Phase Transitions. Phys112 (S2012) 8 Phase Transitions 1

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Florence Shields
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1 Phase Transitions cf. Kittel and Krömer chap 10 Landau Free Energy/Enthalpy Second order phase transition Ferromagnetism First order phase transition Van der Waals Clausius Clapeyron coexistence curve Phys112 (S2012) 8 Phase Transitions 1

2 Landau Free Energy n particular Let us consider a system S exchanging energy with a large reservoir R. Constant volume σ tot ( U S ) = σ R U U S F L σ S σ R = σ R U ( ) + σ S ( U S ) ( U ) σ R U U S + σ S ( U S ) ( ) 1 ( U S ) U S + σ S = σ R ( U ) 1 F L ( U S ) with ( U S ) U S σ S ( U S ) Landau Free Energy ( U S ) = 1 U S σ S U S Most probable configuration maximizes σ tot i.e. minimizes F L e.g. for a perfect gas 3/2 MU S 3π 2 N + 5 N 2 V ( ) = N log U S = 3 2 N Phys112 (S2012) 8 Phase Transitions 2 F L R, U-U s S,U S (Sackur Tetrode with τ S = 2U s 3N ) ( U S ) U S

3 Landau Free Enthalpy Similarly, in situation where the system S exchanges both energy and volume with the reservoir, we can define a Landau Free Enthalpy Constant pressure σ tot ( U S, ) = σ R ( U U S,V ) + σ S ( U S, ) σ R ( U,V ) σ R U U σ R S V V + σ U S S (, S ) = σ R U,V ( ) 1 ( ) 1 U S + p R + σ S ( U S,V s ) ( ) with G L ( U S ) U S σ S ( U S, ) + p R = σ R U,V G L U S Most probable configuration maximizes σ Tot i.e. minimizes G L ( U S, ) σ in particular S ( U S, ) = 1 σ S ( U S, ) = p R G L U S e.g. for a perfect monoatomic spinless gas 3/2 MU S 3π σ S ( U S,V s ) 2 N = N log + 5 N (Sackur Tetrode with τ S = 2U s 2 3N ) U S U S = 3 2 N p R = N Phys112 (S2012) 8 Phase Transitions 3

4 More generally F L G L Order parameter We could define the configuration of the system S with order parameters ξ ι, in which case U S, are functions of ξ: U S ( ξ ), ( ξ ) Depending whether we are working at constant volume or constant pressure, the most likely configuration of the ensemble of the system and reservoir corresponds to the minimum of the Landau free energy or free entropy ( ξ ) U S ( ξ ) σ S ( ξ ) F L ξ i ξ ( ) ( ξ ) U S ( ξ ) σ S ( ξ ) + p R ξ = 0 at equilibrium (constant volume) ( ξ ) = 0 at equilibrium (constant pressure) ξ i ( ) G L Phys112 (S2012) 8 Phase Transitions 4

5 Second Order Phase Transitions Below a certain temperature some order appears τ < τ c F L τ = τ c τ goes down High Temperature ξ Does not happen for ideal gas For this to happen, we need interactions => correlation and non linearities Example: ferromagnetism Phys112 (S2012) 8 Phase Transitions 5

6 Ferromagnetism Ideal spin gas= paramagnetism In a magnetic field B, ε = mb if m is the magnetic moment. For 1 spin Z 1 = exp mb + exp mb = 2 cosh mb For N spins Z = Z N 1 since they are distinguishable mb Z = 2 N cosh N F = τ log( Z ) = Nτ log 2 cosh mb log Z U = τ 2 = NmB tanh mb τ Magnetization (moment/unit volume)m = U mb = nm tanh VB σ V = 1 ( τ log Z ) = n log 2cosh mb V τ mb mb tanh τ n log mb mb 3 Ferromagnets: The spins are producing a magnetic field which tends to orient the spins Mean field approximation B int = λm M. B int = λ 2 M 2 In absence of external field, energy per unit volume is u = 1 2 where the factor 1/2 accounts for the fact that we should not double count and only account for pair interactions Phys112 (S2012) 8 Phase Transitions 6

7 Landau Free Energy Ferromagnetism (2) Natural to use as the order parameter M: Replace in σ, mb int by tanh 1 M τ nm F L ( M ) σ = u( M ) V V = λ 2 M M 2 n log 2 cosh tanh 1 nm M M nm tanh 1 nm n log2 + λ 2 M 2 + τ n 2 R M 2 nm + no M 4 nm Critical temperature τ c = λnm 2 Curie temperature M No discontinuity! 2nd order T decreases Another point of view M = nm tanh mb int mλm = nm tanh if λnm 2 = τ c only one solution M = 0 if < τ c one non trivial solution M 0 Phys112 (S2012) 8 Phase Transitions 7 τ M nm tanh mλm M

8 First Order Transition So far F L (or G L ) was even in ξ No discontinuity in physical variables, no coexistence of 2 phases, no latent heat First order transition: e.g. Van der Waals gas Phys112 (S2012) 8 Phase Transitions 8

9 First Order Transition So far F L (or G L ) was even in ξ No discontinuity in physical variables, no coexistence of 2 phases, no latent heat First order transition: e.g. Van der Waals gas Ideal gas approximation is not good at high density ( ) φ r 1 r 2 hard core Van der Waals attraction 2 effects The volume available is not V: in the hard core approximation V V N 4πr 3 c V Nb 3 The energy of the molecules is decreased by Van der Waals attraction Mean field approximation U U N ( N 1) φ U N φ U N 2 a V φ( r) d 3 r r where φ = c 2a V V Phys112 (S2012) 8 Phase Transitions 9 r c r 1 r 2

