emmanuel trizac lptms / university paris-saclay · 2020-01-13 · introduction to phase transitions...
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Statistical physics Statistical physics
Emmanuel TrizacLPTMS / University Paris-Saclay
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I Introduction to phase transitions and critical phenomena1- The problems raised by phase transitions, from a statistical mechanics perspective2- Classification of phase transitions3- The drosophila of phase transitions4- Order parameter and symmetry breakdown5- Local order and correlation functions : from magnets to liquids
II First order phase transitions1- Unstable isotherms, double-tangent and Maxwell construction2- Spinodal and binodal3- Changing ensembles4- van der Waals equation5- The case of mixtures
III Critical phenomena : qualitative approaches1- Weis molecular field2- Variational mean-field3- Critical exponents4- Landau theory5- Correlation functions and Ginsburg-Landau functional6- Validity of mean-field
OutlineOutline
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Statistical mechanics : considered by some as one of the most intellectually challenging subject.
Noted for a high suicide rate among founders
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PrerequisitesPrerequisites
Elementary statistical mechanics
Honest person toolbox in mathematicsLinear algebra
Probability theory
Complex analysis
f (x)=(x2
−1)( x−2−i)2
x2+2+2 i
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Introduction to phase transitions and critical phenomena
1- Problems raised by phase transitions, from a stat mech perspective2- Classification of phase transitions3- The drosophila of phase transitions4- Order parameter and symmetry breakdown5- Local order and correlation functions : from magnets to liquids
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Surfactant moleculesSurfactant moleculesand micelle formationand micelle formation
Basic mechanism
Opens for considerable richness
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Various surfactant phasesVarious surfactant phases
(a) An amphiphilic molecule(b) Spherical micelle(c) Cylindrical micelle(d) Cubic phase(e) Inverse micelle
(f) Hexagonal phase(g) Bicontinuous cubic structure(h) Vesicle(i) Lamellar phase(j) Sponge phase
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pressure
density
Identify an Identify an ORDER PARAMETERORDER PARAMETER
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Complexity / variety of phase changesComplexity / variety of phase changes
From J. Yeomans, Statistical mechanics of phase transitions (Oxford)
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Non spherical colloids...Non spherical colloids...
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Phase behaviour of Phase behaviour of hard sphero-cylindershard sphero-cylinders
P. Bolhuis, 1996
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Introduction to phase transitions and critical phenomena
1- Problems raised by phase transitions, from a stat mech perspective2- Classification of phase transitions3- The drosophila of phase transitions4- Order parameter and symmetry breakdown5- Local order and correlation functions : from magnets to liquids
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Examples and exceptionsExamples and exceptions
Generic case
Water
dPdT
(coex)=1T
Lliq→ gas
v gas−v liq
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What about magnets ?What about magnets ?
magnetization
magnetic field
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MAGNETS
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Analogy between magnets and liquids ?Analogy between magnets and liquids ?
pressure
magnetizationdensity
magnetic field
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fluids
magnets
Definition of critical exponentsDefinition of critical exponents
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Why are critical exponents interesting ?Why are critical exponents interesting ?→ several layers of universality→ several layers of universality
E. A. Guggenheim, J. Chem. Phys. 13, 253 (1945)
Tc (K) P
c (atm)
Ne 45 26
Ar 150 48
Kr 209 54
Xe 290 58
N2
126 33
O2
154 50
CO 133 34
CH4
190 45
Corresponding states for liquid-gas transition
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There is more to universalityThere is more to universality
Liquid-gas transition:
Magnets with uniaxial anisotropy (MnF2):
Phase separation in binary mixture (CCl4+C7F16):
3d Ising model on cubic lattice, fcc etc…:
→ all belong to the same universality class
What matters is space dimension + symmetry of order parametere.g. for Ising in 2D :
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Introduction to phase transitions and critical phenomena
1- Problems raised by phase transitions, from a stat mech perspective2- Classification of phase transitions3- Ising model : the drosophila of phase transitions4- Order parameter and symmetry breakdown5- Local order and correlation functions : from magnets to liquids