Simulations of Slow Condensed Phases

Transcription

1 Simulations of Slow Condensed Phases Part 1: Metallic Glasses, Alloying & Vibrating Nanoparticles J. Daniel Gezelter Department of Chemistry & Biochemistry University of Notre Dame The role of the experimentalist is to perform crude analog simulations of theoretical predictions. - attributed to Frank Stillinger

2 Overview Two very simple models for diffusion Simulating metallic systems Results from simple glass simulations Nanoparticle dynamics 1. Alloying dynamics of core-shell particles 2. Vibrational and Elastic properties of Au nanoparticles

3 Non-Arrhenius diffusion and non-exponential decay Empirically, self-diffusion in fragile glasses exhibits a crossover from Arrhenius behavior to super-exponential ( e (T 0/T ) 2 ), power law ((T T c ) α ), or Vogel-Fulcher-Tamman ( e B/(T T 0) ) temperature dependence when approaching T g. At these temperatures, time correlation functions (van Hove, cage correlation, etc.) also exhibit stretching with 0 < β < 1: C(t) = e (γt)β Are these observations connected? Will simple models for diffusion capture the phenomena that lead to these behaviors?

4 Zwanzig s interrupted oscillation model Starting with the Green-Kubo formula, D = v(t) v(0) dt Expand v(t) v(0) in terms of the harmonic oscillations of the normal modes of the inherent structures in the liquid: v(t) v(0) = ρ(ω) cos(ωt)e t/τ dω The normal modes are damped by a characteristic lifetime, e t/τ Swap the order of integration, and the resulting expression is: D = k BT m ρ(ω) τ 1 + ω 2 τ 2 dω R. Zwanzig, J. Chem. Phys. 79, 4507 (1983) Stillinger & Weber, Phys. Rev. A 25, 978 (1982) Stillinger & Weber, Phys. Rev. A 28, 2408 (1983) Weber & Stillinger, J. Chem. Phys. 80, 2742 (1984) Stillinger & Weber, J. Chem. Phys. 83, 4767 (1985)

5 Continuous Time Random Walk (CTRW) model Random walks take place on a regular lattice with spacing σ 0 between lattice points, and with a distribution of waiting times, ψ(t) = 1 τ e t/τ γ = 1 ( t τ ) 1 γ 0 < γ < 1 In the γ = 1 limit, this model returns the expected diffusive behavior, D = σ2 0 6τ When γ < 1, however, it is also possible to derive a long-time approximate solution for the mean square displacement, r 2 (t) = 4σ2 0 3 π ( t τ ) γ ( 2 γ γ ) γ 1 (2 γ) 2 Γ(7 2 γ) The CTRW model predicts sub-diffusive transport when the waiting time distribution is pathological. Blumen, Klafter & Zumofen, Phys. Rev. B 27, 3429 (1983) Klafter & Zumofen, J. Phys. Chem. 98, 7366 (1994) Shlesinger, Klafter & Zumofen, Am. J. Phys. 67, 1253 (1999)

6 Common Features & Questions Both transport models require what looks like the same time constant (τ) which measures the characteristic time between diffusive hops. Which model is a better approximation to the transport dynamics in a super-cooled liquid? Does sub-diffusive behavior even make sense on a longer time scale? Figure from: Walter Kob, J. Phys. Condens. Matter 11, R85 (1999) r 2 (t) (a) T=5.0 T=0.466 A particles time t

7 Why metallic systems? They are good experimentally-known glass-formers (e.g. Ag 6 Cu 4 ). Alloys of different sized atoms have built-in lattice frustration. They form glassy systems relatively quickly. There are no orientational degrees of freedom. The potentials are simple interesting.

