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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
PHY579 – JF Guillemoles 4
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)
PHY579 – JF Guillemoles 12
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
PHY579 – JF Guillemoles 14
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)
PHY579 – JF Guillemoles 15
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
PHY579 – JF Guillemoles 16
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)
PHY579 – JF Guillemoles 21
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
PHY579 – JF Guillemoles 25
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
PHY579 – JF Guillemoles 26
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):
PHY579 – JF Guillemoles 36
Continu
Partitionné
NB: Onsager relationships
Coupled fluxes and forces
PHY579 – JF Guillemoles 38
• 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)
PHY579 – JF Guillemoles 39
Dissipative Transport
• Transport is dissipative
o e.g. RI²
PHY579 – JF Guillemoles 40
Coupled Transport
PHY579 – JF Guillemoles 41
• 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?
PHY579 – JF Guillemoles 42
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
PHY579 – JF Guillemoles 43
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
Thermoelectric effect
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
PHY579 – JF Guillemoles 69
• 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
PHY579 – JF Guillemoles 72
• 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
PHY579 – JF Guillemoles 74
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
PHY579 – JF Guillemoles 75
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
Thermoelectric effect
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
Thermoelectric effect
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
Thermoelectric effect
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)
Thermoelectric effect
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
Thermoelectric effect
• 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)
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
83
E = electro-chemical field
Thermoelectric effect
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
Thermoelectric effect
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
Thermoelectric effect
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
Thermoelectric effect
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
Thermoelectric effect
TP
TZT
2
88
α = Seebeck coefficientκ = thermal conductivity(sometimes noted λ)σ = electrical conductivityP = power factor
Thermoelectric effect
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
Thermoelectric effect
90
Thermoelectric effect
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
Thermoelectric effect
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