A General Kinetic-Optical Model for Solid-State Reactions Involving

Mar 10, 2016 - A General Kinetic-Optical Model for Solid-State Reactions Involving the Nano Kirkendall Effect. The Case of Copper Nanoparticle Oxidati...
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A General Kinetic-Optical Model for Solid-State Reactions Involving the Nano Kirkendall Effect. The Case of Copper Nanoparticle Oxidation Mariano D. Susman, Alexander Vaskevich,* and Israel Rubinstein* Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 7610001, Israel S Supporting Information *

ABSTRACT: Oxidation of copper nanoparticles (NPs) and other solidstate reactions have been shown recently to involve the nanoscale Kirkendall effect (NKE). The process, resulting in the formation of internal voids, has been rarely considered in most of the simple reaction kinetic models currently available. Here we present a general solid-state reaction kinetic model based on the assumption of steady-state diffusion profiles for rationalizing the evolution of the optical behavior observed in plasmonic Cu NPs undergoing solid-state oxidation to Cu2O. While the model is applied here to an oxidation process, it is applicable in principle to any system showing Kirkendall voiding. An analytical expression of the rate law for spherical NPs is derived, and implications of the model are discussed. By combining the general kinetic model with Mie scattering solutions under the quasi-static approximation, the extinction cross section of Cu NPs as a function of the Cu-to-Cu2O conversion fraction is modeled, for NPs of different initial sizes and for different degrees of shrinkage upon formation of a Cu2O shell layer. The experimental observation of an initial increase in the surface plasmon (SP) band extinction intensity, followed by a decrease, is qualitatively reproduced using the combined model. This characteristic behavior of Cu nanoplasmonic systems is not expected theoretically in sufficiently small NPs; however, NPs of > ∼6 nm in diameter are expected to exhibit this type of behavior. Our results suggest that the NKE may be important in describing the optical behavior of plasmonic NPs subject to solid-state oxidation.

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INTRODUCTION While the Valensi−Carter (VC) model1,2 is the most complete reaction model providing a simple reaction rate law for solidstate reactions under steady-state diffusion in the spherical geometry, the recent observation of the Kirkendall effect in a large variety of micro- and nanostructured systems3,4 suggests that the VC model may be insufficient for describing reaction kinetics involving formation of hollow interiors in reacting particles. Notably, a simple diffusion-controlled kinetic model describing the kinetics of solid-state reactions in spherical NPs undergoing the nano Kirkendall effect (NKE), affording an explicit rate law with an analytical solution, has not been fully produced.5,6 While some basis was provided by Yin et al.,7 the kinetic rate law was not fully derived. Similarly, Gusak et al.8,9 extensively modeled the NKE, combining steady-state diffusion arguments with curvature (Gibbs−Thomson) effects leading to the suppression or shrinkage of the voids.10 However, simple explicit rate laws have not been provided. Here we introduce a simple theoretical model that describes the kinetics of such processes based on common assumptions used in the VC contracting-core model, and described by Yin et al.7 and Gusak et al.8 The model is general and may be applied to any solid-state reaction wherein a homogeneous product © XXXX American Chemical Society

shell forms on a spherical particle. An extended version of the VC rate law is presented. Most diffusion-controlled reaction models have been developed in the context of oxidation of metals in planar geometries,11 such as the Wagner12−14 and Cabrera−Mott15 models, allowing microscopic interpretation of the parabolic kinetic laws observed in numerous metal oxidations,16 or certain deviations.17 Adaptation of these models to spherical geometry naturally followed.18−20 However, even when these models regard vacancy diffusion as an important factor in the oxidation process, accumulation of vacancies in specific regions of the reacting particles has usually been neglected. Certain sophisticated models discussing the NKE in NPs were recently proposed on the basis of the thermodynamic extremal principle,21 short-circuit diffusion,22 and others.23,24 Although a model under the space-charge regime can be developed for describing the initial oxidation stages,12,15,25 we prioritize the possibility of qualitatively describing the full Special Issue: Kohei Uosaki Festschrift Received: January 6, 2016 Revised: February 16, 2016

