1. 1 Fall, 2013 Instructor: J.-W. John Cheng Mech. Engr. Dept., Nat ’ l Chung Cheng Univ. A2. Boltzmann Distribution Interfacial Physics and Thin-Film Processing

2. JCheng201210 2 Statement of Boltzmann Distribution Consider a system in contact with a heat bath and maintaining a constant number of particles Let  denote a configuration of the system and w(  ) be the associated potential function Boltzmann distribution theorem for discrete systems Probability the system being in configuration , P(  ), is proportional to Boltzmann distribution theorem for continuous sys Probability the system configuration being in ( ,  +d  ), p(  )d , is proportional to

3. JCheng201210 3

4. 4 Application: Air Density in Atmosphere Let  (z) denote the air density at altitude z Show

5. JCheng201210 5 Application: Distribution of Charged Particles Let  i &  i respectively be the electrical potential & number density of charged particles at region i, i = 1, …, n Show

6. JCheng201210 6 Application: Dissolution Let  1 &  2 : the intermolecular energies of a molecule in solid and solution phases, respectively X 1 & X 2 : the molar fractions of molecules in solid and solution phases, respectively (Note X 1 + X 2 = 1) X 2 is called the solubility of the solid Show

7. JCheng201210 7 Application: Dissolution An Example Consider two immiscible liquids such as water & oil. Assume a spherical oil is taken out of the oil phase and placed in the water phase This creates an unfavorable energy increase proportional to the area of the spherical oil multiplied by the interfacial energy, , of the oil-water interface Consider water and cyclohexane (an oil) with  = 50 mJ/m 2, radius(cyclohexane) = 0.28 nm Estimate solubility of cyclohexane in water at 25 o C (Ans: X 2 /X 1 = 6x10 -6, X i in mole fraction) contd

8. JCheng201210 8 Application: Orientation Distribution of Molecules For example, Consider pair potential of two anisotropic molecules, w(r,  ) with r denoting the distance and  the orientation Then the angular distribution of the molecular pairs of two molecules at a fixed distance r apart will be where with (Boltzmann dist. Theorem)