Atomistic Wulff construction and the shape of nanoparticles

of noble metals

Ioannis N. Remediakis

Department of Materials Science and Technology, University of Crete, Greece

International Conference on Applied MathematicsACMAC, Heraklion, September 16 - 20, 2013

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Outline

● Equilibrium shape: 20th century and the nano-era (1995-).

● Equilibrium shape from first principles (2005-).

● Au nanoparticles from first principles and the atomistic Wulff construction method.

● Ag nanoparticles as catalysts for hydrogenation of triple bonds.

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Equilibrium shape (1878)

G = Gb u l k + Σ γh k l Ah k l = min.

Surface tension γ = (Surface energy) / (area)(Surface energy) = (Energy )- (Energy of bulk)

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Wulff's theorem(Wulff 1901) Minimum energy polyhedron:

dh k l / γh k l = constant

Low-energy -> large area -> closer to center.

First Proof: M. Von Laue (1943).

Short proof by R. F. Strickland-Constable (1968):

G=ΣγhklAhkl minimized for constant V=Σ(1/3)dhklAhkl :

δ(G-λV)=0 => Σ(γhkl-cdhkl)δAhkl=0 => γhkl=cdhkl (QED).

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An example of Wulff construction dh k l/γh k l = constant (d=distance from center).

Orthorhombic material (e.g aragonite CaCO3) with

γ100=γ110=γ=0.5γ010 << γ001 << γhkl.

Crystal habit: elongated rods (Cross section below).

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Properties of equilibrium shape

● The shape depends on ratios between surface tensions. ● (hkl) planes with high surface tension (usually high-indexed ones) are less likely to appear in the equilibrium shape.● Being steeper, high-index faces are usually hidden behind low-index ones, even if γhkl is not very high.

● The extra energy associated with the formation of edges be-tween two surfaces is neglected.● The Wulff polyhedron has the same point group (or its super-group) as the crystal structure of the material.

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Examples of Oh polyhedra (fcc metals)● Symmetry is necessary (but not sufficient) condition that the system has reached equilibrium. A non symmet-ric (large) particle is most likely out of equilibrium.

● Face-Centered Cubic (FCC) is the most symmetric close-packed structure.

● Oh = {E, 8C3, 6C2, 6C4, 3C42,

I, 6S4, 8S6, 3σh, 6σd}

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Equilibrium shapes in nature

G = Gb u l k + Σ γh k l Ah k l

Equilibrium shape: minerals (billions of years to equilibrate) or nanoparticles (small size).

www.mindat.org Turner et al., Adv. Func. Mater. 2009

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Wulff's theorem and nanoparticles

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Wulff's theorem and nanoparticles

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Wulff's theorem and nanoparticles

P. L. Hansen, J. B. Wagner, S. Helveg, J. R. Rostrup-Nielsen, B. S. Clausen, H. Topsøe, Atom-Resolved Imaging of Dynamic Shape Changes in Supported

Copper Nanocrystals, Science 295 2053 (2002).

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Surface tesion of Ru(hkl) from DFT

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Virtual catalyst for NH3 synthesis

K. Honkala, A. Hellman, I. N. Remediakis, A. Logadottir,A. Carlsson, S. Dahl, C.H. Christensen and J. K. Nørskov,

Science, 307 558 (2005);Surf. Sci., 600, 4264 (2006);Surf. Sci., 603, 1731 (2009).

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Equilibrium shape of Au nanoparticles from first principles

● Using Density Functional Theory, we calculate the surface tension of different Au surfaces.

● Using Wulff construction, we find nanoparticles with minimum en-ergy.

● Take into account interactions of nanoparticle with its environment.

● Converge the method with respect to the number of orientations.

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First-principles atomistic calculations

Input: Coordinates of the nuclei plus numerical parameters.

Output: equilibrium structure for given external fields, and/or response functions (usually).

Advantage: Understanding and design of materials at their ultimate level: elecrtrons and nuclei (or ions).

Density-functional Theory (DFT): Hψ=Εψ; H=K+Ve-ion+Ve-e+Vx-c : Ve-ion:pseudopotential; Ve-e: mean-field electron-electron; Vxc: many body plus self-interaction correction.

Here: DACAPO, GPAW and ASE (GPL'd open source software) http://www.camd.dtu.dk/Software

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Modeling Au surfaces with DFT● Slab model.

● Parameters chosen for convergence of surface tension within 0.01 J/m2.

● Calculate total energy of slab and bulk Au.

● Eslab = N Ebulk+2Aγ

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γhkl ~ Nhkl ; Νhkl=2h+k or 4h+2k.

(Mackenzie et al., J. Phys. Chem. Solids (1962))

a: MAEAM, Wen & Zhang, (2007);

b: LMTO+Mckenzie, Galanakis et al. (2002).