8 Simulating Transition Metals Spherically symmetric electron density + Positively charged core Potential for Embedding an atom at this location V i = F [ ρ i ] + j i φ ij (r ij ) Embedding Functional Pairwise terms (mainly repulsive) There are three empirical functions that are required for this approach: Electronic Density Function: ρ i = j i ρ j (r ij ) Effective Charge Distribution Function: Density or Embedding Functional: φ ij (r ij ) = Z i(r ij )Z j (r ij ) r ij F [ρ]

9 Other Embedded Atom Methods Finnis-Sinclair: Empirical (but simple) functions for densities, pair potentials and functionals Parametrized for: Cu, Ag, Ni half as fast as LJ M.W. Finnis and J.E. Sinclair, Philos. Mag. A. 50, (1984). EAM: ( ) Densities from Roothan-Hartree-Fock ρ i ρi F i [ ρ i ] = A i E 0 Rose s Universal Embedding Functional ρ 0 ln i ρ 0 i Parameterized for: Cu, Ag, Au, Ni, Pd, Pt and their alloys half as fast as LJ S.M. Foiles, M.I. Baskes, and M.S. Daw, Phys. Rev. B 33, (1986). MEAM: Densities include non-spherical symmetry (s, p, d, and f terms) Analytic Density Functional Screening Function one tenth the speed of LJ Parameterized for: Most transition metals + C, Si, Ge, H, N, O, and compounds M.I. Baskes, Phys. Rev. B 46, (1992). All three methods share the same basic potential structure: V = F [ ρ i ] + i i j>i φ ij (r ij )

10 Results: Structural Features K Below 540K, the second peak in g(r) is split K g(r) 600 K 550 K K K 300 K 0 10 R(Å) 0.3 This coincides with the temperature predicted by the (archaic) Wendt- Abraham measure for Tg. (i.e. when the system crosses an ad hoc value of for the ratio between the first minimum and first peak height in g(r)...) R WA The liquid still has appreciable transport at that temperature, however. T g WA = 547K Temperature (K)

11 Dynamics!% This temperature also appears to mark the onset of fragile behavior in the alloy ::.;9<72!5!4!3!"!!!#!"!!$#!"!!%# $&'()*(+,-.+(/012 ",# ", (768953:4 D )"++ ;<94=3>(?(0:<67(106E57FA()! " )"++ G'HI(?(G9>6!E: :4.()/! (01!& + The CTRW model has the correct temperature dependence of the selfdiffusion constant. Two separate versions of the Zwanzig model show agreement at very low temperatures, but incorrect high-temperature limiting behavior: ",! ",& "!"" #"" $"" %"" &""" &!"" &#"" '()*+

12 Deeper into the CTRW model If we want exact behavior from the CTRW model, we can fix τ and vary σ 0 to get our first unphysical result: the average jump distance increases with decreasing temperature! 4 3! 0 (Å) T (K)

13 Can we extract meaningful waiting time information? Fix σ 0 = Å and γ = 1, use <r 2 (t)> to obtain hopping times: Relax linearity in <r 2 (t)> fits to yield γ for CTRW Cage-Correlation Function CTRW fits to <r 2 (t)> ! hop (fs) ! from Cage Correlation function " from CTRW expression for <r 2 (t)> T (K) T (K)

14 Summary Even with the pathological distribution of hopping times, the CTRW model appears to do a better job than Zwanzig s model at capturing the temperature dependence of the diffusion constant. Structural measures like the Wendt-Abraham estimate for Tg may tell us more about where to look for the mode coupling temperature. They don t appear to be useful at finding Tg. Open Questions: Is there a theory for the hopping distance, σ 0? Does the waiting time distribution require a long-time pathology to result in stretched-exponential correlation functions? If we observe anomalous dynamics at intermediate times, will these become linear at longer times? Are other (more reasonable) waiting time distributions amenable to analytical treatment?

15 Before we started calling them nanoparticles...

16 Preparation of Core-Shell nanoparticles T. Shibata, B.A. Bunker, Z. Zhang, D. Meisel, C.F. Vardeman II, and JDG, J. Am. Chem. Soc. 124, (2002). Spherical Metallic nanoparticles were prepared in sizes ranging from nm in diameter via the reduction of metal salts (Au(CN) 2 and AgClO 4 ) using an organic reducing agent (citrate), or by generating the reducing organic radical radiolytically by γ-irradiation A. Henglein and D. Meisel, Langmuir 14, 7392 (1998).

