Diffusion In Solids The Kirkendall Effect And Fabrication Of Coreshell Nanoparticles

Although our current repertoire of RD fabrication techniques is quite flexible in terms of the reacting chemicals, it is limited by the required porosity/permeability of the supporting medium (a gel, a polymer, or an elastomer). In this section, we will study - and later apply in nanofabrication - phenomena that overcome this limitation with the help of so-called solid interdiffusion.

Can atoms of crystalline solids such as metals or semiconductors diffuse into one another? Indeed, they can. The first experimental demonstration of such interdiffusion dates back to 1896 when Sir Roberts-Austen observed it in a couple of gold and lead metals. For a long time afterward, scientists believed that interdiffusion occurs by a direct exchange of atoms (Figure 10.10) or via a 'ring diffusion' mechanism, in which the atoms swap their positions in a concerted fashion.13 It was not until 1947 when a young American scientist, Ernest Kirkendall, performed his famous experiment suggesting that interdiffusion is instead due to the swapping of atoms and crystal vacancies (Figure 10.10). Interestingly, this explanation had originally met with much disbelief or even hostility since the concentrations of vacancies (about one in a million lattice sites of a metal)13 were historically considered too low for the atom-vacancy exchange to be the dominant effect. Nevertheless, Kirkendall stuck to his explanation tenaciously and after some bitter struggles (involving repeated rejection of his paper13) had the effect confirmed and, ultimately, approved by the community.14

Kirkendall Effect Alloy
Figure 10.10 Plausible mechanisms of interdiffusion between solids A and B. Left: direct atom-atom exchange/'swapping'. Middle: ring diffusion. Right: Kirkendall atom-vacancy exchange

Let us briefly describe the 'canonical' Kirkendall experiment, in which metals/ alloys, A and B, are brought into contact, and a thin wire is placed carefully between them. The wire is attached to a laboratory bench and acts as a point of reference (Figure 10.11). Now the metals are welded together by placing them in

Kirkendall Effect
Figure 10.11 Scheme of Kirkendall's experiment, in which a heated AB diffusion couple translates with respect to a fixed point of reference

an oven at a temperature high enough to speed up interdiffusion but lower than the melting temperature (usually, one-third of the melting temperature).15 When the specimen is removed from the oven, it is observed that the wire has 'traveled' laterally through the specimen, or rather, since the wire is fixed to the laboratory bench, the specimen has traveled past the wire. This net translation cannot be a result of either atom-atom exchange or the ring diffusion mechanism, for in such cases the metal slabs should have changed only their relative compositions, not their dimensions and locations. In the words of Kirkendall himself, 'diffusion formulas based on an equal interchange of solute and solvent [i.e., of A and B] atoms and a substantially stationary interface will be in error'.13 The only explanation, with reference to Figure 10.10, is that metal A acquired a net amount of vacancies, or empty space, while metal B acquired a net amount of mass.

Since its discovery, the Kirkendall effect has been observed in materials other than the brass and copper used in the original experiments. The effect has also found numerous applications, most recently in nanotechnology where it can be cleverly applied to fabricate hollow and core-and-shell nanostructures.

Let us consider an illustrative experimental example (reported by the Berkeley-based group of Paul Alivisatos), in which a nanocrystal of cobalt is immersed in a boiling solution of sulfur and solvent (Figure 10.12).16 At elevated temperatures, cobalt atoms diffuse rapidly into the solution leaving behind vacancies in the crystal. Since isolated vacancies on the crystal lattice are energetically unfavorable, they 'aggregate' near the crystal's center to ultimately produce the so-called Kirkendall void. Simultaneously, sulfur atoms from solution impinge onto the crystal surface where they react with cobalt to produce cobalt sulfide, Co3S4, arranged in the shell-like 'reaction zone'. The end result of this reaction-diffusion process is a hollow, compositionally homogeneous particle of Co3S4.

This process lends itself to an interesting theoretical RD analysis. Historically, models of the Kirkendall effect have considered the process to be strongly diffusion-limited. This assumption is justified for macroscopic systems, in which atoms diffuse over large distances; however, it may fail for nanoscale systems, where the time scales of reaction and diffusion are often commensurate.

