Phase Equilibria and Molecular Solutions Jan G. Korvink and Evgenii Rudnyi IMTEK Albert Ludwig University Freiburg, Germany

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1 Phase Equilibria and Molecular Solutions Jan G. Korvink and Evgenii Rudnyi IMTEK Albert Ludwig University Freiburg, Germany

2 Preliminaries Learning Goals Phase Equilibria Phase diagrams and classical thermodynamics Simulations of phase equilibria Molecular solutions Quantum chemistry: molecule in the solution Simulating molecule in the solution References Leach, A.R., Molecular modelling: principles and applications, 2001

3 On-line resources Reviews of A.Z. Panagiotopoulos Monte Carlo Methods for Phase Equilibria of Fluids Gibbs Ensemble Techniques Molecular simulation of phase equilibria Free Energy Simulations Introduction to Continuum Electrostatics, with Molecular Applications (Mike Gilson)

4 Phase diagrams Introduction to phase diagrams Equilibria criteria Phase molar Gibbs energy as a thermodynamic model Thermodynamics simulation CALPHAD Problem to compute a partition function

5 Introduction to phase diagrams One-component phase diagram: S - solid, L - liquid, V - vapor Temperature-density and temperature-pressure T Τ ρ p

6 Equilibria criteria U, V const -> max S T, V const -> min F T, p const -> min G Phase 1: Phase 2: T 1 S 1 = du 1 + p 1 dv 1 µ 1 dn 1 T 2 S 2 = du 2 + p 2 dv 2 µ 2 dn 2 U - internal energy, V - volume, n - number of moles, T - temperature, p - pressure, µ - chemical potential Panagiotopolus, Maximize S = S 1 + S 2 at U 1 + U 2 = const and V 1 + V 2 = const Result: thermal T 1 = T 2, mechanical p 1 = p 2, and phase µ 1 = µ 2 equilibria criteria

7 Phase molar Gibbs energy If the molar Gibbs energy function G m ( T, p, x) is known for all the phases in question, it is possible to compute all the equilibrium properties {mole fraction x i = n i ( n i )} G G Gibbs energy G = ng m, entropy S =, volume V =, T p enthalpy H = G + TS, internal energy U = H - pv, Helmholtz energy F = G - pv Chemical potential G µ i = n i Phase equilibria by employing the equilibrium criterion

8 Thermodynamics simulation Two component systems Two phases: solid solution + liquid solution Regular solution model mix G = RT( x A ln x A + x B ln x B ) + Ωx A x B Each phase characterized by an interaction parameter - Ω. If it is zero we have an ideal solution Continuous change in Ω leads to different topology reference?

9 CALPHAD Community to produce the databases of molar Gibbs energies to compute the chemical and phase equilibria in the multicomponent systems Phenomenological approach: to choose an analytical form for the molar Gibbs energy to determine unknowns parameters from all available experimental results

10 Simulating partition function 1 Partition function: Z = , h dn exp{ βe} dp N dr N β = 1 k B T N! Helmholtz energy: F = U o k B TlnZ Impossible to compute, Metropolis algorithm can estimate only D = exp{ βe} D( p N, r N ) dp N dr N exp{ βe} dp N dr N Trick 1 = exp{ βe} exp{ βe}, then we can write 1 exp{ βe} exp{ βe} dp N dr N -- = where D = exp{ βe} Z exp{ βe} dp N dr N Does not help: it is necessary to sample high energy regions

11 Simulations of phase equilibria Computing difference between two free energies Particle insertion Direct simulation Gibbs ensemble method More tricks Example: water

12 Free Energy Difference, I Two systems: reference U 1 and in question U 2 Coupling parameter between two systems U( λ) = ( 1 λ)u 1 + λu 2 Partition function depend on λ Z = a exp{ βu( λ) } d r N F λ NVT = exp{ βe}( U( λ) λ) dr N = exp{ βe} dr N U λ, <>-ensemble average Then, thermodynamic integration: F 2 F 1 = U dλ λ Run a series simulation at different λ, then estimate integral numerically

13 Free Energy Difference, II Two systems: reference U 1 and in question U 2 F 2 F 1 = k B Tln( Z 2 Z 1 ) Write expression for Z and use 1 = exp{ β 1 U 1 } exp{ βu 1 } Thermodynamic perturbation Monte-Carlo over the system 1: Monte-Carlo over the system 2: F 2 F 1 = k B T F 2 F 1 = k B T U 2 U ln exp k B T U 1 U ln exp k B T 1 2 In principle, this is equivalent, but in practice, this is not.

