Ensembles II: The Gibbs Ensemble (Chap. 8) Bianca M. Mladek University of Vienna, Austria

1. Ensembles II: • The Gibbs Ensemble (Chap. 8) • Bianca M. Mladek • University of Vienna, Austria • bianca.mladek@univie.ac.at

2. Our aim: We want to determine the phase diagram of a given system. gas temperature liquid density For this, we need to know the coexistence densities at given temperature.

3. In experiments: first order phase transition is easy to locate: at right density and temperature  phase separation (two distinct phases devided by interface) In simulations: ???

4. Find points where: Gibbs …which is the condition for phase coexistence in a one-component system.

5. Problem in simulations: Hysteresis: pressure density

6. NVT Ensemble fluid fluid Let‘s lower the temperature…

7. NVT Ensemble gas gas liquid liquid Problem: The systems we study are usually small large fraction of all particles resides in/near interface.

8. Possible solution #1 larger systems:  we need huge systems  computationally expensive

9. Possible solution #2: “ μPT ”-Ensemble • Problem: no such ensemble exists! • μ, P and T are intensive parameters • extensive ones unbounded • We have to fix at least one extensive variable (such as N or V )

10. Possible solution #3:The Gibbs ensemble gas liquid achieve equilibrium by coupling them A. Z. Panagiotopoulos, 1987.

12. Overall system: NVT ensemble N = N1 + N2 V = V1 + V2 T1 = T2

13. distribute N1 particles • change the volume V1 • displace the particles partition function:

14. partition function: Distribute N1 particles over two volumes:

15. partition function: Integrate volume V1

16. partition function: Displace the particles in box1 and box2

17. partition function: Displace the particles in box 1 and box2 scaled coordinates in [0,1)

18. partition function: probability distribution:

19. 3 different kinds of trial moves: Particle displacement Volume change  equal P Particle exchange  equal μ

20. Acceptance rules Detailed Balance:

21. Displacement of a particle in box1

22. Displacement of a particle in box1

23. Volume change

24. Volume change More efficient: random walk in ln [V1/(V-V1)]

25. Moving a particle from box1 to box2 acceptance rule:

26. Moving a particle from box1 to box2

27. Moving a particle from box1 to box2

28. detailed balance!!!

34. Analyzing the results (1) well below Tc approaching Tc

35. Analyzing the results (2) well below Tc approaching Tc

36. Analyzing the results (3) well below Tc approaching Tc

37. Advantages: • single simulation to study phase coexistence: system „finds“ the densities of coexisting phases • free energies/chem. potentials need not be calculated • significant reduction of computer timeDisadvantages: • only for vapor-liquid and liquid-liquid coexistence • not very successful for dense phases (particle insertion!)

38. “C“ IS FOR COFFEE BREAK!

39. Ensembles II: 2. Free energy of solids (Chap 7 & 10) 3. Case study: cluster solids

40. Gibbs ensemble for solid phases? problem: swap moves are unlikely to be accepted. μ1 ≠ μ2 Gibbs ensemble does not work What other method?

41. Problem: With normal Monte Carlo simulations, we cannot compute “thermal” quantities, such as S, F and G, because they depend on the total volume of accessible phase space. For example: with

42. Solution: thermodynamic integration system of interest Coupling parameter reference system Free energy as ensemble average!

43. F() F(1) F(0) 0 1  Why ? The second derivative is ALWAYS negative: Therefore: Good test of simulation results…!

44. Words of caution Integration has to be along a reversible path. interested in solid  reference system has to be solid Why? Because there is no reversible path coming from the ideal gas temperature liquid solid gas density

45. Standard reference system: Einstein solid Einstein crystal: non-interacting particles coupled to their lattice sites by harmonic springs

46. Einstein solid: recipe • For fixed crystal structure: • At fixed T and : • make simulations at different values of  • measure • Numerically integrate using e.g. Gauss-Legendre quadrature • determine

47. Standard reference system: Einstein solid • special care for discontinuous potentials: not possible to linearly switch off interaction • diverging short-range repulsion: for=1, particles can overlap  weak divergence in • choose ‘s such that • Frenkel - Smit: „Thermodyn. integration for solids“ • ~everyone else: „Frenkel-Ladd“ (JCP, 1984)

48. Common tangent construction Now that we have F(V;T) at hand: common tangent construction: P = - (F/ V)N,T equal pressure  equal slope F = μ – PV equal chemical potentials  equal intercept free energy/ particle volume/ particle

49. But now consider the following systems: Coarse graining: effective particleeffective interaction mesoscopic particlecomplex interactions Effective interactions can be tuned  bounded, purely repulsive potentials

50. Effective interactions: soft and bounded Gaussian Core Model Penetrable Spheres potential potential distance distance temperature temperature density density … are models for effective interactions of polymers etc.