**7. Self-diffusion of atoms over the surfaces of metal nanoparticles**

The diffusion of atoms over the surfaces of small metal particles is of great scientific and practical interest. This interest is related to searching for ways to create nanostructures with the necessary properties controlled by the dimensions and shape of the structure's elements. Studies of self-diffusion of atoms over the surfaces of metal nanoparticles, which form island films, have already been started. These studies have shown, however, that the long-standing previous investigations of atomic diffusion over the surfaces of macroscopic bodies have proven to be inappropriate for the explanation of the mass transfer over the surfaces of nanoparticles. The same proved to be true with regard to standard methods of studying the surface diffusion based on electron and autoionization microscopy.

In recent years, optical methods for studying the reshaping of metal nanoparticles have entered into practice. In these methods, the optical characteristics of islands of films and, in particular, the extinction spectra, were connected with the morphology of film-forming particles. In (Warmack & Humphrey, 1986) it was found that changes in the extinction spectrum of a heated thin golden film whose nanoparticles were 5–100 nm in size are directly related to changes in the nanoparticle shape. Similar studies were performed with a silver film (Wenzel et al., 1999). In what follows the results obtained by the application of the optical technique for monitoring the processes that redistribute positions of atoms in solid silver particles ~10 nm in size are presented. The idea of the method is based on the relation between positions of plasma resonances of islands and the island shapes. The changes in the shapes of islands were induced by heating of a film and were revealed as a result of the modification of the extinction spectrum of the film. These changes were attributed to self-diffusion of atoms over the surfaces of nanoparticles. Spontaneous reshaping of isolated islands is related to the method of their creation in a non-equilibrium state when the metal is deposited onto the substrate. Transition from non-equilibrium shapes to equilibrium ones, referred to as the shape relaxation, is of thermal nature. It was found that the duration of the observed shape relaxation was determined not by the rate of the surface self-diffusion, but rather by the time of restructuring of the nanoparticles' facets.

#### **7.1 Optical monitoring of silver nanoparticle annealing and aging**

Granular silver films were obtained in a deep vacuum by deposition of silver atoms onto a quartz or sapphire substrate. Immediately after the deposition the extinction spectrum of the as-grown film was recorded in air. Then, the film was placed into the vacuum chamber and heated for a certain time. After that the spectrum was recorded again. This cycle of actions was repeated several times.

Fig. 4 shows extinction spectra of silver films after their heating at two fixed temperatures T= 473 and 553 K during different time intervals. The spectra are seen, first, to exhibit substantial changes for a short time; then, the rate of changes decreased. It should be emphasized that the detected changes in the extinction spectrum are not connected with the evaporation of the silver heated in a vacuum. The estimates have shown that no more than 10–3 of the total amount of the substance of the island film can be evaporated during this

density of adsorbed particles on a surface (i.e., both to increase and to decrease it). Since the processes of origination of a new phase on a surface essentially depend on diffusion processes the use of photodiffusion is promising for controlling processes of nucleation and

The diffusion of atoms over the surfaces of small metal particles is of great scientific and practical interest. This interest is related to searching for ways to create nanostructures with the necessary properties controlled by the dimensions and shape of the structure's elements. Studies of self-diffusion of atoms over the surfaces of metal nanoparticles, which form island films, have already been started. These studies have shown, however, that the long-standing previous investigations of atomic diffusion over the surfaces of macroscopic bodies have proven to be inappropriate for the explanation of the mass transfer over the surfaces of nanoparticles. The same proved to be true with regard to standard methods of studying the

In recent years, optical methods for studying the reshaping of metal nanoparticles have entered into practice. In these methods, the optical characteristics of islands of films and, in particular, the extinction spectra, were connected with the morphology of film-forming particles. In (Warmack & Humphrey, 1986) it was found that changes in the extinction spectrum of a heated thin golden film whose nanoparticles were 5–100 nm in size are directly related to changes in the nanoparticle shape. Similar studies were performed with a silver film (Wenzel et al., 1999). In what follows the results obtained by the application of the optical technique for monitoring the processes that redistribute positions of atoms in solid silver particles ~10 nm in size are presented. The idea of the method is based on the relation between positions of plasma resonances of islands and the island shapes. The changes in the shapes of islands were induced by heating of a film and were revealed as a result of the modification of the extinction spectrum of the film. These changes were attributed to self-diffusion of atoms over the surfaces of nanoparticles. Spontaneous reshaping of isolated islands is related to the method of their creation in a non-equilibrium state when the metal is deposited onto the substrate. Transition from non-equilibrium shapes to equilibrium ones, referred to as the shape relaxation, is of thermal nature. It was found that the duration of the observed shape relaxation was determined not by the rate of the surface self-diffusion, but rather by the time of restructuring of the nanoparticles' facets.

