Size-Dependent Vacancy Concentration in Nickel, Copper, Gold, and Platinum Nanoparticles † † † † ‡ † Panpan Gao, Hongxin Ma, Quan Wu, Lijie Qiao, Alex A

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Size-Dependent Vacancy Concentration in Nickel, Copper, Gold, and Platinum Nanoparticles † † † † ‡ † Panpan Gao, Hongxin Ma, Quan Wu, Lijie Qiao, Alex A. Volinsky, and Yanjing Su*, † Corrosion and Protection Center, Key Laboratory for Environmental Fracture (MOE), University of Science and Technology Beijing, Beijing 100083, China ‡ Department of Mechanical Engineering, University of South Florida, Tampa, Florida 33620, United States

ABSTRACT: Size-dependent thermodynamic model is presented to simultaneously analyze stress and vacancy concentration, along with their coupling behavior in spherical metal nanoparticles. A nanoparticle is treated as a composite with a core lattice and a surface shell, which both have their corresponding thermodynamic properties. Thermodynamic analysis of Ni, Cu, Au, and Pt particles shows that the vacancy concentration in the core and the surface shell could be signiﬁcantly decreased for smaller particles. The apparent vacancy concentration in the whole nanoparticle reaches the maximum at the critical particle size of about 100 nm and is lower for larger or smaller particles. This result provides comprehensive understanding of the vacancy concentration in nanomaterials, which had been represented by two opposite views in the past decades.

− 1. INTRODUCTION than in the corresponding bulk metals.28 31 Ouyang et al. Nanosized and nanostructured materials possess many unique theoretically predicted that the vacancy formation energy properties, which differ from their bulk counterparts due to the decreases with decreasing particle size and that the vacancy formation energy on the surface and at the interface is smaller high surface-to-volume ratio. In nanomaterials and nanoporous 32−34 materials, in particular, the surface-to-volume ratio is extremely than in the interior core. Gladkikh et al. proposed large, which makes surface stress contributions quite substantial analogous idea that cohesive energy and melting temperature − − 35 for both physical1 4 and chemical5 9 properties. Recent have similar dependence on particle size. They calculated the − research10 13 showed that both vacancy formation energy and size-dependent vacancy formation energy in nanomaterials concentration change significantly with the material size, using proportionality between the vacancy formation energy especially at the nanometer scale. and the melting temperature, concluding that the vacancy The vacancy defects in metallic nanoparticles play an formation energy is reduced in smaller particles.35,36 Guisbiers important role in many physical−chemical properties, including et al. developed a theoretical model to account for the size- mechanical, optical and electrical properties, along with dependent vacancy formation energy and entropy based on the 14−20 catalytic activity. Although the role of thermal vacancies universal equation, in which they found that the vacancy in nanoparticles is vital for understanding these properties and concentration increases with decreasing sample size and ffi 37,38 related behavior, it has been quite di cult to investigate increasing temperature. However, Müller et al. simulated vacancies experimentally until now. In recent decades, positron an opposite result where the surface energy and surface stress annihilation spectroscopy has been used to study intrinsic 21−24 contributions in metallic nanoparticles can actually increase the defects in nanostructured materials. However, vacancy vacancy formation energy as the system size decreases.11 They concentration measured by positron annihilation can be fl also reported that the vacancy concentration decreased in the strongly in uenced by experimental conditions. crystal core with smaller particle size.11 Recently, Salis et al. Theoretically, many researchers have attempted to inves- approximated surface stress as a constant.12 Total vacancy tigate the size-dependent vacancy formation energy in nano- concentration has maximum at a critical particle radius due to materials to understand the underlying physical mechanisms. Since the vacancy formation process involves breaking up the surface energy, which increases both the core and surface atomic bonds, it was thought that the vacancy formation energy vacancy formation energy. However, the surface-to-volume − could be related to the crystal cohesive properties.25 27 Qi and ratio increases the relative concentration of surface defects for Wang gave the relationship between the vacancy formation smaller particle size. energy and the cohesive energy of small particles by considering the change of the atomic radius induced by the surface stress.28 Received: June 6, 2016 They found that the vacancy formation energy depends on the Revised: July 11, 2016 lattice strain and that the vacancy formation energy is lower Published: July 13, 2016

