Thermodynamics of phase transitions Katarzyna Sznajd-Weron Institute of Physics Wroc law University of Technology, Poland 7 Oct 2013, SF-MTPT Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 1 / 25
Thermodynamics of phase transitions
Katarzyna Sznajd-Weron
Institute of PhysicsWroc law University of Technology, Poland
7 Oct 2013, SF-MTPT
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 1 / 25
Literature
H. B. Callen, Thermodynamics and Introduction toThermostatistics, John Wiley & Sons, Inc. (1985)
J. J. Binney, N. J. Dowrick, A. J. Fisher, and M. E. J. Newman,The Theory of Critical Phenomena. An Introduction to theRenormalization Group, Clarendon Press (1992)
H. E. Stanley, Introduction to Phase Transitions and CriticalPhenomena, Oxford University Press (1971)
K. Christensen and N. R. Moloney, Complexity and Criticality,Imperial College Press (2005)
S. Salinas, Introduction to Statistical Physics (1997)
M. Plischke and B. Bergersen, Equilibrium Statistical Physics(1989)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 2 / 25
Phase transitions - amazing!
Sea level
Tibet
Ice Water Steam
Figure : A part of the phase diagram of water. Source:http://www.chemicalogic.com.
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 3 / 25
Critical Point
Triple Point
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
0 100 200 300 400 500 600 700 800
Pre
ssu
re (
bar)
Temperature (K)
Phase Diagram: Water - Ice - Steam
Saturation Line
Sublimation Line
Ice I Line
Ice III Line
Ice V Line
Ice VI Line
Ice VII Line
Copyright © 1998 ChemicaLogic Corporation.
Vapor
Solid
Liquid
Sublimation Line
Saturation LineMelting Line
(Ice I)
Melting Line(Ice III)
Melting Line(Ice V)
Melting Line(Ice VI)
Melting Line(Ice VII)
Figure : The complete phase diagram of water. Source:http://www.chemicalogic.com., W. Wagner, A. Saul, A. Pru:International Equations for the Pressure along the Melting and along theSublimation Curve for Ordinary Water Substance, J. Phys. Chem. Ref.Data 23, No 3 (1994) 515
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 4 / 25
Continuous and discontinuous phase transitions
Figure : A schematic phase diagram gas-liquid-solid, and the relationshipbetween the heat supplied to the system and the temperature (CoolingCurve).
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 5 / 25
Experiments - critical Curie Point and supercritical
fluid
M.I.T. - Walter Lewin - Ferromagnetic Curie Point[http://www.youtube.com/watch?v=X8ZHQQUusGo]
Poliakoff - Supercritical Fluids[http://www.youtube.com/watch?v=yBRdBrnIlTQ]
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 6 / 25
Metastable states and hysteresis
Hysteresis – the dependence of a system not only on its currentenvironment but also on its past environment.
Supercooled and superheated states.
In the solid-liquid phase transition hysteresisoccurs when the temperature of melting and freezing are different.
Agar melts at about 850C and freezes in the range of320C to 400C. This means that agar melted at 850Cremains in a liquid state up to 850C. On the otherhand, if it is initially in the solid state it remains in thisstate up to 850C. Therefore, at temperatures40−−850C agar may be in a liquid or a solid state,depending on the history (initial state).
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 7 / 25
Hand warmer - how does it work?
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 8 / 25
Hand warmer - Supersaturated solution
Generate heat through the exothermic crystallisation ofsupersaturated solutions (typically sodium acetate)
The release of heat is triggered by flexing a small metal disk,which generates nucleation centers that initiate crystallization
Can be recharged by immersing the hand-warmer in very hotwater until the contents are uniformly fluid and then allowing itto cool
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 9 / 25
Modern classification of phase transitions
Latent heat - heat released or absorbed by a system during aconstant-temperature process (phase transition)
Latent heat - energy required to transfer a particle from onephase to another
Latent heat - allows to distinguish between continuous anddiscontinuous phase transitions
Continuous phase transitions - without latent heat, no phasecoexistence, no hysteresis
Discontinuous phase transitions - latent heat, phase coexistence,metastable states (hysteresis)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 10 / 25
Fluctuations and critical point - critical opalescence
Phenomenon which arises in the region of a continuous phasetransition
At critical point density fluctuations become of a sizecomparable to the wavelength of light
The light is scattered and causes the normally transparent liquidto appear cloudy
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 11 / 25
Order parameter
Spontaneous symmetry breaking at critical pointOrder parameter φ – a measure of the degree of order in asystemφ 6= 0 below the critical pointφ = 0 above the critical point
Source:
www.nobelprize.org/nobel prizes/physics/laureates/2008/popular-
physicsprize2008.pdfKatarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 12 / 25
Order parameter - examples
Phase transition Order parameterliquid-gas density
ferro-paramagnetic magnetizationantyferro-paramagnetic sublattice magnetization
Bose-Einstein condensate wave functionsuperfluidity wave function of He4
superconductivity wave function of Cooper pairUniverse ω+−ω−
ω++ω−
Life ωL−ωR
ωL+ωR
Table : Broken symmetry – ω± denotes the number of particles andantiparticles, ωL,R the number of left-handed and right-handed aminoacids. The proteins in living creatures consist only of left-handed aminoacids.