10 G L Van der Waals phase transition Let us take as one order parameter (and kinetic energy U K ) Constant pressure Let us compute G L U S = U K N 2 a σ S ( U S,V s ) = N log ( U S, ) U S 3/2 MU K 3π 2 N N Nb Very different behavior liquid Coexistence p R < p C = a ( ) σ S ( U S, ) + p R 27b 2 τ goes down + 5 (Sackur Tetrode with τ S = 2U K 2 3N,V = Nb) or equivalently counting number of states G L p R p C = a 27b 2 τ goes down gas Metastability Super-heating/cooling gas Phys112 (S2012) 8 Phase Transitions 10

11 Van der Waals Equation of State G L = 0 = 1 3 U K 2 τ N 1 R U K = 3 U K 2 Nτ U = 3 R S 2 Nτ N 2 a R G L = 0 = N 2 a 1 τ 2 R N Nb + p R p R + N 2 a 2 ( Nb) = N Van der Waals equation of state Droping the R and S indices and defining V c = 3Nb p c = a 27b τ = 8a 2 c (they verify the equation of state) 27b the equation of state can be put in the form p p 3 + V 2 p c V 1 V c 3 = 8 p τ c 3 τ c V c τ=τ c Law of corresponding states p c (unified but not very good description) τ goes down It is easy to check that at p = p c V = V c τ = τ c liquid gas p V = 0! V V c V c Phys112 (S2012) 8 Phase Transitions 11

12 Van der Waals G L n Q ( τ c ) ( ) = 100 n V c Phys112 (S2012) 8 Phase Transitions 12 log

13 P-V Relationship ( ) Van der Waals equation of state G L U K, = 0 p R + N 2 a 2 ( ) = N = Nτ G L ( U K, ) V Nb U K = 0 Phys112 (S2012) 8 Phase Transitions 13

14 Recall At equilibirium G = Nµ ( p,τ ) Proof: Divide the system into two subsystems. The subsystems are in equilibrium G 1 = µ ( p,τ, N 1 ) = G 2 = µ ( p,τ, N 2 ) µ N 1 N 2 N = 0 When expressed as a fuction of p,τ, N G N = µ p,τ ( ) µ is not a function of N µ ( p,τ ) ( ) G = Nµ ( p,τ ) + f ( p,τ ) N = 0,G = 0 f ( p,τ ) 0 Phys112 (S2012) 8 Phase Transitions 14

15 G L liquid Coexistence Coexistence p R < p C = a 27b 2 p τ goes down p c critical point liquid gas gas τ For a given p R smaller than p c, there is a temperature c τ where the two minima are the same: coexistence of liquid and gas They are in equilibrium (same chemical potential) ( ) µ l p,τ G( V l ) = G V g ( ) = µ g ( p,τ ) since G( p,τ ) = µ ( p,τ ) N Latent heat The two phases have very different entropy. Gas has a much higher entropy than the liquid! This corresponds to a difference of enthalpy i.e. there is a latent heat involved in the transformation G l ( p,τ ) = G g ( p,τ ) using G( p,τ ) = F + pv τσ = H ( p,σ ) τσ H l τσ l = H g τσ g H l H g = τ ( σ g σ l ) = LN > 0 where L is the latent heat per particle released in thermal bath when going from gas to liquid needed from thermal bath to go from liquid to gas Phys112 (S2012) 8 Phase Transitions 15

16 Clausius Clapeyron ( ) = G V g ( ) µ l p,τ Coexistence (2) Phys112 (S2012) 8 Phase Transitions 16 ( ) = µ g ( p,τ ) since G( p,τ ) = µ ( p,τ ) N G V l G l ( p,τ ) = G g ( p,τ ) using G( p,τ ) = H ( p,σ ) τσ H l τσ l = H g τσ g ΔH = τδσ The shape of the coexistence curve p τ G l ( p( τ ),τ ) = G g ( p( τ ),τ ) G l p dp + G l τ dτ = G g p dp + G g τ G l dp dτ = ( ) can be obtained easily G τ, p, N dτ with ( ) τ = σ G ( τ, p, N ) p τ G g τ G g p G = σ g σ l = 1 LN 1 L l V g V l τ V g V l τ v g v l p where v g v l is the difference of volumes/particle If v g >> v l v g v l v g τ dp ( Ideal gas) p dτ = p τ L 2 In the case where L is weakly dependent on the temperature p τ ( ) = p o exp L τ = V

17 Conclusions Correlations between particles =>Non linear behavior=>phase transitions Mean field approximation as a first order 2 types of phase transitions First order coexistence of phases latent heat metastability (bulk, bubble/droplet) Second order continuous transformation (discontinuity in derivative) In 1st order, critical point Singularity! Large fluctuations Modern field of critical phenomena (Landau s approach is mathematically too regular) Phys112 (S2012) 8 Phase Transitions 17

5. Systems in contact with a thermal bath

5. Systems in contact with a thermal bath So far, isolated systems (micro-canonical methods) 5.1 Constant number of particles:kittel&kroemer Chap. 3 Boltzmann factor Partition function (canonical methods)