A

DOI: 10.1021/acs.jpcc.6b00137 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C extent of the Kirkendall process using a simple model that can be applicable to spherical particles of any size. As our main goal is to interpret the optical behavior of metal NPs during such reactions, we adopted the assumption of an ion-governed diffusion-controlled process under the presence of steady-state concentration gradients. A known example of a nanostructured system undergoing NKE is copper NP oxidation, demonstrated in solid-state oxidations26−29 as well as in oxidation of Cu NPs in organic media.30 Cu NPs exhibit localized surface plasmon resonance (LSPR) bands in the visible spectral range,28,31 which are sensitive to the NP size and shape, as well as to the properties of the dielectric medium; the LSPR response is therefore highly sensitive to the degree of surface oxidation of the NPs. Compared to other, more stable coinage metals (Au, Ag), Cu has been less studied and applied in the nanotechnology arena. Cu NPs are known to undergo oxidation and corrosion processes, often considered sufficiently fast to restrict their use in a variety of applications, particularly in the field of nanoplasmonics, thus favoring the use of expensive Au or Ag NPs. The limited research implies a seriously deficient understanding of the kinetics of Cu NP oxidation and corrosion, how these processes can be hindered, and their effect on the plasmonic properties of reacting Cu NPs. The complex process of Cu oxidation, studied extensively for decades, was found to be diffusion-controlled.12 A scheme presenting the main elementary steps involved in Cu NP oxidation is shown in Figure 1A, where diffusion of Cu+ and O2− ions controls the rate of oxide formation.

An initial plateau or slight increase in the plasmonic extinction intensity was described, followed by a decrease. This behavior was attributed to an initial damping effect exerted by the ligand used for the NP synthesis, which, following removal or isolation from the metal NP accompanying oxide layer formation during the annealing process, results in a weak increase of intensity, later turning into a decrease. However, in the latter study, electron microscopy characterizations of initial or intermediate oxidation stages were not reported, as well as LSPR spectra of the freshly prepared colloids; hence, the initial degree of oxidation of the particles is not clear. We have performed LSPR measurements during thermal annealing of Cu NPs prepared by chemical deposition on glass substrates. A substantial increase in the LSPR extinction intensity was observed at initial stages of the oxidation, followed by a gradual decrease,29 resembling results obtained in several Cu colloid oxidations.32,36 A simple theoretical description of the effect of surface oxidation on the optical properties of Cu NPs was also reported.37 Our observations are in qualitative agreement with this theoretical model; however, the numerical modeling of the optical spectra is oversimplified, not considering the change in NP size resulting from the difference in the metal and oxide molar volumes, as well as the hollowing process associated with the NKE. The aim of the present work is to study, by theoretical means, the effect of oxide formation and the NKE on the kinetics and optical properties of Cu NPs undergoing solidstate oxidation, and the relevance of the NP size to the observed spectral trends. For modeling the optical properties of the reacting NPs, we employ Mie scattering solutions under the quasi-static approximation. The dielectric constant of the hollowing metal core is estimated using an effective medium approximation (EMA); for convenience, vacancies are assumed to be distributed homogeneously in the core.

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THEORETICAL MODEL We first define general terms and assumptions. Two limiting cases, equivalent to the VC model and a model denoted here as the NKE model, are presented as the basis for the development of a general model, which considers cases where vacancy diffusion into the reacting particle core plays a role.

Figure 1. Schemes describing (A) the elementary steps most relevant to the oxidation mechanism of Cu NPs and (B) a NP of initial radius a0 undergoing oxidation. Boundaries I and II at radii a1 and a2, respectively, define the oxide volume formed after a given oxidation time.

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BASIC DEFINITIONS AND ASSUMPTIONS Let us consider the oxidation of a spherical Cu NP of an initial radius a0 in air, described by the oxidation reaction

While the optical behavior of Cu nanoplasmonic systems under oxidation (or corrosion) has been documented in various chemical environments, the dependence of the optical evolution on the reaction conditions and particle size is barely understood. For example, oxidation of Cu NPs in several solvents was reported,32,33 including water,34,35 DMF,33 benzene,32 cyclohexane,32 chloroform,30 and toluene,32 among others. While in some solvents the oxidation process was claimed to lead to damping of the Cu SP intensity (as in water,34 hexane,30 chloroform,30 and DMF33), in others (as in benzene32 and cyclohexane32), an initial increase of the SP extinction was seen, followed by a decrease and eventual disappearance of the plasmonic band.32,36 While the latter behavior was interpreted as being related to a solvent or ligand effect,32,36 the reasons underlying a certain type of behavior are yet unclear. Recently, in situ LSPR measurements during ∼10 nm diameter Cu NP oxidation in air were reported by Rice et al.28