17 Alloying Dynamics in Core-Shell Nanoparticles Room-temperature alloying with a time scale of approximately 3 days was observed using X-ray absorption fine structure (XAFS) data for the Au(core)-Ag(shell) 12.5 Å-19.8Å particles. Larger particles alloyed on a substantially slower time scale. x Ag Au (A) (B) (D) (C) (E) (F) (H) (G) 2.5nm 3.5nm 4.6nm 8.3nm 20 nm Total Nanoparticle Size (nm) (I) (J) (K) (L) (M)

18 What s the mechanism? Solid-state diffusion constants are 7-9 orders of magnitude too small to explain alloying on this timescale. (A) (B) (C) Diffusion of atoms in a liquid droplet should alloy the metals in a matter of microseconds! (E) (F) (G) What can simulation tell us about the mechanism for alloying? (H) 5nm (J)

19 Initial Nanoparticle Geometries r core 2Å Interface r shell

20 Analysis of Mixing Times In the infinite time limit, the mean square displacement of particles confined to a spherical volume of radius R asymptotically approaches a value of 6R 2 /5. Similarly, the mean square displacement of the radial coordinate (relative to the center of the spherical volume) asymptotically approaches 3R 2 /40. The solutions to the diffusion equation for particles with reflecting boundary conditions at R are: sin(nπr/r) ρ(r, t) = a n r n=1 Many diffusional modes contribute to the particle dynamics. e n2 π 2 Dt/R The KWW function (which is often used to approximate sums of exponentials) and the infinite time limit are helpful in predicting a mixing time: r(t) r(0) 2 (1 ) 6R2 e ( t/τ)β 5 r radial (t) r radial (0) 2 (1 ) 3R2 e ( t/τ)β 40 In these equations, β is not the result of anomalous dynamics. It simply reflects the contributions of many diffusional modes to the overall shape of the mean square displacement. r(t) θ(0,t) r(0)

21 Projected Dynamics at 300K /.!6*7189:3;<7;0*=;<;1<794.6*7189:3;<7;0*=;<;1<794 $!6*7189:3;<7;0*=;<;1<794 /! 01+!*2345- %. Projected Alloying Times at 300K: 0% vacancies: ~ years 5% vacancies: 2.9 days 10% vacancies: 1.5 hours %!!"!!!#!"!!$!"!!$%!"!!$&!"!!$'!"!!$#!"!!% $()*+,!$ -

22 Structural Features!(r)/! 800K, structures sampled from last 1 ns of a 24 ns trajectory 3 0% interfacial vacancies 2 Ag Au % interfacial vacancies!(r)/! r (Å)

23 Summary of Interdiffusional Dynamics Vacancy migration seems to be required to explain the fast alloying dynamics. We expect that a few percent of interfacial vacancies can lead to the dramatic speedup observed via XAFS. Dissimilar metals in core-shell particles may be even more prone to lowtemperature interdiffusion due to lattice mismatch.

24 Elastic Properties of Au Nanoparticles P. Conforti, M. Sprague, C. F. Vardeman II and JDG, in press in J. Phys. Chem. A (2003). Hartland and Hu have shown that the time scale for laser heating (determined by the e-ph coupling constant) is faster than a single period of the breathing mode for spherical nanoparticles. Hot-electron pressure and direct lattice heating contribute to the thermal excitation of the atomic degrees of freedom, and both mechanisms are rapid enough to coherently excite the breating mode of the spherical particles. Intensity (a.u.) 0 Au 15 nm Diameter!"!"!#$!# e-ph coupling T e = T l ph-ph coupling Delay Time (ps) Is the dephasing seen following coherent excitation due to inhomogeneous size distributions in the sample? Or is it due to softening of the breathing mode vibrational frequency following a melting transition? Period (ps) Is the flattening in the period vs. laser intensity due to melting? Anharmonicity? Something else? Intensity (µj) 8 10

25 Simulating Laser Heating We have looked at the initial (40 ps) response to instantaneous heating in a range of spherical nanoparticles: 20 Å (1926 atoms), 25 Å (3884 atoms), 30 Å (6602 atoms), 35 Å (10606 atoms) hν The initial heating is easy: simply resample the velocities from a Maxwell- Boltzmann distribution at twice the target temperature.