© 2016 American Chemical Society 17613 DOI: 10.1021/acs.jpcc.6b05712 J. Phys. Chem. C 2016, 120, 17613−17619 The Journal of Physical Chemistry C Article ⎛ ⎞ Although nanoparticles can have various morphology due to 00⎜⎟1 2 ff uss()εσεε=+×uY 2 ( s ) +× 2 s di erent preparation methods, in the present theoretical work, ⎝ 2 ⎠ (2a) we discuss spherical metal nanoparticles as an example. A 1 1 nanoparticle is treated as a composite with a core lattice and a u ()εε=+uKeuK0202 =+ (3 ) surface shell to study the size-dependent concentration of ccccc2 2 (2b) thermal vacancies in nanomaterials. In this paper, thermody- where u0 and u0 are the generalized surface and core energy namic model is established by considering the surface energy s c densities in the strain-free state, respectively; σ0 is the and surface stress effects on the Gibbs free energy of vacancy s eigenstress in the surfaces shell and Y is the surface biaxial formation. An accurate solution of size-dependent vacancy s Young’s moduli, while K is the bulk modulus of the core; ε is concentration containing the eigen parameters is provided, c the biaxial strain, and e =3ε is the volumetric strain. including biaxial surface eigenstress and biaxial Young’s moduli Substituting eqs 2a and 2b into eq 1, the total potential of the core and the surface. Surface stress, vacancy energy of a nanoparticle is given by concentration, and their coupling behavior in surface shell ⎛ ⎞ and the core lattice are analyzed. 2 00 24 3 029 U()επ=Ω+++ 4Ru ( 2 σεε Y ) π RuK⎜⎟ + ε csss0 0 3 c0 ⎝ c 2 c⎠ 2. ANALYSIS (3) ΩΩ2 2.1. Deformation and Surface Stress of Nanoparticles. where VR=Ω++41π 2 0 0 is the surface shell sc0 0 R 2 Usually, there are three common approaches to study the ()c0 3Rc0 Ω ≪ properties of surfaces: the sharp surface approach, the diffusive volume. When 0/Rc0 1, the surface shell volume is 39 ≈ π 2Ω surface approach, and the interphase approach. Both diffusive approximately reduced to Vs 4 Rc0 0. The nanoparticle core surface and interphase approaches treat surfaces as three- volume is VR= 4 π 3. At equilibrium, the minimum energy cc3 0 dimensional (3D) objects. In the sharp surface approach, a ∂ ε ∂ε| ini requires U( )/ ε=ε = 0, which yields the generalized single dividing interface of zero thickness is used to separate a Young−Laplace equation42 to describe the mechanical force studied system from its environment, where the surface balance between the surface and the core contribution to thermodynamic properties is defined as excess 0 ini ini values that would be obtained if the studied system and the 2σεssΩ+00023YRK Ω+cc ε =0 (4) environment retained their properties constant up to the dividing interface.40 The diffusive interface is described by a From eq 4, the initial strain in the core induced by the surface gradient term, i.e., the concentration gradient.41 The interphase stress is given by approach treats an interface as a thermodynamic phase, and is 0 2σs Ω0 usually chosen to be at locations where the properties are no εini =− Ω+ longer changing significantly with the position. The interphase 23YRKscc00 (5) surface has a finite volume (thickness) and may be assigned The biaxial surface and lattice stress of the nanoparticle are thermodynamic properties in a normal way. then, respectively given by The present study focuses on the fundamental elastic 0 properties of a solid surface and considers an isotropic 0 ini σs σσss=+Y s ε = nanoparticle to simplify the theoretical analysis. A nanoparticle 12+ΩYRKscc00 /(3) (6a) is modeled to be a spherical composite of a core and a surface shell coherently bonded to the core. In general, various defects 6σ 0K Ω σε==−3K ini sc0 may be introduced in nanoparticles during fabrication. In the cc 23YRKΩ+ present study, we consider only the core and the surface shell, scc00 (6b) surface eigenstress and stresses induced by vacancy concen- Here, we consider that the nanoparticle consists of pure tration in the core and the shell, meaning that other defects are components, which have only vacancy defects. At a relatively not analyzed here for simplicity. Furthermore, the core and the small number of vacancies the crystal can be regarded as an surface shell are assumed to be both mechanically isotropic. ideal solid solution. At constant temperature, the solid solution Since vacancies randomly appear in materials, it was assumed sample volume can expressed as a function of stress and that the vacancies are uniformly distributed in the nano- composition. For a given number of moles of the solvent in an particles. Without deformation and any defects, the core has an ideal solid solution, its volume is a function of the number of σ original radius Rc0 and the original thickness of the surface shell vacancies, nc,v, and the hydrostatic stress, c at constant Ω is denoted by 0. temperature. In differential form, one has: When a nanoparticle is formed, its volume will change due to ⎛ ⎞ ⎛ ⎞ ∂V ∂V surface stress and/or all the defects. In this case, it is convenient ddV = ⎜ c ⎟ n + ⎜ c ⎟ dσ to take the stress-free and defect-free nanoparticle as a c ⎝ ∂n ⎠ cv, ⎝ ∂σ ⎠ c cv, σ c n reference. The total potential energy of a nanoparticle is c cv, (7a) given by The vacancy formation volume can be defined as ⎛ ⎞ U()εεε=+Vu () Vu () (1) ss cc ⎜ ∂Vc ⎟ Vcv̅ , ≡ ⎜ ⎟ ε ⎝ ∂ncv, ⎠ Here, denotes strain, us is the surface energy density, uc is the σc (7b) core energy density, while V and V is the surface and core s c ’ σ Δ volume of the nanoparticle, respectively. Hooke s law states that c = Kc,vec = Kc,v( Vc/Vc0)nc,v with Kc,v For a nanoparticle with an original radius Rc0 and the original being the isothermal vacancy-related bulk modulus. Thus, one Ω thickness of the surface shell 0, the energy density is given by has