In developing a zero-order RD model, we will make two simplifications. First, based on previous studies,16-20 we will assume that the concentration profile of cobalt atoms within the cobalt sulfide shell adjusts rapidly relative to the speed of growth of the layer itself. Under such circumstances, we can use the quasi-steady approximation, in which the concentration profile of cobalt within the 'shell' obeys the steady-state diffusion equation. Second, we will assume that the diffusion of cobalt atoms into the liquid sulfur is much faster than the diffusion of sulfur atoms into the bulk cobalt. This assumption rests on the physical basis that there are more vacancies per unit volume in a liquid than in a solid. Hence, we neglect the diffusion of sulfur atoms into the cobalt nanocrystal entirely.

Figure 10.12 Top: a cobalt nanocrystal (represented as an aggregate of blue spheres) is immersed in a solution of sulfur (yellow spheres) and ortho-dichlorobenzene (not shown). Middle: when the temperature is raised to 400 K, cobalt atoms begin to diffuse into the solution leaving behind vacancies, which aggregate into one large void. Bottom: at the same time, the incoming sulfur atoms react with cobalt to give a hollow Co3S4 nanocrystal

Figure 10.12 Top: a cobalt nanocrystal (represented as an aggregate of blue spheres) is immersed in a solution of sulfur (yellow spheres) and ortho-dichlorobenzene (not shown). Middle: when the temperature is raised to 400 K, cobalt atoms begin to diffuse into the solution leaving behind vacancies, which aggregate into one large void. Bottom: at the same time, the incoming sulfur atoms react with cobalt to give a hollow Co3S4 nanocrystal

With reference to Figure 10.13, the concentration of cobalt atoms, c, in the shell is governed by the steady-state diffusion equation V2c = 0. At the interior boundary of the shell, the concentration of cobalt is equal to that of the bulk solid, c(Rj) = 1/vCo, where vCo is the molar volume of cobalt; at the exterior boundary, conservation of

Figure 10.13 Growth of hollow Co3S4 nanocrystals modeled by RD. (a) Initially, the radius of a particle is equal to that of the cobalt crystal, Ri — R2. The diffusive fluxes of cobalt, JCo (dominant), and sulfur, JS (negligible), are also indicated by the arrows. (b) As cobalt atoms diffuse outwards, they leave behind crystal vacancies and react with sulfur to create a shell of outer radius R2, which increases with time. (c) Ultimately, the vacancies aggregate into a Kirkendall void surrounded by a shell of Co3S4

Figure 10.13 Growth of hollow Co3S4 nanocrystals modeled by RD. (a) Initially, the radius of a particle is equal to that of the cobalt crystal, Ri — R2. The diffusive fluxes of cobalt, JCo (dominant), and sulfur, JS (negligible), are also indicated by the arrows. (b) As cobalt atoms diffuse outwards, they leave behind crystal vacancies and react with sulfur to create a shell of outer radius R2, which increases with time. (c) Ultimately, the vacancies aggregate into a Kirkendall void surrounded by a shell of Co3S4

cobalt atoms requires that — D(dc/dr)R — kc(R2) (Section 8.1), where D is the diffusion coefficient and k is the surface reaction rate constant of the reaction producing Co3S4. To simplify the solution, we introduce the following dimensionless variables: r — r/Ri, c — c/cb and R — R2/Ri. Thus, our dimensionless problem becomes V2c — 0 with boundary conditions R(l) — I and — (dR/dr)- — DaR(R), where Da — kRi/D is the familiar Damkohler number characterizing the ratio of the diffusive time scale, Tdiff — R\/D, to the reaction time scale, trxn — Ri/k. Solving the above equation in the relevant spherical coordinate system (Section 2.4) gives the following concentration profile for cobalt within the shell (I <R < R):

Now, to determine how the interface between the Co3S4 shell and the sulfur solution evolves in time, we first equate the number of cobalt atoms diffusing through the interface, JCo4pR2dt, with the number deposited in the product layer, (3/nCo3s4)4pR2dR2, where nCo3s4 is the molar volume of Co3S4 and 3 is a stoichiometric factor. From this relation, we arrive at the following differential equation for the outer radius, R2, of the shell:

Applying Fick's law for the diffusive flux, JCo — — D(dc/dr)Ri, and reintroducing dimensionless quantities, we find that the evolution of R obeys dR

nCo3S4

Here, we see explicitly how the system's dynamic depends on the relevant reactive and diffusive time scales, trxn and tdiff, respectively. Scaling time by the reaction time scale (of course, we could just as easily scale by the diffusive time scale - i.e., both are acceptable!), we find dR nCo3S4

This equation may be readily solved analytically to give the outer radius of the shell R(t) as a function of time; in Figure 10.14, this solution is plotted for several representative values of Da.

