New from Ulf in Berzerkeley

Transcription

1 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 R. Pedersen (UC Berkeley) Roskilde, 21 1 / 34

2 Part 1: Crystallization of a molecular liquid in the highly supercooled regime. Collaboration with Toby Hudson & Peter Harrowell (U. Sydney). Part 2: Statistics of density fluctuations: Is there an underlying liquid-liquid phase transition in the highly supercooled regime? Collaboration with David Chandler (U. C. Berkeley). Ulf R. Pedersen (UC Berkeley) Roskilde, 21 2 / 34

3 A well studied single component glass former. Ortho-Terphenyl model suggested Lewis & Wahnström (1993): Three Lennard-Jones spheres (representing phenyl s) u(r) = 4ε((σ/r) 12 (σ/r) 6 ) connected by rigid bonds. σ =.483 nm, ε = k B 6K 5 kj/mol. NVT, straight forward simulations. N = 324. Why trimer angle ϕ = 75? LW 1993: a value such that crystallization is prohibitively difficult ortho-terphenyl vs. LW-model: ϕ 6, not 75 benzene rings oblat ellipsoids. Crystal: two pairs in orthorhombic cell (different from LW-model, as we shall see) Ulf R. Pedersen (UC Berkeley) Roskilde, 21 3 / 34

4 Spontaneous crystallization of the supercooled liquid Potential Energy [kj/mol] ρ = g/ml, T = 375 K. t cryst = 2 µs Time [s] Positions of Lennard-Jones centers: near body centered cubic (BCC). Ulf R. Pedersen (UC Berkeley) Roskilde, 21 4 / 34

5 Glass forming abilities Time [s] Diffusion vs. crystallization time trimers: t D 324 trimers: t * 972 monomers: t D 972 monomers: t * t D = σ 2 /6D t * = Σ t liq /N non-liquid Trimers: T m = 65 K Monomer: T m = 638 K liquid glassy state crystal ρ = g/ml T m /T T m : Melting temperature. *Anglell et al. J. de Chim 82, 13 (1985) Lewis-Wahnström OTP is a decent glass-former. Why? Bad (high enthalpy) crystal*: Trimer: U cry /U liq =.6 Monomer: U cry /U liq =.9 (at max cryst. rate) Ulf R. Pedersen (UC Berkeley) Roskilde, 21 5 / 34

6 near-bcc structure of LJ centers. Approximate crystal energy [kj mol -1 ] Orientational ordered unit-cell. (note, formed cry. have some disorder - next slide.) Trimer angle [degrees] 75 is near optimal for this structure! Illustrate the stubborn ingenuity of molecules for periodic packing and question our intuition. Ulf R. Pedersen (UC Berkeley) Roskilde, 21 6 / 34

7 Random molecular orientation (within qubatic order) Orientation: Vector joining the two outer atoms: Crystal Liquid Random Distribution.2 o 3 o 6 o 9 o 12 o 15 o 18 o Angle between neighbor molecule orientations Quabatic order : Parallel or perpendicular orientations. Ulf R. Pedersen (UC Berkeley) Roskilde, 21 7 / 34

8 Qubatic orderparameter Potential Energy [kj/mol] Qubatic orderparameter (a) (b) Time [s] In lower panel: Green x s: Q i = Black line: N j i cos 2 (2θ ij ) 7 15 (N 1)( ) Q = Q i Ulf R. Pedersen (UC Berkeley) Roskilde, 21 8 / 34

9 Disorder of the crystal dynamics? Ulf R. Pedersen (UC Berkeley) Roskilde, 21 9 / 34

10 Dynamics Rotational Autocorrelation (a) Liquid Crystal First Legendre Second Legendre Time [s] Fast dynamics of crystal (plastic crystal): D cry = m 2 /s vs. D liq = m 2 /s. (D water = m 2 /s) Mean Square Displacement [nm 2 ] Liquid Crystal Lennard-Jones centers Center of mass Time [s] Ulf R. Pedersen (UC Berkeley) Roskilde, 21 1 / 34

11 Dynamics θ(t): Angular displace in time interval t: 18 o 12 o 9 o 6 o o (b) 1 Distribution Liquid Crystal t = 1 ns cos( θ(t)) In crystal, defects are used as intermediates Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

12 ... end of Part 1. Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

13 Part 2: Statistics of density fluctuations. Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

14 ln(p V (N)) Single component Lennard-Jones Liquid Gaussian statistics T =.7-8 3x3x3 sub-volume Number of particles; N Important well-know result: Fluctuations are well approximated by gaussian statistics. However, phase transitions (e.i. vapor/liquid) may give deviations. Gaussian statistics is often used approximation. Including theories of glass-transition. Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

15 Does a phase transition in the supercooled viscous regime give rise to non-gaussian statistics? Liquid-liquid transition?... as in high/low density water transition or the Matyushov-Angle ideal glass -hypothesis. Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

16 Collective density field in k-space The microscopic density field in real space, ρ(r) = N δ(r r n ). n The collective density field in k-space (waves in continuum): ρ k N 1/2 V drρ(r) exp( ik r) = N 1/2 N n exp( ik r n ) S(k) = ρ k 2. Periodic cartesian box: k = 2π(n xˆx/l x + n y ŷ/l y + n z ẑ/l z ). Full knowledge of ρ k statistics contains all (static) information. First order liquid-liquid transition Bimodal P(ρ k ). Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