14 Particle insertion: Widom approach First state - N particles, second state - N+1 particles µ F = = F n 2 F 1 = k B Tln( Z N + 1 Z N ) TV Because of change in the number of particles µ ex = µ µ id = k B T U test ln exp k B T 1 It is possible to derive similar for the particle deletion, but the performance is worse Problem with crystals: no place to insert a particle

15 Direct simulation The main problem is introducing interface Table 1: Percentage of particles in the interface of a cubic domain (from Frencel&Smit) N P int 78% 49% 14% 6% The equilibration time is too long The metastable states (overcooled liquid or oversaturated vapor) Computationally expensive Example: two-phase three component system (Panagiotopoulus, J. Phys.: Condens. Matter 12 (2000) R25)

16 Gibbs ensemble method Simulating two-phase equilibria without interface Two different boxes at given temperature Panagiotopoulus, Chemical potential in the implicit form

17 More tricks Non-Boltzman distributions Sampling high energy states: Umbrella sampling Configuration bias Monte-Carlo Cluster moves Histogram-reweighting method: Grand Canonical Monte-Carlo

18 Example: water It is necessary to derive specialized force fields Panagiotopoulos, 2000

19 QC: Molecule in solution Molecules in the electrical field Extreme case Macroscopic continuum models Electrostatic energy Discrete modeling of the solvation shell

20 Molecules in the electrical field Electrical field leads to the molecule polarization (non-zero net dipole moment) The field of these dipoles runs against the inducing field Macroscopic result - dielectric q constant U 1 q πεε o r 12 hexane - 1.9, benzene - 2.3, water - 78 The electronic distribution in the molecule in the solution is different =

Extreme case Solution chemistry Na 2 SO 4 = Na + + SO 4 2- http://www.chem.msu.

21 Extreme case Solution chemistry Na 2 SO 4 = Na + + SO However, SO 4 2- is not stable in the gas phase Second electron affinity is negative SO 2-4 = SO e Quantum chemistry (or force field) of individual SO 2-4 is of no use

22 Macroscopic continuum models E = E o + E s E s = E es + E disp + E cav, total = isolated molecule + solvation energy, solvation = electrostatic + dispersion + cavity If we can not use E o, then let us solve Schrödinger equation with modified Hamiltonian H = H o + U Problems: Whether we can use macroscopic ε near cavity We take solvent at some temperature (ε depend on T), but we describe ground molecule energy E s - solvation energy or Gibbs solvation energy

23 Electrostatic interaction Kirkwood, molecule - N point charges { Q i, r i }, spherical cavity 1 ( E es --Σ, Σ n + 1) ( 1 ε) n n+ ( n+ 1)ε Q Q ( r r ) n = P i j ij i j a 2n + 1 n ( cosθ ij ) n - multipole expansion, a - cavity radius, P - Legendre polynomial n = 0, monopole, Born solvation energy for the ions, Q 2 E es = a ε n = 1, dipole, Onsager s reaction field 1 E es = -- 2( ε 1) µ2 2ε + 1 a 3 For non-spheric cavities, solve Poisson equation

24 Discrete solvation shell To model solvation shell, add point charges around the molecule in question Advantage More real interactions No cavity geometry No mascroscopic parameters Problems How to construct the solvation shell Solve Schrödinger equation for a cluster of the solute molecule and several solvent molecules chemphys/faeder/gallery/ Optimize geometry

25 Simulating molecule in solution Continuum models: Poisson-Boltzman equation Example: trypsin Stochastic mechanics

26 Continuum models Take protein and make a cavity based on van der Waals radii of atoms Write Poisson equation ε φ = 4πρ Use ε for solvent out of the cavity, and ε=2-4 within the cavity to take into account the polarization of the atoms Model a concentration of the mobile ions by Boltzman equation n = n bulk exp( Qφ ( k B T )) Combine and, to make life simple, linearize k is related to Debye-Hückel radius ε φ kφ = 4πρ Solve by the boundary element method or the finite difference method

27 Results From Leach s book 3D Electrostatic isopotential contours around trypsin [Marquart et al 1983]. Contours are drawn at -1kT (red) and +1kT (blue). The trypsin inhibitor is also shown with its electrostatic potential mapped onto the molecular surface.

28 Stochastic mechanics One solute molecule + a number of solvent molecules Non-periodic, stochastic boundary conditions Langevin dynamics Mẋ + Cẋ + V( x) = DW () t Kinetic energy: mass matrix and acceleration Dissipation energy to surrounding: dumping matrix and velocity Potential function RHS: normalized white noise, W(t) - Wiener process Fluctuation-dissipation theorem DD T = 2k B TC Typically dumping matrix is modeled as C = γm

29 Summary Phase Equilibria Phase diagrams and classical thermodynamics Simulations of phase equilibria Molecular solutions Quantum chemistry: molecule in the solution Simulating molecule in the solution

Statistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany

Statistical Thermodynamics and Monte-Carlo Evgenii B. Rudnyi and Jan G. Korvink IMTEK Albert Ludwig University Freiburg, Germany Preliminaries Learning Goals From Micro to Macro Statistical Mechanics (Statistical