17614 DOI: 10.1021/acs.jpcc.6b05712 J. Phys. Chem. C 2016, 120, 17613−17619 The Journal of Physical Chemistry C Article

⎛ ⎞ 0 ∂VV σσσRR+Ω=222 +Δ+Ω= σσ0 (13a) ⎜ c ⎟ = c0 cc000 s cc s s 0 ⎝ ⎠ ∂σc Kcv, Equation 13a is an explicit form of the so-called generalized ncv, (7c) capillary equation.46 Substituting eqs 12a and 12b into eq 11a We assume the formation volume of a vacancy to be constant yields and independent of the number of vacancies and hydrostatic ⎛ ⎞ ⎛ ⎞ stress. Then, integrating eq 7a leads to the change in volume Δa Vcv̅ , Δa 1 Vsv̅ , K ⎜32− nR⎟ + Y̅ ⎜ −× n ⎟Ω caused by vacancies and hydrostatic stress cv, ⎝ cv,0⎠ c sv ,⎝ sv, ⎠ 0 a0 Vc0 a0 3 Vs0 V 0 c0 +Ω=20σ ΔVVnccvcv≈ ̅ ,,+ σc s 0 (13b) Kcv, (7d) eq 13b gives the solution of

Dividing the core lattice volume change by the original core 0 2σs Ω0 Vc̅ ,v 2 Ys̅ ,vΩ 0 Vs̅ ,v lattice volume gives −+nnc,v + ×s,v Δa KRcc,v 0 Vc0 3 KRcc,v 0 Vs0 ⎡ ⎤ = Δa 1 Vcv̅ , σ a0 Ψ (14a) = ⎢ + c ⎥ ⎢ ncv, ⎥ a0 3 ⎣ Vc0 Kcv, ⎦ (8) Ψ =+32YKRscc̅ ,vΩ 0 /() ,v 0 (14b) Similarly, we may introduce the formation volume of a eq (14) shows that the change in lattice constant depends on ̅ vacancy, Vs,v, inside the surface shell. The vacancy-induced the vacancy concentration inside the core lattice and the surface volumetric change at stress-free condition is given by shell, along with the particle size. Once the lattice strain is known, the lattice stress and the surface stress are calculated ΔVVnsv,,,= sv̅ sv (9) from eq 12a and eq 12b, respectively, yielding Here, ns,v is the vacancy number inside the shell. Because the Ω0 Θ thickness of the surface shell is small, we may assume that the σc =−2 stress field with zero radial stress is homogeneously distributed Rc0 Ψ (15a) there. With a vacancy defect, we have the elastic surface area Δ Θ strain, As,e, from linear elasticity σ = s Ψ (15b) ΔA Δσ se, = 2 s ⎛ ⎞ Vc̅ ,v Vs̅ ,v AYs0,sc̅ (10a) Θ=3σ 0 +Y̅ ⎜ n − n ⎟ ss,v⎝ c,v s,v⎠ Vc0 Vs0 (15c) ΔΩe 2vs,v Δσs =− Here, Θ is the stress parameter. The sample strain is given by Ω0 1 − vYs,vsv̅ , (10b) ⎛ ⎞ ⎛ Ω ⎞ V̅ 2v Δσ Ω 0 Δl Δa ⎜ 0 ⎟ ⎜ 1 s,v s,v s ⎟ 0 where Δσ = σ − σ is the change in surface tangential stress, = 1 − +·⎜ ns,v − ⎟ s s s l aR⎝ ⎠ ⎝ 3 V 1 − vY̅ ⎠ R with σ0 being the surface eigenstress,43 Y̅ = Y /(1 − v ) is the 00 c0 s0 s,vsc ,v 0 s s,v s,v s,v ⎛ ⎞ ’ Δa Ω surface vacancy-related biaxial Young s modulus, and Ys,v and vs,v = ⎜1 − 0 ⎟ ⎝ ⎠ are the surface vacancy-related Young’s modulus and Poisson’s aR0 c0 ratio, respectively. The total volume change of the surface shell ⎡ ⎢ induced by the stress and vacancies can be separated into the 1 Vs̅ ,v 2vs,v +·⎢ n − normal and tangential components as ⎢ 3 V s,v 1 − v ⎣ s0 s,v V̅ V̅ 2v 1 sv, ΔΩe 1 sv, sv, Δσs V̅ V̅ ⎤ × n + =× n − −Ω2/()σ0 KR +c,v n −s,v n ⎥ sv, sv, scc0 ,v 0V c ,vV s ,v Ω 3 Vs0 Ω0 3 Vs0 1 − vYsv,,sv̅ ()c0 s0 ⎥ 0 Ψ ⎥ (11a) Rc0 ⎦ (16) V ΔA V 2 sv̅ , se, 2 sv̅ , Δσs Equation 16 describes the relationship between the sample × nsv, + =× nsv, + 2 3 Vs0 As0 3 Vs0 Ysv̅ , (11b) strain and the lattice strain, showing that the changes in sample dimensions stem from the interdependent contributions of the In the interphase approach, the surface shell is coherently surface and lattice core. For vacancy-free nanoparticle, the bonded with the core. Thus, the surface area strain should be lattice strain calculated by eq 14 is simplified to eq 5. Then the equal to twice the linear strain of the core, i.e., lattice stress and the surface stress calculated from eq 15 is V̅ Δσ 2 sv, s Δa simplified to eq 6. ×+=nsv, 22,or 3 Vs0,0Ysv̅ a 2.2. Vacancy Concentration in Nanoparticles. For ⎛ ⎞ Δa 1 Vsv̅ , investigating the vacancy concentration in nanoparticles, it is Δσ = Y̅ ⎜ −× n ⎟ ssv, ⎝ sv, ⎠ helpful to start from the definition of vacancy formation in bulk a0 3 Vs0 (12a) materials. For a large enough system (N >103 atoms), which On the other hand, eq 8 gives consists of N atoms and n vacancies, according to the general fi ⎛ ⎞ de nition of the chemical potential, the chemical potential of Δa Vcv̅ , σ = K ⎜3 − n ⎟ vacancies is determined as ccv, ⎝ a V cv, ⎠ 0 c0 (12b) ()n ∂ΔG n μ = =ΔHTSkTff − Δ + B ln The force balance requires44,45 V ∂n Nn+ (17)