17 (NVT) Umbrella sampling of ρ k By adding a harmonic umbrella potential, U = U + κ 2 ( ρ k a) 2, distribution function at ρ k a can be investigated: P( ρ k ) P ( ρ k ) exp(β κ 2 ( ρ k a) 2 ). MD-simulations of a series of umbrella potentials (a s and/or κ s). Proportionality constants are determined by solving the MBAR equation iteratively [Shirts & Chodera, JCP 129, (28)]. Computations scales with N (not N 2 ). Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

18 ρ k statistics: Kob-Andersen binary Lennard-Jones mixture Last half of t sim 5e5 used for analysis. N.B. for large umbrellas, crystallization is easy (these trajectories are discarded from analysis). Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

19 ρ k statistics: Wahnström binary Lennard-Jones mixture WABLJ S(k) 2-1 k = 5.7 k = k ln(p) -2-3 Amorphous Frank-Kasper cluster. A new phase? [PRL 14, 1571 (21)] ρ k No sign of new phase. Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

20 ρ k statistics: Water. Ulf R. Pedersen (UC Berkeley) Roskilde, 21 2 / 34

21 ρ k statistics: Lewis-Wahnström o-terphenyl ln(p) -1-2 (c) k =.78 nm -1 S(k) k [nm -1 ] k = 2.3 nm -1 k = 1.4 nm -1-3 LWOTP ρ k Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

22 Crystallization Qubatic order U [Kcal/mol] U u = 5 ( ρ k - 8 ) 2 U u = 5 ( ρ k - 9 ) 2 U u = 5 ( ρ k - 1 ) 2 LWOTP Time [s] Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

23 Systems size dependence Dashed: Gaussian statistics -1 N = 324 ln(p) -2 N = 2592 LWOTP ρ k Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

24 What about short length-scale density statistics? Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

25 Sampling P V (N): probability of having N particles in a subvolume V. Periodic in x and y, and vapor/liquid coexistence in z. Sampling a range of umbrellas u: U u = U + κu 2 (Ñ a u ) 2 where Ñ is smoothed number of particles in subvolume [Patel, Varilly & Chandler, JPC B, 114, (21)] N Ñ = dr (2πσ 2 ) 1/2 exp( r n r 2 /2σ 2 ) V n P V (N) P u (A) exp(β κu 2 (Ñ a u) 2 ) Prefactors are determines using MBAR [Shirts & Chodera, JCP 129, (28)]. β = (k B T ) 1 Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

26 Number density [σ A -3 ] In slab: Kob-Andersen binary Lennard-Jones mixture All A s B s Mole fraction of A s Probe volume: -5 < z < 5 T = z [σ A ] Structural relaxation time, τ α [LJ units] Kob-Andersen mixture at vapor/liquid coex. N=3, A=1.3 Number density Structural relaxation time is measured from incoherent-isf as F s ( q = 2π/σ ΑΑ, τ α ) = 1/e of A s located at -5<z<5 at time t = Temperature.4 Temperature [LJ units] Model in short: B particles are 2% smaller than A s. Strong AB affinity. Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

27 2 KABLJ, T =.4 Raw Umbrella Histogram Number of particles in 3x3x3 subvolume Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

28 KABLJ mixture vapor/liquid equlibrium at T =.55 KABLJ mixture vapor/liquid coex. at T =.4 ln(p) ln(p) -2-4 Dashed: Gaussian statistics N Number of particles in 3x3x3 subvolume ln(p) ln(p) -2-4 Dashed: Gaussian statistics N Number of particles in 3x3x3 subvolume Observation: Gaussian approximation is good. At low temperature, it is more difficult to empty the volume than expected from gaussian approximation. Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

29 Composition fluctuations? E.g. crystallization is accompanied by a large composition fluctuation: 6% FCC (A) and 4% CsCl (AB). Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

30 Kob-Andersen Lennard-Jones mixture in slab. T = subvolume. P gauss V exp( α( N A ) 2 β( N B ) 2 γ N A N B ) Some non-gaussian features. Fat tails, crystal? Ulf R. Pedersen (UC Berkeley) Roskilde, 21 3 / 34

31 ln(p) Free energy: F-F min N B Dashed: Gaussian statistics T =.3 T =.4 T = Number of B s in 3x3x3 subvolume Composition fluctuations becomes larger upon cooling. Fat tail, crystal? Structural origin of increasing heat capacity? Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

32 Crystallite in sub-volume: Strong umbrella potential, Uumb = 2.4 N B 7 2 in sub-volume. T =.4. CsCl crystallite in umbrella volume. Strong AB affinity stabilizes this crystal. Red: Smaller B s. Green: Larger A s. Crystal-liquid surface tension inhibits this crystallite to grow (Ask the person next to you if you are red-green color blind.) Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

33 Conclusions, and No sign of an underlying phase transition other than crystal Gaussian description is good. Ulf R. Pedersen (UC Berkeley) Roskilde, / 34

34 Funding: The Danish Council for Independent Research in Natural Sciences. Collaborators: U. Sydney: Toby Hudson, Peter Harrowell. UC Berkeley: David Chandler. Thank you for your attention. Ulf R. Pedersen (UC Berkeley) Roskilde, / 34