ηCu 0 +

η′ O2 → Cu ηOη′ 2

(1)

where η and η′ are the respective stoichiometric coefficients of cations and anions, for the formation of a mole of oxide product. All phases described by eq 1, particularly the oxide product, are considered to be in chemical equilibrium or represent kinetically persistent states (as is the case of Cu2O at low temperatures).29 In our description, we consider the oxidation of Cu NPs at low temperatures, implying the formation of Cu2O as the main oxidation product. Throughout the solid-state reaction, a homogeneously thick, nonporous, and adherent oxide layer is assumed to form on the reacting metal NP. Two sharp moving boundaries exist as the oxidation proceeds, i.e., (I) the metal-oxide and (II) the oxideair boundaries, located at radii a1 and a2, respectively. These B

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possible diffusion paths (e.g., through grain boundaries or dislocations).39 The solid-state kinetic models below for particle oxidation are based on the assumption that, at any time, steady-state concentration profiles are achieved for the diffusion of the ions through the transient oxide shells (∂ci/∂t = 0). The ionic concentrations at both transient oxide boundaries are assumed to remain constant during the reaction (the equilibrium concentrations for the contact with the metal, ci,1, or with air, ci,2), while the equilibrium concentration of O2− may depend on the O2 partial pressure. The steady-state equation for diffusion through a spherical shell was described by Barrer.40 Regardless of the initial profile, it can be shown that the same steady-state concentration profile and ionic flow will be achieved after a sufficiently long time (t → ∞):

boundaries vary with the conversion fraction (Figure 1B). For simplicity, formation of voids at the metal−oxide interface, resulting from the NKE, is not considered in the present model; this may imply certain deviations of the kinetic laws at high degrees of conversion. In the Cu oxidation mechanism, the half-reactions responsible for the formation of the ionic species are considered to be fast but spatially separated, as follows: ηCu 0 → ηCu+ + ηe− η′ O2 + 2η′e− → η′O2 − 2

(at I ) (at II )

(2)

(3)

These half-reactions are responsible for the formation of the oxide layer (and variation of its volume, dVOx) by addition of individual ions into the growing ionic oxide structure. Another fast step is charge-transport through the oxide layer. Cu2O is a p-type semiconductor, and actual charge transport is provided by hole movement. For convenience, we formally describe the charge balance as electronic −

−

ηe (I ) → ηe (II )

ci(r ) =

(4)

Ji =

η′ O2 → ηCu+(I ) + η′O2 −(II ) 2

(6)

Note that these fast steps are limited by accumulation of ions at the oxide boundaries. If the reaction is diffusion controlled, the rate-controlling step in the oxide formation process (with the formula CuηOη′) is diffusion of ions from one oxide layer boundary toward the other. Oxide growth at a particular interface, where one of the half-reactions occurs, can only be sustained if a continuous supply of the corresponding counterion is provided. The latter occurs by diffusional transport. The oxide formation can then be reduced to a process taking place at the oxide layer boundaries, where ions are being added to the oxide lattice. Nevertheless, given the counter diffusion of ions along the entire oxide volume, it is convenient (see below) to consider a certain probability of oxide formation at any point in the layer, and, in the extreme case, that at some radius (aOx) a sharply defined steady-state oxidation f ront (of) exists where all the oxide is being formed: ηCu+(I ) + η′O2 −(II ) → Cu ηOη ′(of )

(8)

dQ i dt

= −4πDi*(ci ,2 − ci ,1)

a1a 2 a 2 − a1

(9)

where Qi is the quantity of i (in moles) and Di* is the tracer diffusivity of the ith species in m2 s−1 units, assumed not to depend on the species concentration.41 Note that by species i we may refer indistinctly to the ion or the corresponding reacting element in its zero oxidation state, as the half-reaction relating these species is fast. This is particularly important when describing vacancy diffusion. It is useful to express the progress of the reaction by the dimensionless conversion fraction (δ), herein defined as the fraction of the initial material converted to product