17615 DOI: 10.1021/acs.jpcc.6b05712 J. Phys. Chem. C 2016, 120, 17613−17619 The Journal of Physical Chemistry C Article ⎛ ⎞ The standard chemical potential of the vacancies is determined Δ+μσ0 2/3V̅ ﬁ bp bp ⎜ ssv, ⎟ as the in nite crystal free energy change per 1 added vacancy, Xs = Xc exp⎜ ⎟ disregarding the mixing entropy: ⎝ kTB ⎠ (25b) (0) where the superscript “bp” denotes “big particle”. Eq 25b gives μ =ΔHTSvf − Δ (18) V the expression of the surface vacancies concentration in big And consequently particles, showing that the surface vacancies could be aﬀected by the surface stress. As stated in the McLean adsorption ()n (0) n (0) μ0 μ =+μμkTB ln=+kTB ln Xv isotherm, the reference chemical potential s in the surface V VV+ (19) μ0 Nn must be much lower than c to cause higher vacancy It is taken into account here that if n is a small quantity, the concentration in the surface shell. It shows large gradients in concentration of vacancies is vacancy concentrations in the surface shell, and the lateral gradients in the core lattice are small and can be neglected.47 Xv ≈+nN/( n ) (20) In this case, the total number of vacancies in the nanoparticle can be given as The free energy of the crystal containing n vacancies can be represented in the form of a sum n =+nncv,, sv (26) GNn=+∑ μμ()n and the nanoparticle consists of N atoms, with Ns atoms lying i iv on the surface and N atoms located in the core: i (21) c μ N =+NNc s (27) Here, Ni and i are the number of atoms of type i and their chemical potential, respectively. Consequently, the change of n Then the vacancy concentration in the core and the surface may lead to equilibrium shell can be respectively given by

