Size-dependent diffusion of supported metal nanoclusters: mean-field-type treatments and beyond for faceted clusters
Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The...
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| Published in | Nanoscale horizons Vol. 8; no. 11; pp. 1556 - 1567 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
England
Royal Society of Chemistry
23.10.2023
Royal Society of Chemistry (RSC) |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2055-6756 2055-6764 2055-6764 |
| DOI | 10.1039/d3nh00140g |
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| Abstract | Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity,
D
N
, on NC size,
N
(in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for
D
N
versus N
assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that
D
N
∼
hN
−4/3
n
eq
. Here,
h
=
ν
exp[−
E
d
/(
k
B
T
)] denotes the hop rate, and
n
eq
= exp[−
E
form
/(
k
B
T
)] the density of those adatoms. The adatom formation energy,
E
form
, approaches a finite large-
N
limit, as does the effective barrier,
E
eff
=
E
d
+
E
form
, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of
E
form
and
n
eq
versus N
for our model. MF theory then captures the occurrence of local minima in
D
N
versus N
at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an
E
eff
which is well above MF values, and which increases with
N
, features captured by a beyond-MF treatment.
Size-dependent diffusion of supported faceted nanoclusters is mediated by disassembly & reassembly of outer layers of facets. A mean-field picture (random independent motion of surface atoms) fails to capture behavior. |
|---|---|
| AbstractList | Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity,
D
N
, on NC size,
N
(in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for
D
N
versus N
assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that
D
N
∼
hN
−4/3
n
eq
. Here,
h
=
ν
exp[−
E
d
/(
k
B
T
)] denotes the hop rate, and
n
eq
= exp[−
E
form
/(
k
B
T
)] the density of those adatoms. The adatom formation energy,
E
form
, approaches a finite large-
N
limit, as does the effective barrier,
E
eff
=
E
d
+
E
form
, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of
E
form
and
n
eq
versus N
for our model. MF theory then captures the occurrence of local minima in
D
N
versus N
at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an
E
eff
which is well above MF values, and which increases with
N
, features captured by a beyond-MF treatment. Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, D N , on NC size, N (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for D N versus N assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that D N ∼ hN −4/3 n eq . Here, h = ν exp[− E d /( k B T )] denotes the hop rate, and n eq = exp[− E form /( k B T )] the density of those adatoms. The adatom formation energy, E form , approaches a finite large- N limit, as does the effective barrier, E eff = E d + E form , for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of E form and n eq versus N for our model. MF theory then captures the occurrence of local minima in D N versus N at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an E eff which is well above MF values, and which increases with N , features captured by a beyond-MF treatment. Size-dependent diffusion of supported faceted nanoclusters is mediated by disassembly & reassembly of outer layers of facets. A mean-field picture (random independent motion of surface atoms) fails to capture behavior. Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, DN, on NC size, N (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for DN versus N assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that DN ~ hN–4/3neq. Here, h = νexp[–Ed/(kBT)] denotes the hop rate, and neq = exp[–Eform/(kBT)] the density of those adatoms. The adatom formation energy, Eform, approaches a finite large-N limit, as does the effective barrier, Eeff = Ed + Eform, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of Eform and neq versus N for our model. MF theory then captures the occurrence of local minima in DN versus N at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an Eeff which is well above MF values, and which increases with N, features captured by a beyond-MF treatment. Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, DN, on NC size, N (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for DNversus N assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that DN ∼ hN−4/3neq. Here, h = ν exp[−Ed/(kBT)] denotes the hop rate, and neq = exp[−Eform/(kBT)] the density of those adatoms. The adatom formation energy, Eform, approaches a finite large-N limit, as does the effective barrier, Eeff = Ed + Eform, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of Eform and neqversus N for our model. MF theory then captures the occurrence of local minima in DNversus N at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an Eeff which is well above MF values, and which increases with N, features captured by a beyond-MF treatment. Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, , on NC size, (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that ∼ . Here, = exp[- /( )] denotes the hop rate, and = exp[- /( )] the density of those adatoms. The adatom formation energy, , approaches a finite large- limit, as does the effective barrier, = + , for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of and for our model. MF theory then captures the occurrence of local minima in at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an which is well above MF values, and which increases with , features captured by a beyond-MF treatment. Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, DN, on NC size, N (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for DNversus N assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that DN ∼ hN-4/3neq. Here, h = ν exp[-Ed/(kBT)] denotes the hop rate, and neq = exp[-Eform/(kBT)] the density of those adatoms. The adatom formation energy, Eform, approaches a finite large-N limit, as does the effective barrier, Eeff = Ed + Eform, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of Eform and neqversus N for our model. MF theory then captures the occurrence of local minima in DNversus N at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an Eeff which is well above MF values, and which increases with N, features captured by a beyond-MF treatment.Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, DN, on NC size, N (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for DNversus N assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that DN ∼ hN-4/3neq. Here, h = ν exp[-Ed/(kBT)] denotes the hop rate, and neq = exp[-Eform/(kBT)] the density of those adatoms. The adatom formation energy, Eform, approaches a finite large-N limit, as does the effective barrier, Eeff = Ed + Eform, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of Eform and neqversus N for our model. MF theory then captures the occurrence of local minima in DNversus N at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an Eeff which is well above MF values, and which increases with N, features captured by a beyond-MF treatment. |
| Author | Evans, James W Campbell, Charles T Lai, King C |
| AuthorAffiliation | Ames National Laboratory - USDOE Chemistry Department Division of Chemical & Biological Sciences Department of Physics & Astronomy Iowa State University University of Washington |
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| Author_xml | – sequence: 1 givenname: King C surname: Lai fullname: Lai, King C – sequence: 2 givenname: Charles T surname: Campbell fullname: Campbell, Charles T – sequence: 3 givenname: James W surname: Evans fullname: Evans, James W |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/37574918$$D View this record in MEDLINE/PubMed https://www.osti.gov/biblio/1995032$$D View this record in Osti.gov |
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| Snippet | Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion... |
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| SubjectTerms | Adatoms Diffusion barriers Free energy Heat of formation Nanoclusters Ostwald ripening Reforming |
| Title | Size-dependent diffusion of supported metal nanoclusters: mean-field-type treatments and beyond for faceted clusters |
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