Granular silver films were obtained in a deep vacuum by deposition of silver atoms onto a quartz or sapphire substrate. Immediately after the deposition the extinction spectrum of the as-grown film was recorded in air. Then, the film was placed into the vacuum chamber and heated for a certain time. After that the spectrum was recorded again. This cycle of actions

Fig. 4 shows extinction spectra of silver films after their heating at two fixed temperatures T= 473 and 553 K during different time intervals. The spectra are seen, first, to exhibit substantial changes for a short time; then, the rate of changes decreased. It should be emphasized that the detected changes in the extinction spectrum are not connected with the evaporation of the silver heated in a vacuum. The estimates have shown that no more than 10–3 of the total amount of the substance of the island film can be evaporated during this

**7. Self-diffusion of atoms over the surfaces of metal nanoparticles** 

surface diffusion based on electron and autoionization microscopy.

**7.1 Optical monitoring of silver nanoparticle annealing and aging** 

was repeated several times.

growth of surface nanostructures.

time at T= 553 K. A certain decrease in the area under the curves of the extinction spectra (this quantity characterizes the total amount of substance in the film) is connected, as shown by numerical estimates, with the type of the frequency dispersion (differing from that of Drude–Lorenz) of the silver complex permittivity in the region of 2–4 eV.

Fig. 4. Annealing kinetics of the granular silver films (a) 1 – as prepared, 2 to 4 - after annealing at 473 K for 8, 16 and 56 minutes. (b) 1 – as prepared, 2 to 5 - after annealing at 553 K for 2, 4, 8 and 20 minutes

Fig. 5 shows the transmission electron microscopy images of (a) an as-grown film and (b) of the film annealed during 30 minutes at T= 553 K. One can see that agglomerates formed on the as-grown film (except for the smallest ones) are diverse in shape. As a result, the

Light-Induced Surface Diffusion 429

 *T T* = *N T*κ γν

Here, N is the number of atoms in a granule and γ is the surface tension coefficient (the energy of a single bond of an atom in the crystal equal, by the order of magnitude, to the fusion heat per atom). The frequency of jumps νT that determines the surface diffusion

where *E* is the activation energy of the surface diffusion, *k* is the Boltzmann constant, and *ν*•

For the size of granules of about 10±2 nm and the cell size of 0.5 nm an estimate of *N*, yields *N*≈104. Assuming that *E*= 0.36 eV, *v*● =1013 с-1, *γ* = 0.1 eV и *kТ* = 0.04 eV we obtain from the above relationship that *τ*≈10–3, which is by six orders of magnitude smaller than the observed quantity. This, a large discrepancy cannot be substantially corrected by either taking into account the dependence of the relaxation time on the initial particle shape or by

In conformity with the commonly accepted ideas (Ehrlich, 1974; Naumovets & Zhang, 2002), atoms are heat transferred over surfaces from places with a higher chemical potential to those with a lower chemical potential. At temperatures much lower than the melting point, most atoms of the surface are located in terraces, i.e., facets with small indices. The atoms transferred over the terraces, torn away from the steps (boundaries of terraces) reach other steps and get stuck there. Thus, the sources of the transferred atoms are considered to be

The shape of microcrystals (granules) also changes upon variation of temperature due to the surface selfdiffusion. In small rounded granules (smaller than 100 nm in size), the facets are rather small (smaller than the surface area of the granule by a factor of 10–20). The facets with the area of ~100 surface cells are separated from the neighboring facets by transition regions, i.e., by rough imperfect facets [8] referred to as atomic rough. The structure of surfaces of rounded crystal nanoparticles is mosaic, i.e., each facet borders several other with different structures. As the temperature changes, the free energies of the facets change differently and, as a result, the shape of the granules noticeably changes with no changes in

With increasing temperature, the granules are getting rounder and their surface area decreases. Indeed, the increasing temperature of the granules leads to their melting, i.e., to total elimination of their flat terraces. For this reason, it is natural to assume that the area of the terraces, in the process of rounding, decreases, and the rough regions become wider. This process is possible when the chemical potential of atoms on the terraces (μf) is greater than the potential (μr) of atoms in the roughed regions. Changes in the shape of the granule

In the continuum model of the crystal shape relaxation (Mullins, 1957; Nichols & Mullins, 1965), the shape of granules is characterized by the curvature of the surface. The curvature in the places where the roughed facets are localized is considered to be higher than in the places with flat facets. This simplification makes it possible to describe the mass transfer as a frictional flow of atoms over the surface induced by the capillary forces of surface tension.

