Katarzyna Sznajd-Weron - if.pwr.wroc.plkatarzynaweron/students/TPF_ang/20131007_TPF1.pdf ·...

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)


Phase transitions - amazing!

Sea level

Tibet

Ice Water Steam

Figure : A part of the phase diagram of water. Source:http://www.chemicalogic.com.


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


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).


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]


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).


Hand warmer - how does it work?


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


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)


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


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.


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)


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)


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


Universality

Source: H. E. Stanley, Introduction to Phase Transitions and CriticalPhenomena, Oxford University Press (1971)


Phase transitions can be studied at two levels:

Macroscopic - Thermodynamics (How?)

Microscopic - Statistical Physics (Why?)


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)


What is the equilibrium state for ...?


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)


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)


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?


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 ...


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)


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