(5)

Combining eqs 2−4, these series of fast steps can be summarized as ηCu 0 +

r(a 2 − a1)

The steady-state flow of species i (Ji, in mol s−1) across the oxide shell is given by

which ensures coupling between the half-reactions, and, in the case of the formation of Cu2O as the final oxidation product, fulfillment of the stoichiometric relationship: η = 2η′

a1ci ,1(a 2 − r ) + a 2ci ,2(r − a1)

δ(t ) ≡

VOx(t ) ZV0

(10)

where VOx is the volume of oxide formed after time t on a particle of initial volume V0 = VCu0(t = 0) = (4/3)πa03, and Z is the dimensionless coefficient of expansion of the metal upon oxidation, also known as the Pilling−Bedworth ratio,42 defined as Z=

dV 1 ρCu0 FWOx = − Ox η FWCu0ρOx dVCu0

(11)

where FWOx and FWCu0 and ρOx and ρCu0 are the formula weights and the bulk mass densities of the oxide and metal, respectively. Densities at the nanoscale are assumed to be equal to the bulk densities. In eq 10, VOx/Z represents the volume of metal consumed to form the volume VOx of the oxide, and therefore, δ represents the fraction of the initial volume consumed in the process.

(7)

Namely, oppositely charged ions diffuse to this hypothetical oxidation front in order to form the stoichiometric oxide. The diffusion behavior of the oppositely charged ions is assumed to be uncorrelated, equivalent to the behavior of neutral entities. No assumptions regarding charge conservation are required; charge neutrality of the system arises from the fast coupling between the two half-reactions according to the stoichiometric considerations. For describing the ionic diffusion process, electric field enhanced transport (Cabrera−Mott model)15 is neglected, as well as strain effects38 and other

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VALENSI−CARTER (VC) MODEL: INWARD O2− DIFFUSION The VC (core-contraction or core-shrinking) model1,2 is seemingly the most complete existing model for describing diffusion-controlled reactions in layered spheres. This model is applicable to particles of any size at any conversion fraction. It considers the variation in diffusion profile with the system’s C

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is the inward oxidation rate constant, in m2 s−1. A scheme describing this model is presented in Figure S1 (Supporting Information).

spherical volume change, thus overcoming limitations in the Jander and Ginstling−Brounshtein models.43 For Cu NP oxidation, let us consider the hypothetical case wherein the incoming diffusion of O2− ions to interface I is the rate-controlling oxidation step, while the diffusion of Cu+ through the entire transient oxide is assumed to be unimportant. Under these conditions, the oxidation front is located at the metal−oxide interface (aof = a1). (Note that this assumption implies a faster flow of O2− compared to Cu+, which is incompatible with cases where the Kirkendall effect is observed.) At interface I, a stoichiometric consumption of metal cations has to occur in order to balance the incoming flow of O2− anions: 1 dQ Cu+ 1 dQ O2− =− η dt η′ d t

(at I )

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NANO KIRKENDALL EFFECT (NKE) MODEL: OUTWARD CU+ DIFFUSION A similar model can be produced for the opposite case, where Cu+ diffusion outward is the oxidation rate-controlling step, while the inward diffusion of anions is considered unimportant. Under these conditions, a sharp oxidation front is assumed to be located at the external oxide−air interface (aof = a2).44 If no inward diffusion of anions occurs, the outward diffusion of metal leads to loss of core material without a change in core volume (V1). Therefore, a net inward diffusion of metal vacancies (Jv) has to compensate for the variation in metal volume. Interface I does not change its position as the oxidation advances, while only the reacting interface II represents a moving boundary. The corresponding equations for a1 and a2 as a function of δ would be

(12)

The oxidation rate is defined by the inward flow of oxygen anions, for which eq 9 can be written as dQ O2− dt