26 Extracting the Dynamics of the Breathing Mode How does one measure the volume of a collection of atoms (without a periodic box)? We use the volume of the convex hull as an approximate measure of particle volume Convex hulls overestimate the volume (and can fail when the particles split). But they are better estimates than the minimum enclosing sphere!

27 Breathing Dynamics 2.0e Å particle Hulll Volume (Å 3 ) 1.9e e e e e e e Å particle time (ps)

28 Beat frequencies in the volume fluctuations 3 6")7)+89%&:;( ρ V (ω) = V (t) V (0) e iωt dt / #6!! 0!1*%-!1*!-2. $! /. $! $!)7)+89%&:;( Beat pattern is striking at 800K #$!! ##!! #!!! 5!! 3!! 4!! #6!! #$!! ##!! #!!! 5!! 3!! 4!!!$! " #! #" $! %&'()*+,-

29 Dependence on Temperature and Radius Å Radius 25 Å Radius 30 Å Radius 35 Å Radius Period (ps) Period (ps) Intensity (µj) Experiments show that increasing the laser intensity increases the period of the oscillation up to a threshold T (K) Simulations Show: Period is increasing with increasing temperature even beyond the melting transition. Beat Frequencies.

30 Three approaches to calculating the Bulk Modulus The Thermodynamic Approach: K = 1 κ = V ( P V ) The Linear Response Approach: The Extended System Approach: lim δρ( k) 2 = k BT ρ k 0 V K 2 δρ( k) = V e i k r (ρ( r, t) ρ ) dv H NV T = U + fk B T ext ( τ 2 T χ(t)2 2 + t 0 χ(s)ds ) This is, to within a constant, the Helmholtz free energy, A. Since P K = V K V P = T ( S V ( ( T ( = ( A V ) T ) T T ( U V ) T ) ( 2 S V 2 U 2 V T )T 2 2 U V 2 )T ) K = V ( 2 H NV T V 2 )T H NVT is nearly (but not perfectly) conserved in NVT simulations.

31 The bulk modulus and the surface-melting transition (## '"#,*+789- '## "# 0 1 *+2*345!' *,!' - # $#!# "# /#.#."*6.#*6 ("*6 (#*6 (#!"# $"# %"# &"# '#"# ''"# '("# )*+,-

32 The Lamb Model for Elastic Spheres H. Lamb. Proc. London Math. Soc. 13, (1882). G.V. Hartland, M. Hu and J.E. Sader, J. Phys. Chem. (2003). τ t = 2πR θc t where tan θ = 3θ 3 θ 2 τ l = 2πR ηc l where tan η = 4η 4 η 2 c 2 l /c2 t In a liquid with surface stress, γ, the transcendental equation that must be solved is: τ = 2R ηc l (l) where η cot η = 1 η2 c 2 l 4c 2 l 2γ ρr In the 35 Å particle, the Lamb theory (and liquid extension) give a reasonable explanation for the splittings at low and high temperatures: Period of Toroidal Mode (at 300K) Period of Longitudinal (breathing) mode (at 300K) Period of Liquid droplet breathing mode (at Tm) 2.46 ps 2.56 ps 2.73 ps

33 Conclusions The initial drop-off in the bulk modulus can be attributed to surface melting The peak in the bulk modulus can be explained by lattice reformation upon compression. The volume fluctuation power spectra are picking up both longitudinal and toroidal modes at low temperatures. At higher temperatures, we see the melting take place during the duration of the simulation as evidenced by the two frequency components present in the volume fluctuation power spectra. Since core-shell melting will only drop the frequency of the volume fluctuations (and won t give two frequency components), we can safely say that we see both the crystalline and liquid droplet forms of the nanoparticles in the first 40 ps.

34 Acknowledgments Support From: DOE / NDRL NSF CAREER (CHE ) NSF MRI (DMR ) Dreyfus New Faculty Award Alfred P. Sloan Foundation Megan Sprague Computational Geometry & Vacancy Migration Charles F. Vardeman II All of the Above Patrick Conforti Elastic Nanoparticles