Having taken the time to derive this equation in a most general fashion (i.e., for any Damkohler number, Da), let us briefly examine the limiting cases of reaction control and diffusion control. For example, when the characteristic reaction time scale is much larger than that of diffusion (i.e., trxn ^ tdiff, corresponding to reaction-limited process), Equation (10.5) simplifies to give a constant rate of growth for the Co3S4 shell (cf. Figure 10.14; Da = 0):

dt 3nCotrxn 3nCoR1

Conversely, for diffusion control (i.e., tdiff ^ Trxn), the growth of the shell becomes independent of the reaction rate and slows in time as the shell becomes thicker (due to increasingly shallow concentration gradients):

dR nco3s4 _ Dnco3s4

These limiting cases are of practical importance when one would like to control the thickness of the alloyed shell by tuning the experimental soaking times. In particular, the reaction-limited behavior is ideal for such control since the shell thickness increases as a simple linear function of time.

Figure 10.14 Growth of Co3S4 product layer versus time. The four lines correspond to different values of Da
Table 10.1 Examples of nanostructures synthesized with the help of the Kirkendall effect. From Fan et al. Formation of Nanotubes and Hollow Nanoparticles Based on Kirkendall and Diffusion Processes: A Review, Small, 3,10, 1660-1671 copyright (2007) Wiley-VCH

Material

Morphology"

Growth process^

Ref.

Year of publication

Co3S4, CoO,

Hollow nanocrystals

Wet sulfidation or oxidation

17

2004

CoSe

of Co nanocrystals

Co3S4, CoSe2,

Hollow nanochains

Solution reaction of

26

2006

CoTe

Co nanonecklace

27b

ZnO

Microcages

Dry oxidation of Zn polyhedra

2004

ZnO

Dandelion

Hydrothermal reaction

28

2004

CU2O

Hollow NPs

Low-temperature dry oxidation

29

2007

ZnS

Hollow nanospheres

Wet sulfidation of ZnO

30

2005

nanospheres

PbS

Hollow nanocrystals

Reaction of Pb NPs with vapor S

31

2005

CuS

Octahedral cages

Sulfidation of Cu2O octahedra

32

2006

Cu7S4

Polyhedron nanocages

Sulfidation of Cu2O nanocube

33

2005

FexOy

Porous thin film

Hydrothermal reaction

34

2005

FexOy

Hollow NPs

Room-temperature oxidation

35

2005

of <8 nm particles

ZnO

Hollow NPs

Low-temperature oxidation

36

2007

of <20 nm particles

AlxOy

Amorphous hollow

Low-temperature oxidation

29

2007

NPs

of <8 nm Al particles

AuPt

Hollow NPs

Solution reaction

37

2004

MoS2

Cubic microcages

Solution reaction

38

2006

MoO2

Hollow microspheres

Hydrothermal reaction

39

2006

Ni2P, Co2P

Hollow NPs

Wet phosphidation of Ni NPs

40

2007

[email protected]