()n ∂ΔG n μ = = 0 Xnp = cv, V (22) c ∂n nNcv, + c (28a) ﬀ i.e., the equilibrium concentration of the vacancy di ers from n zero and is equal to Xnp = sv, s nN+ ⎛ ⎞ sv, s (28b) μ(0) ⎛ ΔS ⎞ ⎛ ΔH ⎞ 0 ⎜ v ⎟ ff X =−exp = exp⎜ ⎟ exp⎜− ⎟ Thus, the total number of vacancies is v ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ kTBBB⎠ k kT (23) Xnp Xnp n = c N + s N ΔS is the vacancy formation entropy, which is very small, so we np c np s v 11− Xc − Xs (29) ignore it in the analysis. For our model, in nanoparticle systems, based on the Gibbs-based 3D model, the chemical potential of The concentration of vacancies in the nanoparticle is vacancy in the surface shell equals to the chemical potential of np np Xc Xs vacancy in the core. In the following, we determine the vacancy np NNc + np s np 11−−Xc Xs concentration inside the surface shell by utilizing equal vacancy X = v Nc Ns np + np chemical potentials. 11−−Xc Xs (30) The chemical potentials of vacancies at constant temperature ≪ ≪ in the core and the surface shell are respectively given by Because of Xc 1 and Xs 1, eq 30 can be reduced to 0 NXnp + NXnp μ =+μσkTB,ln Xccc − V̅ v np cc ss cc (24a) Xv = NNcs+ (31) μ =+μσ0 − sskTB,ln Xsscv 2 V̅ /3 (24b) The number of atoms in the core and the surface shell is calculated from N = V /Ṽand N = V /Ṽ, respectively, where Here, k is the Boltzman’s constant, T is the absolute c c0 c s s0 s B Ṽand Ṽdenote the vacancy-free atomic volume in the core temperature, X and X denote the concentration of vacancies c s c s and the surface shell. inside the core and the shell, respectively, with μ 0 and μ 0 being c s The vacancy concentrations in the core and the surface shell the corresponding reference chemical potentials. Equation 24 are calculated from the equivalent chemical potentials and are indicates that the vacancy concentrations and stresses in the given by core and the shell play an important role in the chemical potentials. On the basis of the equal vacancy chemical potential ⎛ ⎞ ⎛ ⎞ σσccvV̅ , 2Vcv̅ ,0 sΩ in the core and the surface shell, we have, from eqs 24a and Xnp = Xbp exp⎜ ⎟ =−Xbp exp⎜ ⎟ c c ⎝ ⎠ c ⎝ ⎠ 24b: kTB kTRB0c (32a)

⎛ 0 ⎞ ⎧ 0 ⎫ Δ−μσVV̅ + 2/3 σ̅ ⎪⎪ np np ⎜ ccv,, ssv ⎟ np np Δ+μσ2(scvVRV̅ ,0Ω+ / c 0 sv̅ , /3) Xs = Xc exp⎜ ⎟ X = X exp⎨ ⎬ ⎝ kT ⎠ s c ⎪⎪ B (25a) ⎩ kTB ⎭ Δμ0 − μ0 − μ0 σ → (32b) with = ( s c ). Eq 15a indicates that c 0 when the Ω ﬁ ratio of 0/Rc0 approaches zero, which is satis ed in bulk where the generalized capillary equation is used. The surface materials. For this extreme case, eq 25a is reduced to stress is calculated from

17616 DOI: 10.1021/acs.jpcc.6b05712 J. Phys. Chem. C 2016, 120, 17613−17619 The Journal of Physical Chemistry C Article

0 ⎛ ⎞ 3σ YVsc̅ ,v̅ ,vVs̅ ,v between the reference chemical potential in the surface shell s ⎜ np np⎟ 0 σs = + Xc − Xs and the core is Δμ = 100 meV. Ψ Ψ ⎝ Ṽ Ṽ ⎠ c s (33) Parts a and b of Figure 1 show that the vacancy Solving eqs 32 and 33 determines the surface stress and the concentration in the core and the surface shell present the concentration simultaneously, when the surface eigenstress and same trend for all metallic particles, which decreases when the ff fi the material properties are available. particle size is reduced, and this di erence is signi cantly larger when the particle size is reduced to few tens of nanometers. 3. RESULTS AND DISCUSSION The vacancy concentration approaches an infinitesimal value The intrinsic surface elastic parameters48 and bulk vacancy when the particle size is lower than 10 nm. Figure 1c shows that formation energy49 of different materials are summarized in the vacancy concentration in the surface shell is much higher Table 1. The following figures illustrate analytical results at than that in the core, and this phenomenon becomes more significant when the particle size gets smaller, which is in 50 0 agreement with experimental measurements. Although the Table 1. Surface Eigenstress σs , Surface Young’s Modulus Ys, vacancy concentration in the core and the surface both decrease Bulk Young’s Modulus Yc, and the Vacancy Formation with the particle size, the apparent concentration in the whole Energy ΔHf for Ni, Cu, Au, and Pt Nanoparticles particle exhibits a different behavior. When the particle size is σ 0 Δ materials s , GPa Ys, GPa Yc, GPa Hf,eV reduced, the apparent concentration first increases and reaches Ni 1.31 205.32 183.65 1.6 a maximum value at dozens of nanometers particle size, and Cu 1.38 125.43 107.23 1.22 then rapidly decreases to a very small value. As mentioned Au 1.56 81.07 66.81 0.96 above, the influence of surface stress for metal nanoparticles Pt 2.6 109.8 84.08 1.45 considered in this paper will result in a decrease of both core and surface vacancy concentration. However, the fraction of the room temperature T = 300 K. Atomic volume for all metals in surface shell in total particle gets larger when particle size is the core and the surface shell used in calculating these plots is reduced, which leads to variation of apparent vacancy ̃ × −29 3 ̃ × −29 3 approximately Vc =1 10 m and Vs = 1.1 10 m , concentration. Thus, the appearance of the peak value of the respectively. Although vacancy concentration can vary with the apparent vacancy concentration is due to competition of the surface shell thickness, the vacancy concentration variation surface stress and the surface fraction in total particle. trend with the particle size remains almost the same. It is widely Obviously, the Pt particles show the largest difference in accepted that the interfacial thickness can vary from 0.5 to 2 vacancy concentration compared to other particles. nm. In the present work, we assume that the surface shell Parts a and b of Figure 2 show that the lattice and the sample thickness is 1 nm. For simplicity, the modulus and Poisson’s strains vary with the particle size for different metallic particles. ratio of the core and the shell are assumed to be independent of The strains increase significantly when the particle size the vacancy concentration. The Poisson’s ratio of the surface decreases below 100 nm. From eqs 14 and 16, both strains shell is assumed to be 0.3. The vacancy formation volume in the are caused by the surface stress and the vacancy concentration. core and the surface shell is assumed to be 0.7 × 10−29 m3 and Vacancy concentration is very small at 300 K, so the strain 0.66 × 10−29 m3, respectively. We assumed that the difference caused by vacancy concentration can be ignored. The strains in

