Thermodynamics of phase transitions

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1 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

2 Literature H. B. Callen, Thermodynamics and Introduction to Thermostatistics, 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 the Renormalization Group, Clarendon Press (1992) H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, 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

3 Phase transitions - amazing! Sea level Tibet Ice Water Steam Figure : A part of the phase diagram of water. Source: Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 3 / 25

4 Pressure (bar) Critical Point Phase Diagram: Water - Ice - Steam 1.E+06 1.E+05 Saturation Line Sublimation Line Melting Line (Ice VI) Melting Line (Ice VII) 1.E+04 Ice I Line Ice III Line Ice V Line Melting Line (Ice V) 1.E+03 Ice VI Line Ice VII Line Melting Line (Ice III) Liquid 1.E+02 Melting Line (Ice I) Saturation Line 1.E+01 1.E+00 Solid 1.E-01 1.E-02 Triple Point 1.E-03 1.E-04 1.E-05 Sublimation Line Vapor 1.E-06 1.E Temperature (K) Copyright 1998 ChemicaLogic Corporation. Figure : The complete phase diagram of water. Source: W. Wagner, A. Saul, A. Pru: International Equations for the Pressure along the Melting and along the Sublimation 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

5 Continuous and discontinuous phase transitions Figure : A schematic phase diagram gas-liquid-solid, and the relationship between the heat supplied to the system and the temperature (Cooling Curve). Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 5 / 25

6 Experiments - critical Curie Point and supercritical fluid M.I.T. - Walter Lewin - Ferromagnetic Curie Point [ Poliakoff - Supercritical Fluids [ Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 6 / 25

7 Metastable states and hysteresis Hysteresis the dependence of a system not only on its current environment but also on its past environment. Supercooled and superheated states. In the solid-liquid phase transition hysteresis occurs when the temperature of melting and freezing are different. Agar melts at about 85 0 C and freezes in the range of 32 0 C to 40 0 C. This means that agar melted at 85 0 C remains in a liquid state up to 85 0 C. On the other hand, if it is initially in the solid state it remains in this state up to 85 0 C. Therefore, at temperatures C 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

8 Hand warmer - how does it work? Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 8 / 25

9 Hand warmer - Supersaturated solution Generate heat through the exothermic crystallisation of supersaturated 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 hot water until the contents are uniformly fluid and then allowing it to cool Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 9 / 25

10 Modern classification of phase transitions Latent heat - heat released or absorbed by a system during a constant-temperature process (phase transition) Latent heat - energy required to transfer a particle from one phase to another Latent heat - allows to distinguish between continuous and discontinuous phase transitions Continuous phase transitions - without latent heat, no phase coexistence, 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 phase transition At critical point density fluctuations become of a size comparable to

11 Fluctuations and critical point - critical opalescence Phenomenon which arises in the region of a continuous phase transition At critical point density fluctuations become of a size comparable to the wavelength of light The light is scattered and causes the normally transparent liquid to appear cloudy Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 11 / 25

Order parameter Spontaneous symmetry breaking at critical point Order parameter φ a measure of the degree of order in a system φ 0 below the critical point φ = 0 above the critical

12 Order parameter Spontaneous symmetry breaking at critical point Order parameter φ a measure of the degree of order in a system φ 0 below the critical point φ = 0 above the critical point Source: prizes/physics/laureates/2008/popularphysicsprize2008.pdf Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 12 / 25

13 Order parameter - examples Phase transition Order parameter liquid-gas density ferro-paramagnetic magnetization antyferro-paramagnetic sublattice magnetization Bose-Einstein condensate wave function superfluidity wave function of He 4 superconductivity wave function of Cooper pair ω Universe + ω ω ++ω Life ω L ω R ω L +ω R Table : Broken symmetry ω ± denotes the number of particles and antiparticles, ω L,R the number of left-handed and right-handed amino acids. The proteins in living creatures consist only of left-handed amino acids. Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 13 / 25

14 Correlation function Correlation function: φ(r i ) = φ + δφ(r i ). (1) G(r i, r j ) = φ(r i )φ(r j ) = φ 2 + δφ(r i )δφ(r j ). (2) First term i.e. φ 2 describes long-range order and the second term describes short-range order G f (r i, r j ) = δφ(r i )δφ(r j ) (3) For T = T c : G f (r) 1. (4) r d 2 η Beyond the critical point: ( G f (r) exp r ). (5) ξ Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 14 / 25

15 Critical point and correlations In general ( ) exp r ξ G f (r). (6) r d 2 η Correlation length ξ characteristic length of a correlated region (very important in modern theory of phase transition!!!). Critical state: T T c ξ. (7) Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 15 / 25

16 Critical exponents and universality classes critical exponent dependence value for Fe α c T T c α 0.03 β φ T T c β 0.37 γ κ T T T c γ 1.33 η G(r) r 2 d η 0.07 ν ξ T T c ν 0.69 Table : Critical exponents Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 16 / 25

17 Universality Source: H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press (1971) Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 17 / 25

18 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

19 Equilibrium Macroscopic phenomenon refers to a time scale much larger than the scale of the microscopic movement The particular state of motion (microscopic) during observation the macroscopic state is constant Thermodynamics is based on the assumption that under given environmental conditions the system has clearly defined the equilibrium properties External conditions are determined by external parameters such as temperature, pressure, magnetic field, etc. Different environmental conditions different equilibrium properties of the system State function describes the equilibrium state of a system (a property of a system that depends only on the current state of the system) Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 19 / 25

20 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

21 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 under the circumstances (from microscopic point of view) Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 21 / 25

22 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 by the length dx : dw = fdx. (8) External magnetic field h does the work (an increase of magnetization): dw = hdm. (9) Pressure p is the force that changes the volume by dv, and a corresponding work: dw = pdv (10) Chemical potential is the force that changes the number of particles N: dw = µdn (11) Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 22 / 25

23 Thermodynamic (macroscopic) parameters Generalized coordinates defining the state of the system: distance X, magnetization M, volume V, the number of particles N Generalized external forces: tension f, magnetic field h, pressure p, 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

24 Extensive and intensive parameters 1 Extensive (additive) the value of such a parameter for the whole system is equal to the sum of the parameters for subsystems making up the system. Examples of such parameters are distance X, magnetization M, volume V, the number of particles N 2 Intensive the value for the whole system is equal to the value of that parameter for each of the identical subsystems making up a given system. Examples of such parameters are temperature T, pressure p, chemical potential µ and... Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 24 / 25

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 all particles) to describe these changes? We introduce a new generalized coordinate to describe the microscopic motion in a collective way Entropy S is a new generalized coordinate and corresponding generalized force? dq = TdS (12) Katarzyna Sznajd-Weron (WUT) Thermodynamics of phase transitions 7 Oct 2013, SF-MTPT 25 / 25

Thermodynamics of phase transitions

Thermodynamics of phase transitions Katarzyna Sznajd-Weron Department of Theoretical of Physics Wroc law University of Science and Technology, Poland March 12, 2017 Katarzyna Sznajd-Weron (WUST) Thermodynamics