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 13 / 25
Correlation function
φ(ri) = φ + δφ(ri). (1)
Correlation function:
G (ri , rj) = 〈φ(ri)φ(rj)〉 = φ2 + 〈δφ(ri)δφ(rj)〉 . (2)
First term i.e. φ2 describes long-range order and the second termdescribes short-range order
Gf (ri , rj) = 〈δφ(ri)δφ(rj)〉 (3)
For T = Tc :
Gf (r) ∼ 1
rd−2−η . (4)
Beyond the critical point:
Gf (r) ∼ exp
(− r
ξ
). (5)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 14 / 25
Critical point and correlations
In general
Gf (r) ∼exp
(− rξ
)rd−2−η . (6)
Correlation length ξ – characteristic length of a correlated region(very important in modern theory of phase transition!!!). Criticalstate:
T → Tc ⇒ ξ →∞. (7)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 15 / 25
Critical exponents and universality classes
critical exponent dependence value for Fe
α c ∼ |T − Tc |−α ∼ −0.03β φ ∼ |T − Tc |−β ∼ −0.37γ κT ∼ |T − Tc |−γ ∼ 1.33η G (r) ∼ r 2−d−η ∼ 0.07ν ξ ∼ |T − Tc |−ν ∼ 0.69
Table : Critical exponents
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 16 / 25
Universality
Source: H. E. Stanley, Introduction to Phase Transitions and CriticalPhenomena, Oxford University Press (1971)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 17 / 25
Phase transitions can be studied at two levels:
Macroscopic - Thermodynamics (How?)
Microscopic - Statistical Physics (Why?)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 18 / 25
Equilibrium
Macroscopic phenomenon refers to a time scale much largerthan the scale of the microscopic movement
The particular state of motion (microscopic) – duringobservation the macroscopic state is constant
Thermodynamics is based on the assumption that under givenenvironmental conditions the system has clearly defined theequilibrium properties
External conditions are determined by external parameters suchas temperature, pressure, magnetic field, etc.
Different environmental conditions — different equilibriumproperties of the system
State function – describes the equilibrium state of a system (aproperty of a system that depends only on the current state ofthe system)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 19 / 25
What is the equilibrium state for ...?
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 20 / 25
Properties of equilibrium
Macroscopic state is independent of time
Independent on the history, unequivocal
May be described by small number of macroscopic parameters
Macroscopic state in equilibrium - the most random state underthe circumstances (from microscopic point of view)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 21 / 25
The energy of the system can be changed by
external forces that perform work
Work dW performed by tension f that extends a metal rod bythe length dX :
dW = −fdX . (8)
External magnetic field h does the work (an increase ofmagnetization):
dW = −hdM . (9)
Pressure p is the force that changes the volume by dV , and acorresponding work:
dW = pdV (10)
Chemical potential is the force that changes the number ofparticles N :
dW = −µdN (11)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 22 / 25
Thermodynamic (macroscopic) parameters
Generalized coordinates defining the state of the system:distance X , magnetization M , volume V , the number ofparticles N
Generalized external forces: tension f , magnetic field h, pressurep, chemical potential µ
Can you find some common property of generalized coordinates?
Can you find some common property of generalized forces?
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 23 / 25
Extensive and intensive parameters
1 Extensive (additive) – the value of such a parameter for thewhole system is equal to the sum of the parameters forsubsystems making up the system. Examples of such parametersare distance X , magnetization M , volume V , the number ofparticles N
2 Intensive – the value for the whole system is equal to the valueof that parameter for each of the identical subsystems makingup a given system. Examples of such parameters aretemperature T , pressure p, chemical potential µ and ...
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 24 / 25
Is the change of energy possible without work
(change in the macroscopic coordinates)?
Heat - the result of changes in microscopic motion
Do we need all microscopic coordinates (coordinates of allparticles) to describe these changes?
We introduce a new generalized coordinate to describe themicroscopic motion in a collective way
Entropy S is a new generalized coordinate and correspondinggeneralized force?
dQ = TdS (12)
Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 25 / 25