τ

ν ν

varying numerical values of the quantities entering this equation.

steps whose density characterizes facets of bulk crystals.

shape stop when the potentials become equal.

coefficient at the temperature T is given by the relationship

is the Debye frequency.

**7.2 Theoretical estimates** 

their volumes.

4 3 .

(26)

*<sup>T</sup>* = − • exp , ( *E kT*) (27)

extinction spectrum acquires a strong inhomogeneous broadening, which decreases under heating because islands are getting apart and more uniform in shape. It is likely that, at the initial stage of heating, the links between islands in agglomerates are being rapidly destroyed or separate islands are being formed due to merging of the smallest ones. Then, separate islands are being rounded (Fig. 5b). At the final stage, this transformation occurs at T= 553 K only by a factor of 3. 3 faster than at T= 473 K.

If the rounding of the islands at the final stage is related to the surface diffusion, then the above acceleration in the shape relaxation is provided by the activation energy E=0.36 eV. This agrees with the experimental (0.4 and 0.33 eV) and theoretical (from 0.5 to 0.25 eV) data (Brune, 1998).

In this case, according to (Mullins, 1957; Nichols & Mullins, 1965) the shape relaxation time τT can be estimated by the formula

extinction spectrum acquires a strong inhomogeneous broadening, which decreases under heating because islands are getting apart and more uniform in shape. It is likely that, at the initial stage of heating, the links between islands in agglomerates are being rapidly destroyed or separate islands are being formed due to merging of the smallest ones. Then, separate islands are being rounded (Fig. 5b). At the final stage, this transformation occurs at

a)

b)

If the rounding of the islands at the final stage is related to the surface diffusion, then the above acceleration in the shape relaxation is provided by the activation energy E=0.36 eV. This agrees with the experimental (0.4 and 0.33 eV) and theoretical (from 0.5 to 0.25 eV) data

In this case, according to (Mullins, 1957; Nichols & Mullins, 1965) the shape relaxation time

Fig. 5. TEM micrographs of the granular silver films, a – as grown, b – after annealing

(Brune, 1998).

τT can be estimated by the formula

T= 553 K only by a factor of 3. 3 faster than at T= 473 K.

$$
\pi\_T = N^{4\mathcal{V}3} \,\,\kappa T \!\!/ \mathcal{V} \nu\_T \,. \tag{26}
$$

Here, N is the number of atoms in a granule and γ is the surface tension coefficient (the energy of a single bond of an atom in the crystal equal, by the order of magnitude, to the fusion heat per atom). The frequency of jumps νT that determines the surface diffusion coefficient at the temperature T is given by the relationship

$$\nu\_T = \nu\_\bullet \exp\left(-E/kT\right),\tag{27}$$

where *E* is the activation energy of the surface diffusion, *k* is the Boltzmann constant, and *ν*• is the Debye frequency.

For the size of granules of about 10±2 nm and the cell size of 0.5 nm an estimate of *N*, yields *N*≈104. Assuming that *E*= 0.36 eV, *v*● =1013 с-1, *γ* = 0.1 eV и *kТ* = 0.04 eV we obtain from the above relationship that *τ*≈10–3, which is by six orders of magnitude smaller than the observed quantity. This, a large discrepancy cannot be substantially corrected by either taking into account the dependence of the relaxation time on the initial particle shape or by varying numerical values of the quantities entering this equation.

#### **7.2 Theoretical estimates**

In conformity with the commonly accepted ideas (Ehrlich, 1974; Naumovets & Zhang, 2002), atoms are heat transferred over surfaces from places with a higher chemical potential to those with a lower chemical potential. At temperatures much lower than the melting point, most atoms of the surface are located in terraces, i.e., facets with small indices. The atoms transferred over the terraces, torn away from the steps (boundaries of terraces) reach other steps and get stuck there. Thus, the sources of the transferred atoms are considered to be steps whose density characterizes facets of bulk crystals.

