Computing phase diagrams of model liquids & self-assembly of (bio)membranes

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1 Computing phase diagrams of model liquids & self-assembly of (bio)membranes Ulf Rørbæk Pedersen Crystal structure of a simplistic model of a molecule. What is the melting temperature?

2 A typical phase diagram Phase boundaries of first-order transitions µ sl = µ solid µ liquid = 0 : µ = chemical potential = Gibbs free energy per particle

3 Supercooling of water freezing rain (isslag in Danish) Water droplets does NOT* freeze at Tm = 0 C but can be super cooled. This result in freezing rain: Super-cooling is a general phenomena. *: Your mother have properly been lying to you.

4 First order phase transitions are easily bypassed. Why? A small crystallite within classical nucleation theory [Becker & Döring (1935)]: G c = +G(interface) G(volume) G c = 4πγr 2 c 4 3 πµ slρ s r 3 c γ = surface tension At T = T m crystallization time is There is a free energy barrier associated with the formation of an interface. : Probability to be inulf macro-state R. Pedersen web: is P(r c ) e Gc(rc)/k ulf@urp.dk BT

5 At T = 0 the crystalization time is also. Thus, there is an optimal crystallization temperature T.

6 Cooling ramps gives an underestimate of the melting temperature T m or boiling point T b (or nothing at all).

7 One solution, the interface pinning method. In short: create interface by hand and compute thermodynamic force on interface Two-phase configuration of the Lennard-Jones model. [U. R. Pedersen, J. Chem. Phys. 139, (2013)]

Adding an artificial interface pinning spring-like potential Solid-liquid system (Lennard-Jones): The idea: Measure µ sl = µ solid µ liquid analogous to a fish

8 Adding an artificial interface pinning spring-like potential Solid-liquid system (Lennard-Jones): The idea: Measure µ sl = µ solid µ liquid analogous to a fish scale measuring F g = mg. Add unphysical spring to the potential part of the Hamiltonian H(R, Ṙ) = U(R) + K(Ṙ): Q(R) = crystalinity U(R; κ, Q) = U 0 (R)+ κ 2 [Q(R) Q] 2.

9 The Lennard-Jones melting line Simplistic model of atoms (no explicit electrons): 30 p 20 solid liquid 10 0 p m +p tail [Mastny 2007] [Sousa 2012] fit T The IP method reproduce melting line computed with other methods (thermodynamic integration).

10 More playing around: Self-assembly of biological membranes

11 Biomembranes are made of amphiphilic* molecules *Amphiphilic: both water-loving (hydrophilic) & fat-loving (hydrophobic).

12 A physicist simple model (a new model!) Find the model in the directory lipid. + harmonic bonds Can you make a membrane?

13 Back-up slides

14 A practical order-parameter of crystallinity Long-range order=saxs or SANS (Dorthe Posselt) Q(R; k) = ρ k where ρ k = N exp( i k r i ) k Bragg peak; 1/ k =distance between crystal planes. Computational efficient = scales as O(N) (not O(N 2 )): F i (R; κ, Q, k) = i [U 0 (R) + κ 2 [Q(R) Q] 2 ] = F (0) i (R) κ[q(r) Q] i Q(R) i Q(R) = j ρ k = k R[ρ k] sin(k r j ) + I[ρ k ] cos(k r j ) ρ k N Algorithm: 1 Loop particles to compute ρ k. 2 Loop particles to compute forces. i

15 Locate µ sl = 0 with Newtons root finding method Textbook thermodynamics: (µ sl ) slope: T = s p Entropy difference: s = ( u + p v µ sl )/T Find melting temperature T m (µ sl = 0) by iterating µ s (a) (b) T (c) p = 1 T = 0.8 (d) µ p v T (i+1) = T (i) + µ(i) sl s (i) Lennard-Jones model N N ( σ ) 12 ( σ ) 6 U 0 (R) = 4ε r r i j>i

16 General applicable: Ab initio computations (DFT) Collaboration (G. Kresse [Wien U.]): Number of particles inf T m (experimental) = 1687 K % Silicon (Si); Cubic diamond; 1180 Red: Z = 4; Blue: Z Inverse system size 1/N Melting temperature [K] T m (computed) T m (experimental) is a DFT problem. [Pedersen et al., Phys. Rev. B 88, (2013)] 1% 2% 3% 4% 5% Relative deviation

17 Some more period three elements Table: Ab inito melting temperatures of period 3 elements. Super cell Unit cell Q N T m [experimental] Na BCC ρ k & Q (21) [370] Mg HCP 1 ρ k (20) [923] Al FCC Q (30) [933] Si CD ρ k (20) [1635(2)] 1 orthorhombic cell with four particles located at (0, 0, 0), (0, 1 2, 1 2 ), ( 1 2, 0, 2 3 ) and ( 1, 1, 1 ) in units of cell lengths

New from Ulf in Berzerkeley

New from Ulf in Berzerkeley from crystallization to statistics of density fluctuations Ulf Rørbæk Pedersen Department of Chemistry, University of California, Berkeley, USA Roskilde, December 16th, 21 Ulf