Figure 1. Vacancy concentration (a) in the core and (b) the surface shell to the bulk ratio, (c) the vacancy concentration in surface shell to core ratio, and (d) the apparent vacancy concentration in particles to the bulk ratio, all as functions of the particle size for Ni, Cu, Au and Pt.

17617 DOI: 10.1021/acs.jpcc.6b05712 J. Phys. Chem. C 2016, 120, 17613−17619 The Journal of Physical Chemistry C Article

Figure 2. (a) Lattice and (b) sample strains vs the particle size and (c) the surface and (d) lattice stress vs particle size for Ni, Cu, Au, and Pt. parts a and b of Figure 2 are caused primarily by the surface ■ ACKNOWLEDGMENTS stress. From eqs 15 and 33, the surface stress and lattice stress This work was supported by the National Basic Research in parts c and d of Figure 2 can be caused by the surface stress Program of China (Grant No. 2012CB937502) and the and vacancy concentration, presenting the same trend with the National Science Foundation (IRES 1358088). strain. As expected for all metallic particles, the surface stress decreases when particle size decreases to a few nanometers. The lattice stress shows an opposite sign with the surface ■ REFERENCES eigenstress and increases exponentially with the reduced (1) Zhang, T. Y.; Ren, H. Solute concentrations and strains in particle size. When the particle size is extremely large, the nanograined materials. Acta Mater. 2013, 61, 477−493. lattice stress decreases to zero. Clearly, Pt particles show the (2) Ren, H.; Yang, X. S.; Gao, Y.; Zhang, T. Y. Solute concentrations largest surface stress compared to other particles, causing the and stresses in nanograined H−Pd solid solution. Acta Mater. 2013, largest variation in the vacancy concentration change. 61, 5487−5495. (3) Jiang, Q.; Lu, H. M. Size dependent interface energy and its applications. Surf. Sci. Rep. 2008, 63, 427−464. 4. CONCLUSIONS (4) Fischer, F. D.; Waitz, T.; Vollath, D.; Simha, N. K. On the role of Present work is based on the thermodynamic analysis, which surface energy and surface stress in phase-transforming nanoparticles. requires that the nanoparticles obey basic thermodynamic laws Prog. Mater. Sci. 2008, 53, 481−527. and principles. The intrinsic elastic parameters introduced in (5) Biener, J.; Wittstock, A.; Zepeda-Ruiz, L. A.; Biener, M. M.; ̈ present work are determined from atomistic simulations in our Zielasek, V.; Kramer, D.; Viswanath, R. N.; Weissmuller, J.; Baumer, previous work.48 The apparent vacancy concentration in M.; Hamza, A. V. Surface-chemistry-driven actuation in nanoporous gold. Nat. Mater. 2009, 8,47−51. spherical nanoparticle is calculated by taking into account (6) Jin, H. J.; Wang, X. L.; Parida, S.; Wang, K.; Seo, M.; Weissmüller, both the core and the surface defects concentration. J. Nanoporous Au−Pt alloys as large strain electrochemical actuators. Notwithstanding both the vacancy concentration in the core Nano Lett. 2010, 10, 187−194. and the surface shell for Ni, Cu, Au, and Pt particles, it can be (7) Hughes, M.; Xu, Y. J.; Jenkins, P.; McMorn, P.; Landon, P.; greatly decreased as the particle size is decreased. The apparent Enache, D. I.; Carley, A. F.; Attard, G. A.; Hutchings, G. J.; King, F.; vacancy concentration in the whole nanoparticle reaches the Stitt, E. H.; Johnston, P.; Griffin, K.; Kiely, C. J. Tunable gold catalysts maximum at the critical particle size of about 100 nm and is for selective hydrocarbon oxidation under mild conditions. Nature lower for larger or smaller particles. This phenomenon can be 2005, 437, 1132−1135. explained by the competition eﬀects of the surface stress and (8) Renner, F. U.; Stierle, A.; Dosch, H.; Kolb, D. M.; Lee, T. L.; the surface fraction in the total particle. Zegenhagen, J. Initial corrosion observed on the atomic scale. Nature 2006, 439, 707−710. (9) Jin, H. J.; Parida, S.; Kramer, D.; Weissmüller, J. Sign-inverted ■ AUTHOR INFORMATION surface stress-charge response in nanoporous gold. Surf. Sci. 2008, 602, Corresponding Author 3588−3594. *(Y.S.) E-mail: [email protected]. Telephone: 86-10-62333884. (10) Guisbiers, G. Schottky Defects in Nanoparticles. J. Phys. Chem. C 2011, 115, 2616−2621. Notes (11) Müller, M.; Albe, K. Concentration of thermal vacancies in The authors declare no competing ﬁnancial interest. metallic nanoparticles. Acta Mater. 2007, 55, 3237−3244.