Surface tensions, equilib. shapes

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Equilibrium shapes

● Predicted nanoparticles match observations even at small sizes.

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Environment-dependent shape

● In a reactive environment, Nads/A molecules may

attach to the solid surface and form bonds of energy Eads

each, lowering thus the surface tension (γ) to (γint).

γint=γ+NadsEads/A

● Large difference in γ (and nanoparticle shape) only for strong bindind AND high concentration.

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CO Adsorption on Au

G. Mpourmpakis, A. N. Andriotis, D. G. Vlachos, Nano Lett. 10, 1041 (2010) ; S. R. Bahn, N. Lopez, J. K. Nørskov, and K. W. Jacobsen, Phys. Rev. B 66, 081405 (2002)

The first calculation of CO on every Au(hkl).

Barmparis and INR, PRB 2012.

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Au nanoparticles in CO gas

TEM by K. Ueda, T. Kawasaki, H. Hasegawa, T. Tanji and M. Ichihashi, Surf. Interf. Anal. 40, 1725 (2008).

γint=γ+θEads/Aat

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Thiol adsorption on Au

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Thiol-protected Au nanoparticles

Barmparis, Honkala, Remediakis, J. Chem. Phys. 2013

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Adsorption increases reactivity.

Barmparis, Honkala, Remediakis, J. Chem. Phys. 2013

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Application: active sites for catalysis.

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Warm-up: CO + ½O2 → CO2

● Rate: r=k [CO] [O2]½ – k' [CO2]

● k=k0 exp(-Ea/RT)

● Chemical equilibrium: [CO2] = K [CO] [O2]

½

● K = k / k' = K0 exp(-ΔH/RT)

● [CO] = NCO/N = PCO/P = pCO

● Adsorbed: [CO*] = NCO*/N ≡ θCO

Image from wikipedia

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r=kKCOKO2½pCO(1-pCO)½/(1+KCOpCO+KO2

½(1-pCO)½)

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Turnover frequency (TOF)

TOF = (mols of product per second per gram of catalyst) = (reaction rate per site)×(sites per gram of catalyst) = (reaction rate per site)×(sites per gram of metal)×(wt % metal in catalyst).

Reaction rate: from experiments on single-crystal surfaces or DFT-TST simulations.

Metal content of catalyst from analytical chemistry.

Site density: usually not calculated, but some times compared to the number of atoms per unit mass (equals Vi / ρ ).

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Density of surface sites: octahedra

R. Van Hardeveld, F. Hartog, The statistics of surface atoms and surface sites on metal crystals, Surf, Sci., 15, 189-230 (1969).

n=N surf /N A

M=

AAat

/N A

ρV=

1Aat N A ρ

AV

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Active sites

Logadottir A, Rod TH, Norskov JK, Hammer B, Dahl S, Jacobsen CJH (2001)

J Catal 197:229

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Hyfrogenation of propyne

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Reaction order (1) dissociation mechanism

(2) association mechanism

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Selectivity

The reaction stops at the first hydro-genation:

CH3--̶C≡CH → CH3--̶CH=CH2

CH3--̶CH=CH2 ↛ CH3--̶CH2--̶CH3

From DFT calculations:

Propene desorption: Ea=0.14 eV; propene hydrogenation: Ea=1.01 eV.

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A maximum at 4.5 nm particles

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Equilibrium shape of Ag particles

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B5 sites are the active sites

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B5 sites exists mostly above 4.3 nm

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Wulff construction gives the key

Vile et al., ChemCatChem 2013 (in press)

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Conclusions

Multi-scale model based on quantum-mechanical calculations and the Wulff construction yields accurate equilibrium shapes for metal nanoparticles.

Nanoparticles in weakly interactive environments have similar shapes.

Nanoparticles increase their sphericity and reactivity upon ex-posure to reactive environment.

Atomistic Wulff construction can be used for the accurate counting of active sites in realistic catalysts.

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● C. Bittencourt, (U. Mons, Belgium)

● G. Kopidakis.

● G. Barmparis.

● N. Lopez (Institute of Chemical Research of Catalonia).

● K. Honkala (Jyväskylä University, Finland).

Acknowledgements

● Gianvito Vilé, Javier Pérez-Ramírez (ETH Zurich)

● C. Bittencourt, (U. Mons, Belgium)

● G. Kopidakis.

● G. Barmparis.

● N. Lopez (Institute of Chemical Research of Catalonia).

● K. Honkala (Jyväskylä University, Finland).

● C. Bittencourt, (U. Mons, Belgium)

● G. Kopidakis.

● G. Barmparis.

● N. Lopez (Institute of Chemical Research of Catalonia).

● K. Honkala (Jyväskylä University, Finland).