Yolk-shell NPs

Wet sulfidation of [email protected] NPs

41

2007

Pt-Cu

Core-shell NPs

Solution reaction

42

2005

AlN

Hollow nanospheres

Reaction of Al NPs with

43

2006

NH3CH4 gas

AlN

Hollow nanospheres

Annealing of Al NPs in ammonia

44

2007

SiO2

Hollow nanospheres

Water oxidation of Si NPs

45

2004

Co3O4

Porous nanowires

Oxidation of Co(OH)2 nanowires

46

2006

SrTiO3,

Porous spheres

Hydrothermal reaction of TiO2

47

2006

BaTiO3

spheres

CdS

Polycrystal nanoshell

Reaction of Cd nanowire with r r c

48

2005

ZnAl2O4

Crystalline nanotubes

Solid-state reaction of core-shell

49

2006

nanowires

Ag2Se

Nanotubes

Photodissociation of adsorbed

50,51 2006

CSe2 on Ag nanowires

20b

Zn2SiO4

Monocrystal

Solid-state reaction of core-shell

2007

nanotubes

nanowires

Co3S4

Quasi-monocrystal

Reaction of

52

2007

nanotubes

Co(CO3)0.35Cl0.20(OH)1.10

nanowires in solution with H2S

CuO

Polycrystal nanotubes

Dry oxidation of Cu nanowires

53

2005

CuS

Monocrystal

Reaction of CuCl nanorod

54

2007

nanotubes

with H2S

"NPs = nanoparticles.

bThe Kirkendall effect was not pointed out by the authors but was most likely one of the growth mechanisms.

"NPs = nanoparticles.

bThe Kirkendall effect was not pointed out by the authors but was most likely one of the growth mechanisms.

While the model is general in nature and, in principle, applicable to different materials, it can only be treated as a first-order approximation.20 For one, we do not know the exact diffusion coefficient and reaction rates on/in the nanoscopic crystals, and we cannot be even sure that the diffusion is Fickian at this scale. This leaves plenty of room for future theoretical work including discrete atomic simulations. Even without such quantitative models in place, however, the Kirkendall effect has already proven to be a very useful strategy for nanofabrica-tion. Table 10.1, adapted from a review in the journal Small, lists the types of particles that have been synthesized using the Kirkendall effect: spherical shells (Figure 10.15(a,b)), hollow polyhedra, yolk-shell particles (Figure 10.15(c)),

Kirkendahl Voiding

Figure 10.15 Examples of nanostructures synthesized via the Kirkendall effect. (a) Cobalt nanocrystals used as precursors for (b) cobalt sulfide, Co3S4, nanoshells. (c) Yolk-shell 'nanoreactors' comprising a platinum core enclosed in a spherical shell of cobalt oxide, CoO. The particles were synthesized by first depositing cobalt onto platinum nanocrystal 'seeds' and then voiding the shell by the Kirkendall effect. (d) Chain of cobalt selenide, CoSe2, nanocrystals made by voiding a chain of 'wired' cobalt nanocrystals using selenium. The cobalt particles were initially assembled into a necklace structure by the interactions between their magnetic dipoles. (e, f) Images of ZnAl2O4 nanotubes prepared from ZnO-Al2O3 core-shell nanowires. (a-c) reproduced from reference 17 with permission from AAAS, (d) reproduced from reference 26 with permission, copyright (2006) Wiley-VCH, (f) reproduced, with permission, from reference 49, copyright (2006), Nature Publishing Group

Figure 10.15 Examples of nanostructures synthesized via the Kirkendall effect. (a) Cobalt nanocrystals used as precursors for (b) cobalt sulfide, Co3S4, nanoshells. (c) Yolk-shell 'nanoreactors' comprising a platinum core enclosed in a spherical shell of cobalt oxide, CoO. The particles were synthesized by first depositing cobalt onto platinum nanocrystal 'seeds' and then voiding the shell by the Kirkendall effect. (d) Chain of cobalt selenide, CoSe2, nanocrystals made by voiding a chain of 'wired' cobalt nanocrystals using selenium. The cobalt particles were initially assembled into a necklace structure by the interactions between their magnetic dipoles. (e, f) Images of ZnAl2O4 nanotubes prepared from ZnO-Al2O3 core-shell nanowires. (a-c) reproduced from reference 17 with permission from AAAS, (d) reproduced from reference 26 with permission, copyright (2006) Wiley-VCH, (f) reproduced, with permission, from reference 49, copyright (2006), Nature Publishing Group particle chains (Figure 10.15(d)), dandelions and nanotubes (Figure 10.15(e,f)). The interest in these nanostructures is motivated by their potential uses in biological sensing (e.g., based on optical detection using metallic shells and boxes), in lightweight materials, in drug delivery ('nanocapsules'), and in catalysis as nanoreactors, in which a catalytic core is positioned inside of a noncatalytic shell (Figure 10.15(c)).

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