17618 DOI: 10.1021/acs.jpcc.6b05712 J. Phys. Chem. C 2016, 120, 17613−17619 The Journal of Physical Chemistry C Article

(12) Salis, M.; Carbonaro, C. M.; Marceddu, M.; Ricci, P. C. (35) Gladkikh, N. T.; Kryshtal, O. P. On the size dependence of the Statistical thermodynamics of schottky defects in metal nanoparticles. vacancy formation energy. Funct. Mater. 1999, 6, 823−827. Nanosci. Nanotechnol. 2013, 3,27−33. (36) Gladkikh, N. T.; Kryshtal, A. P.; Bogatyrenko, S. I. Melting (13) Ouyang, G.; Zhu, W. G.; Yang, G. W.; Zhu, Z. M. Vacancy Temperature of Nanoparticles and the Energy of Vacancy Formation formation energy in metallic nanoparticles under high temperature and in Them. Tech. Phys. 2010, 55, 1657−1660. high pressure. J. Phys. Chem. C 2010, 114, 4929−4933. (37) Guisbiers, G.; Buchaillot, L. Universal size/shape-dependent law (14) Look, D. C.; Hemsky, J. W.; Sizelove, J. R. Residual native for characteristic temperatures. Phys. Lett. A 2009, 374, 305−308. shallow donor in ZnO. Phys. Rev. Lett. 1999, 82, 2552−2555. (38) Guisbiers, G. Size-Dependent Materials Properties Toward a (15) Vaccaro, L.; Morana, A.; Radzig, V.; Cannas, M. Bright visible Universal equation. Nanoscale Res. Lett. 2010, 5, 1132−1136. luminescence in silica nanoparticles. J. Phys. Chem. C 2011, 115, (39) Zhang, T. Y.; Ren, H. Solute Concentrations and Strains in − 19476−19481. Nanoparticles. J. Therm. Stresses 2013, 36, 626 645. (16) Ricci, P. C.; Carbonaro, C. M.; Corpino, R.; Cannas, C.; Salis, (40) Gibbs, J. Thermodynamics; Longmans, Green: New York, 1928. M. Optical and structural characterization of terbium-doped Y SiO (41) Cahn, J. W. On spinodal decomposition. Acta Metall. 1961, 9, 2 5 − phosphor particles. J. Phys. Chem. C 2011, 115, 16630−16636. 795 801. (17) Zeng, H. B.; Cai, W. P.; Liu, P. S.; Xu, X. X.; Zhou, H. J.; (42) Chen, T.; Chiu, M. S.; Weng, C. N. Derivation of the Klingshirn, C.; Kalt, H. ZnO-based hollow nanoparticles by selective generalized Young-Laplace equation of curved interfaces in nanoscaled etching: elimination and reconstruction of metal−semiconductor solids. J. Appl. Phys. 2006, 100, 074308. interface, improvement of blue emission and photocatalysis. ACS (43) Zhang, T. Y.; Wang, Z. J.; Chan, W. K. Eigenstress model for surface stress of solids. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, Nano 2008, 2, 1661−1670. 81, 195427. (18) Ding, Y.; Sun, C. Q.; Zhou, Y. C. Nanocavity strengthening: (44) Zhang, T. Y.; Luo, M.; Chan, W. K. Size-dependent surface Impact of the broken bonds at the negatively curved surfaces. J. Appl. stress, surface stiffness, and Young’s modulus of hexagonal prism [111] Phys. 2008, 103, 084317. β-SiC nanowires. J. Appl. Phys. 2008, 103, 104308. (19) Ouyang, G.; Wang, C. X.; Yang, G. W. Surface energy of (45) Weissmuller, J.; Cahn, J. W. Mean stresses in microstructures nanostructural materials with negative curvature and related size − due to interface stresses: A generalization of a capillary equation for effects. Chem. Rev. 2009, 109, 4221 4247. solids. Acta Mater. 