= −4πDO2−*(c O2−,2

aa − c O2−,1) 1 2 a 2 − a1

a13 = a0 3 (13)

a 2 3 = a0 3(1 + Zδ)

which, in terms of cation consumption using relation 12, is dQ Cu+ dt

aa η = 4π DO2−*(c O2−,2 − c O2−,1) 1 2 a 2 − a1 η′

The corresponding outward Cu flow can be written as dQ Cu+ dt

Both a1 and a2 can be expressed as a function of δ by considering the conservation of matter for the core metal (eq 15) and the volume expansion of the sphere during the reaction (eq 16): (15)

a 2 3 = a0 3(1 − δ + Zδ)

(16)

= −4πDCu+*(cCu+,2 − cCu+,1)

a v 3 = a0 3δ

2 Zδ 3

2Z

dt

=−

dQ Cu0 dt

4 ρCu0 dδ = πa 0 3 3 FWCu0 dt

kout = −

(18)

(19)

a0 2

t (26)

FWCu0 DCu+*(cCu+,2 − cCu+,1) ρCu0

(27)

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GENERAL MODEL Although the previous models may represent valid approximations for some empirical observations, it is useful to develop a general model describing realistic cases, where diffusion of both ions is considered. In the following model, it is assumed that the oxidation rate is controlled by diffusion of both Cu+ and O2−. In contrast with the VC and NKE models, where the location of the oxidation front was fixed at one of the moving

k Z − [1 − (1 − Z)δ]2/3 − (Z − 1)(1 − δ)2/3 = in2 t 2(Z − 1) a0 (20)

where η FWCu0 DO2−*(c O2−,2 − c O2−,1) η′ ρCu0

kout

is the corresponding outward oxidation rate constant, in m2 s−1. Interestingly, the stoichiometric condition eq 12 is not needed for solving the diffusion problem. Still, since the oxide forms at interface II, it can be assumed that JCu+/η = JO2−/η′ at the oxidation front (aof = a2). A scheme describing this model is presented in Figure S2 (Supporting Information).

Equations 14 and 19 can be equated, and integrated between conversion fractions 0 and δ and times between 0 and t. The well-known Valensi−Carter (VC) equation is then obtained

k in =

=

where

Conservation of the Cu mass gives dQ Cu+

(25)

− (1 + Zδ)2/3 + 1

Replacing using eq 15 and deriving with respect to time, dQCu0/dt can be expressed as a function of dδ/dt: dt

(24)

The diffusion problem can be solved by equating eqs 19 and 24, using eqs 22 and 23 to describe the geometric evolution, and integrating as above. The following kinetic model is obtained

a13

4 ρCu0 dδ =− πa 0 3 3 FWCu0 dt

a1a 2 a 2 − a1

Equation 22 cannot describe the mass balance for the metal, but eqs 15 and 17−19 still apply. This is because the inward vacancy flow results in a decrease in the metal effective density at the core with conversion. For simplicity, we assume that the metal vacancies are homogeneously distributed throughout the volume V1. An equivalent spherical void radius (av) can be defined as

Therefore, by substituting eqs 15 and 16 in eq 14, the outgoing cation flow can be expressed as a sole function of δ. The total amount of metallic Cu is given by 4 ρCu0 πa13 Q Cu0 = 3 FWCu0 (17)

dQ Cu0

(23) +

(14)

a13 = a0 3(1 − δ)

(22)

(21) D

DOI: 10.1021/acs.jpcc.6b00137 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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described by Yin et al.,7 the inward growth of the oxide shell volume is proportional to the inward flux of O2− at I. The underlying reason is that, in order to sustain the growth at boundary I, a minimal number of anions have to be locally present to allow further incorporation of cations in the oxide lattice. The probability of local growth is therefore proportional to the flux of counterions. Similarly, Cu+ diffusion to interface II leads to outward growth of the oxide layer. At interface I, the inward diffusion of O2− can be compensated by either Cu+ or vacancy diffusion. At interface II, the stoichiometric relationship has to be imposed in order to ensure a stoichiometric balance in the system; the outward diffusion of Cu+, which dictates the outward oxide growth rate, is then compensated by a stoichiometric inward O2− flow at II. The Cu+ diffusion problem can be solved as above, considering the mass balance of Cu, making use of eqs 24 and 19, but in combination with eqs 31 and 32, and integrating within the same limits as before. For 0 < ϕ ≤ 1, one obtains

boundaries as an ad hoc hypothesis, no restriction is imposed on the location of the oxidation front in the general model, such that a1 ≤ aof ≤ a2. We assume that the volume enclosed by the inner oxide interface (V1), containing the unreacted metal, varies as a result of both consumption of the metal (dVCu < 0) and accumulation of vacancies (dVint v > 0) forming an internal void, such that dV1 = dVCu + dV vint

