j ~1

QNUS jT

jT

jT

j

1

Nelec jqj

NUSQ jjjTj

.~

Experimentally : currents jN and jQ are measurable

(heat current)


• We consider an isotherm material driven by an electric field. Link the Onsager’s coefficients to well known coefficients.

We can demonstrate that a linear response of the system can be written using the Onsager’scoefficients with the following expressions:

jQ = LQQÑ1

T

æ

èç

ö

ø÷+ LQN

1

TÑ -m( )

jN = LNQÑ1

T

æ

èç

ö

ø÷+ LNN

1

TÑ -m( )

jQ = LQQÑ1

T

æ

èç

ö

ø÷+ LQN

E®

T

jN = LNQÑ1

T

æ

èç

ö

ø÷+ LNN

E

T

QNNQ LL

• We consider a material with a temperature gradient in the open circuit condition. Link the Onsager’s coefficients to well known coefficients.

jQ = LQN

E®

T=

LQN

LNN

j®

N = p j®

N

jN = LNN

E

T=s E

Loi d’Ohm

Effet Peltier

jQ =1

T2

LNQ

2

LNN

- LQQ

æ

èç

ö

ø÷ÑT = -lÑT

E =LNQ

TLNN

Ñ T( ) =aÑ T( ) Effet Seebeck

Loi de Fourier

83

E = electro-chemical field


E = r jN +aÑT

jQ = p. jN - lÑT

We thus have:

T

dT

dT

Onsager reciprocity

One known parameter The others known as well

rq on entropy: Tjj QS /

84

a =LNQ

TLNN

p =LQN

LNN

QNNQ LL


generation

Generation Efficiency

2

2

1RITITcba D

2IRc

h =PU

Qc

=

Optimisation :

ab

ba

b

a

S

S

c

f

mab

mab

c

fc

T

TTZ

TZ

T

TT

1

11max

2

fc

m

TTT

22/12/1

2

bbaa

baabZ

85

PU = power supplied to the loadQC = heat absorbed at the junction

Tc = Tchaud = Thot

Tf = Tfroid = Tcold


cooling

Coefficient of performance (C.O.P. or φ)

2

2

1RITITfba D

TIRI ba D 2

P

QPOC

f..

Optimisation :ab

ba

b

a

S

S

11

1

.. max

mab

f

cmab

fc

c

TZ

T

TTZ

TT

TPOC

2

fc

m

TTT

22/12/1

2

bbaa

baabZ

86

Tc = Tchaud = Thot

Tf = Tfroid = Tcold


Carnot

ZT : figure of merit

87

fridge

Figure of merit (ZT) and TE performance :

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,00,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

ren

dem

ent

max

ZT

DT = 300°C

DT = 250°C

DT = 200°C

DT = 150°C

DT = 100°C

DT = 50°C

Tf = 200°C


TP

TZT

2

88

α = Seebeck coefficientκ = thermal conductivity(sometimes noted λ)σ = electrical conductivityP = power factor


Charge carriers

Acousticphonons

89

TP

TZT

2

α = Seebeck coefficient = Sκ = thermal conductivity(sometimes noted λ)σ = electrical conductivityP = power factor

• Good electrical

conductivity

• Low thermal

conductivity

• High thermoelectric

power


90


Electrical Conductivity σ

jN =s E

)(2 EfNDe e

coeff. diffusion density of states

s = e.n.me

Einstein’s Relation

e-density

mobility

D

me

=kT

e

91

*

.

m

kTD

Microscopic interpretation:

Related to band bending

elementarycharge

TE material: σ ≈ 103 (Ω.cm)-1


Thermal conductivity (λ or κ)

TjQ

le

κe : conductivity assisted by electrons

κl : conductivity assisted by phonons (through the lattice)

Relation of Wiedmann-Franz TL

e

L = 2.5 108 W.Ω/K2

Lorentz factor

Relation of Debye lvCVl3

1

Cv: heat capacityl: phonons mean free path v: sound velocity – fermi velocity

92

Fourier law

e

l

LZT

1

2

Thermal quanta

Thermal conductivity assisted by phonons (= quantized vibration of the lattice)

Thermoelectric materials

Strategies(1) bulk materials, rattlers, substructures « phonon-glass, electron-crystal » (2) low-dimensionnal systems(3) organic semi-conductors

Criteria / Design for TE materialscarrier concentration ≈ 1019-1021cm-3

Heavily doped semiconductors

High electrical conductivity

Atoms with similar electronegativity (but low ΔΧ low m*)crystalline structure: Mobility

Low thermal conductivity

Phonon scattering within the unit cell: rattling structures, point defects (interstitials, vacancies, alloys)Phonon scattering at interfaces: multiphase composites, superlatticesComplex crystalline structure : ↗ optic modes, ↘ acoustic modesHeavy elements: Reduced atom vibration

Large thermopower Large effective mass,

Thermoelectric effect /materials

95

Evolution of TE materials

Order of magnitude for Seebeck coefficient

• From B. Lenoir

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