1997, 45, 1899−1906. (20) Yates, J. T., Jr. Photochemistry on TiO2: Mechanisms behind − (46) Weissmuller, J.; Duan, H. L.; Farkas, D. Deformation of solids the surface chemistry. Surf. Sci. 2009, 603, 1605 1612. with nanoscale pores by the action of capillary forces. Acta Mater. (21) Mukherjee, M.; Chakravorty, D.; Nambissan, P. M. G. 2010, 58,1−13. Annihilation characteristics of positrons in a polymer containing silver (47) McCarty, K. F.; Nobel, J. A.; Bartelt, N. C. Vacancies in solids nanoparticles. Phys. Rev. B: Condens. Matter Mater. Phys. 1998, 57, and the stability of surface morphology. Nature 2001, 412, 622−625. − 848 856. (48) Ma, H. X.; Xiong, X. L.; Gao, P. P.; Li, X.; Yan, Y.; Volinsky, A. (22) Nambissan, P. M. G.; Upadhyay, C.; Verma, H. C. Positron A.; Su, Y. J. Eigenstress model for electrochemistry of solid surfaces. lifetime spectroscopic studies of nanocrystalline ZnFe2O4. J. Appl. Phys. Sci. Rep. 2016, 6, 26897. 2003, 93, 6320−6326. (49) Kraftmakher, Y. Equilibrium vacancies and thermophysical (23) Chaudhuri, S. K.; Ghosh, M.; Das, D.; Raychaudhuri, A. K. properties of metals. Phys. Rep. 1998, 299,79−188. Probing defects in chemically synthesized ZnO nanostructures by (50) Maier, K.; Peo, M.; Saile, B.; Schaefer, H. E.; Seeger, A. High− positron annihilation and photoluminescence spectroscopy. J. Appl. temperature positron annihilation and vacancy formation in refractory Phys. 2010, 108, 064319. metals. Philos. Mag. A 1979, 40, 701−728. (24) Gao, P. P.; Zhu, Z. J.; Ye, X. L.; Wu, Y. C.; Jin, H. J.; Volinsky, A. A.; Qiao, L. J.; Su, Y. J. Defects evolution in nanoporous Au(Pt) during dealloying. Scr. Mater. 2016, 113,68−70. (25) Yang, C. C.; Li, S. Investigation of cohesive energy effects on size-dependent physical and chemical properties of nanocrystals. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 75, 165413. (26) Attarian Shandiz, M.; Safaei, A.; Sanjabi, S.; Barber, Z. H. Modeling the cohesive energy and melting point of nanoparticles by their average coordination number. Solid State Commun. 2008, 145, 432−437. (27) Vanithakumari, S. C.; Nanda, K. K. A universal relation for the cohesive energy of nanoparticles. Phys. Lett. A 2008, 372, 6930−6934. (28) Qi, W. H.; Wang, M. P. Size dependence of vacancy formation energy of metallic nanoparticles. Phys. B 2003, 334, 432−435. (29) Qi, W. H.; Wang, M. P. Vacancy formation energy of small particles. J. Mater. Sci. 2004, 39, 2529−2530. (30) Qi, W. H.; Wang, M. P.; Zhou, M.; Hu, W. Y. Surface-area- difference model for thermodynamic properties of metallic nanocrystals. J. Phys. D: Appl. Phys. 2005, 38, 1429−1436. (31) Xie, D.; Wang, M. P.; Cao, L. F. Comment on “Vacancy formation energy of small particles. J. Mater. Sci. 2005, 40, 3565−3566. (32) Ouyang, G.; Zhu, W. G.; Yang, G. W.; Zhu, Z. M. Vacancy Formation Energy in Metallic Nanoparticles under High Temperature and High Pressure. J. Phys. Chem. C 2010, 114, 4929−4933. (33) Ouyang, G.; Zhu, W. G.; Sun, C. Q.; Zhu, Z. M.; Liao, S. Z. Atomistic origin of lattice strain on stiffness of nanoparticles. Phys. Chem. Chem. Phys. 2010, 12, 1543−1549. (34) Ouyang, G.; Sun, C. Q.; Zhu, W. G. Atomistic origin and pressure dependence of band gap variation in semiconductor nanocrystals. J. Phys. Chem. C 2009, 113, 9516−9519.

17619 DOI: 10.1021/acs.jpcc.6b05712 J. Phys. Chem. C 2016, 120, 17613−17619