(28)

while the total volume variation of the reacting particle is dV2 = dV1 + dVOx

(29)

A parameter ϕ ∈ (0, 1] is defined, indicating what proportion of the reacted metal volume (dVCu = −dVOx/Z) gives rise to core contraction (dV1): ϕ ≡ −Z

dV1 dVOx

(30)

It is assumed that ϕ is constant during an isothermal oxidation, and is independent of the conversion fraction. On the basis of these definitions and assumptions, the following equations are obtained for the variation of the oxide boundaries a1 and a2:

a13 = a0 3(1 − ϕδ)

(31)

a 2 3 = a0 3(1 − ϕδ + Zδ)

(32)

Z − ϕ[1 − (ϕ − Z)δ]2/3 − (Z − ϕ)(1 − ϕδ)2/3 2(Z − ϕ)ϕ kout = 2t a0

where kout is defined by eq 27. To solve the O2− diffusion problem, the stoichiometric condition is applied at interface II. The diffusion problem is similarly solved using eqs 14 and 19, resulting in the same integral equation, where the process rate constant is kin, defined in eq 21. It can be shown that eq 34 tends to the NKE model (eq 26) for ϕ → 0 and to the VC model (eq 20) for ϕ → 1 (the rate constant should be changed accordingly). Figure 2A−C summarizes the evolution of the integral kinetic model (g), which corresponds to the left-hand side of the time equations (eqs 20, 26, and 34), the location of the oxide layer boundaries and equivalent void radii, and the ratio of oxide thickness to particle radius, as a function of the conversion fraction, for the models discussed above. In general, larger differences between the models are seen at high conversion fractions. Comparing the two limiting cases, the time evolution of the conversion fraction may be up to 4 times faster (Figure 2D) in the context of the NKE compared to the VC case.

which reduce to eqs 22 and 23 for ϕ → 0 (NKE model) and to eqs 15 and 16 for ϕ → 1 (VC model). The dynamics of vacancy coalescence and formation of an internal hollow cavity are usually intricate.4,7,45 For simplicity, we again assume that the vacancies are distributed homogeneously throughout the core volume. Hence, there is no accumulation of vacancies at close proximity to boundary I, ensuring contact between the metal and the oxide, and the attainment of equilibrium concentrations at the interfaces. An equivalent (or virtual) spherical internal void radius (aint v ) can be defined as (a vint)3 = a0 3(1 − ϕ)δ

(33)

which tends to 0 for ϕ → 1 and to eq 25 for ϕ → 0. Table 1 summarizes the equations describing the evolution of relevant volumes, as a function of the conversion fraction, according to the models discussed above. In the general model, two diffusion problems have to be solved, namely, Cu+ and O2− diffusion. Following the premise

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PHYSICAL MEANING OF ϕ The physical meaning of ϕ can be deduced by estimating the oxide growth rate at each oxide layer boundary. To this end, it is convenient to change the frame of reference of the oxide formation from a static frame of reference, fixed with respect to the initial radius a0, to a moving oxide lattice reference. This transformation is commonly performed in order to facilitate the interpretation of the Kirkendall effect.6 In the moving lattice frame of reference, the Kirkendall spherical surface (equivalent to the Kirkendall plane) is allowed to spatially shift from its initial radius a0 to ak. The oxide layer grows in the inward (dVinOx > 0) and outward (dVout Ox > 0) directions with respect to this Kirkendall surface, such that dVOx = dVinOx + dVout Ox . Hypothetical markers initially located at the metal−air interface will change positions following this Kirkendall shift as oxidation advances, if such a shift exists.

Table 1. Actual or Equivalent Spherical Volumes as a Function of the Conversion Fraction, according to the General Model, and the Particular Cases Where ϕ = 0 (NKE) and ϕ = 1 (VC)a ϕ=1

volume VCu Vint v V1 VOx V2 Vext v VT a

0 V0(1 − δ) V0(1 − δ + Zδ) V0δ

0