1st AND 2nd ORDER PHASE TRANSITION

TOPIC:1st AND 2nd ORDER PHASE TRANSITION
SUBJECT:#SOLID_STATE_PHYSICS

1. Topic 1 st Order and 2 nd Order Phase Transition Subject Advance Solid State Physics Class BS Physics (Morning) 7 th Semester

2. 1 st Order and 2 nd Order Phase transtion Phase Transition: Phase transition occur when a substance changes from a solid, liquid, or gas state to a different state. Every element and substance can make transition from one phase to another at a specific combination of temperature and pressure. There are two types of phase transition. ( i )1 st Order Phase Transition (ii) 2 nd order Phase Transition

3. Landau Theory of Phase Transition This theory tells about how phase occur and what happens to the system when phase transition occurs. When phase transition occurs in a system then for this system at constant temperature and volume F (Helmholtz function) is minimum and the system is in equilibrium. In fact, F tend to approach a minimum value for all processes in universe.

4. Landau Theory of Phase Transition F is the thermodynamic potential at constant volume of a system and is given by F = U-TS Here U is internal energy T is temperature and S is entropy. Suppose F is minimum with respect to single order parameter x, which is different for different cases. e.g. x = magnetization of ferromagnetic substance, superconducting electrons in case of superconductor.

5. Landau Theory of Phase Transition The parameter x further depends on temperature. Here is an important thing that for each system to describe it as best as possible the F will be the function of T and x also, the system should be in equilibrium at constant T and V . According to Landau the free energy function or Helmholtz function is given as F L = U( x,τ ) - τ S( x,τ )…………..(1)

6. Landau Theory of Phase Transition F (τ) is minimum at a particular value of x and T at values other than the particular value F (τ) has larger value. F L (x, τ ) <= F L (x, τ) at x ≠ x ……..(2) For most of ferromagnetic substances F L (x, τ) = f (τ) + 1/2f 2 (τ)x 2 + 1/4f 4 (τ)x 4 +……… (3) f (τ) , f 2 (τ) and f 4 (τ) are the coefficients of expansion and they are always function of temperature. For phase change only, f 2 (τ) changes and is given by f 2 (τ) = α (τ- τ )…………………. (5)

7. Landau Theory of Phase Transition By putting the value of f 2 (τ) from equation 5 to 4 we get F L (x, τ) = f (τ) + 1/2 α (τ- τ 0) x 2 + 1/4f 4 (τ)x 4 ……………. (6) At equilibrium change in F with respect to x is zero so above equation becomes α (τ- τ 0) x + f 4 x 3 = 0 from here we can obtain three values of x which are x= 0 and x 2 = α/f 4 (τ – τ) for x = 0 f(τ) = f (τ) , This is valid when temperature is about τ

8. Landau Theory of Phase Transition For x = 0 f(τ) = f (τ) , This is valid when temperature is about τ . x 2 = α/f 4 (τ – τ) this value exist when τ is less than τ 0. When x is imaginary we can’t decide what type transition occurs their. Putting value of x 2 in equation 6 we get F L (x, τ) = f (τ) - 1/2 α 2 (τ- τ ) 2 …………………….(7)

9. First Order Phase Transition A transition is said to be first order phase transition if a finite amount of heat which is called latent heat is supplied for transition. During such a transition, a system releases or absorbs a fixed (typically large) amount of energy per unit volume. This transition is also called discontinuous phase transition. Latent Heat: The heat required to convert a specific amount of solid into liquid and liquid into solid without change in temperature. e.g. for 1g of ice it is 80 calrie .

10. First Order Phase Transition When we want to convert solid into liquid e.g. ice into water then we supply latent heat Solid → Liquid → Gas N 1 N 2 N 3 L= finite latent heat Here N 1 , N 2 and N 3 are number of particles in solid, liquid and gas phase respectively. Then Gibbs energy per particle is given by = g ……………………. (1) Here G is Gibbs energy g is Gibbs energy per particle  

11. First Order Phase Transition If we have N 1 number of particles in solid state and N 2 number of particles in liquid state then total number of particles are N which are remains constant and given by N = N 1 + N 2 …………………. (2) The Gibbs energy for these particles is given by G= G 1 + G 2 ……………………. (3) By using equation 1 and solving for 3 we ge G= N 1 g 1 +N 2 g 2 …………………. (4)

12. First Order Phase Transition From equation 2 we get ∂N = ∂N 1 + ∂N 2 ……………………. (5) As the total number of particles remains constant so, we get ∂N 1 = - ∂N 2 …………………………………… (6) By using equation 6 in 4 we get ∂N 2 (g 1 – g 2 ) = 0 ……………………... (7) As ∂N 2 ≠ 0 so g 1 – g 2 = 0 hence g 1 = g 2 Similarly for liquid to gas transition g 3 = g 2 At a certain temperature and pressure g 1 = g 2 = g 3 at this point three states co-exist and it is called triple point.

13. First Order Phase Transition From above calculations we can say that Gibbs energy per particle for each state is constant in first order phase transition. Here G = H – TS Where H = enthalpy of the system.

14. First Order Phase Transition Enthalpy: The sum of internal energy of a system plus product of pressure and volume is called enthalpy. The graph on previous page shows that Gibbs function decreases when temperature increase. As the temperature increase enthalpy also increases but product of T and S increases sharply as compared to H. So Gibbs function decrease.

15. First Order Phase Transition If in a process pressure is constant then the process is said to be iso – baric. The Gibbs energy is given by G= Vdp – SdT ………………. (8) -( )= 0 – SdT /d S= -( ) P ……………… (9)  

16. First Order Phase Transition This graph shows that entropy shows discontinuity with temperature. In first curve the temperature increases entropy also increases but a time comes when T becomes constant and entropy increases at this time latent heat is used to free the particles from their substance or it is used against the interatomic forces. After this entropy start to increase again.

17. 2 nd Order Phase Transition The transition is said to 2 nd order when no latent heat is supplied to the substance for transition. This transition is also called continuous phase transition. Example: ( i ) Conversion of ferromagnetic substance into paramagnetic and para to diamagnetic substances. If we increase the temperature of a ferromagnetic substance, then at a certain temperature it becomes paramagnetic upon further increase in temperature the para becomes diamagnetic.

18. 2 nd Order Phase Transition (ii)Conversion of ordinary solid into Superconductor and vice versa. When we decrease the temperature of a substance to a certain level it becomes superconductor. No latent heat is supplied in this process. And when we increase the temperature to small extent the superconductor start to loss its superconducting electrons.

19. Graphical Behavior in 2 nd Order Transition The graphical behavior of Gibbs function and entropy for 2 nd order phase transition is shown below the discontinuity of curve is not observed in this case because no latent heat is supplied.

20. References https://chem.libretexts.org/Fundamentals_of_Phase_Transitions https://www.youtube.com/watch?v=TrCCQru2ru0 https://www.youtube.com/watch?v=R9Xj3bII0jg https://www.youtube.com/watch?v=L9WLBgUwvDA