The shape of microcrystals (granules) also changes upon variation of temperature due to the surface selfdiffusion. In small rounded granules (smaller than 100 nm in size), the facets are rather small (smaller than the surface area of the granule by a factor of 10–20). The facets with the area of ~100 surface cells are separated from the neighboring facets by transition regions, i.e., by rough imperfect facets [8] referred to as atomic rough. The structure of surfaces of rounded crystal nanoparticles is mosaic, i.e., each facet borders several other with different structures. As the temperature changes, the free energies of the facets change differently and, as a result, the shape of the granules noticeably changes with no changes in their volumes.

With increasing temperature, the granules are getting rounder and their surface area decreases. Indeed, the increasing temperature of the granules leads to their melting, i.e., to total elimination of their flat terraces. For this reason, it is natural to assume that the area of the terraces, in the process of rounding, decreases, and the rough regions become wider. This process is possible when the chemical potential of atoms on the terraces (μf) is greater than the potential (μr) of atoms in the roughed regions. Changes in the shape of the granule shape stop when the potentials become equal.

In the continuum model of the crystal shape relaxation (Mullins, 1957; Nichols & Mullins, 1965), the shape of granules is characterized by the curvature of the surface. The curvature in the places where the roughed facets are localized is considered to be higher than in the places with flat facets. This simplification makes it possible to describe the mass transfer as a frictional flow of atoms over the surface induced by the capillary forces of surface tension.

Light-Induced Surface Diffusion 431

the granule surfaces, the self-diffusion depends not only on their material, but also on their shape. This results, in particular, in the characteristic nonexponential kinetics of the shape relaxation demonstrated by long intervals of absence of any changes (Combe et al,. 2000).

It follows from the proposed mechanism of the shape relaxation of nanometer-sized granules that the time of relaxation to the equilibrium shape is determined not only by the temperature at which the relaxation occurs, but also by the state from which they relax. In particular, a non-equilibrium granule obtained by cooling a hot equilibrium granule to a certain temperature will relax faster than a granule obtained by heating an equilibrium cold granule to the same temperature. The difference between the two relaxation times results from the fact that, in the first case, the initial surface of the granule is disordered, and the facets on the surface will form faster than in the second case when a slow stage of formation of the critical nuclei is needed for roughening of the initial flat facets. The proposed reason for slowing down the shape relaxation of granular particles with flat facets presents new opportunities for controlling their shape by external perturbations of particles at the stage of

At present, metal nanoparticles are used in various fields of science and engineering. Their optical properties associated with collective electronic excitations are of particular interest. In most cases, ensembles of nanoparticles obtained on dielectric surfaces by means of the self-organization of atoms adsorbed from a vapor phase are investigated and used. The shapes of the particles thus obtained are often nonequilibrium and vary with the time. The shapes vary more rapidly when the substrate is heated (Ivlev et al., 1988). These facts are well known and reported in detail in papers devoted to electron microscopy investigations, atomic force microscopy data, and the optical extinction spectra of metal island films

Although it is clear that the equilibrium shape of nanoparticles should depend on the temperature (Combe et al., 2000, as far as we know, this dependence has not yet been studied systematically. A change in the shapes of the particles is usually treated as an irreversible transition to the equilibrium state, and heating accelerates the transition process. We observed reversible changes in the optical extinction spectra of silver and sodium films on dielectric substrates under repeated cyclic variations of their temperatures. Moreover, it was found that the illumination of sodium films noticeably accelerates the transition of their spectra to a stable state corresponding to room temperature with the negligibly small heating of nanoparticles by light. The nonthermal photoevaporation of atoms from nanoparticles, i.e., photoatomic emission (Abramova et al., 1984; Bonch-Bruevich et al., 1998; Hoheisel 1988; Burchianti et al. 2009) was also insignificant owing to the choice of the wavelength of light near the threshold of this relatively low probable process. Light-induced changes in the shapes of metal nanoparticles are actively investigated at present. The most well-known works in this field are separated into two groups. In the first group(Huang et al., 2005; Stietz, 2000; Habenicht et al., 2005), the effect of light is reduced to the thermal effect, owing to which individual nanoparticles are either rounded or displaced on the substrate and coagulate when meeting each other. In the second group (Sun et al., 2003; Jin et al., 2001; Kim et al., 2009), light induces physicochemical processes in colloids of metal nanoparticles, which result first in their transformation usually from spheres to prisms and,

The proposed optical method makes it possible to reveal these features.

**8. Illumination-stimulated reshaping of metal nanoparticles** 

their relaxation.

(Warmack & Humphrey, 1986).

This model provides the relationship (26), which we used to estimate the shape relaxation time.

The continuum model is evidently too rough to describe the phenomenon under study. This is obvious from the fact that the difference between flat facets cannot be characterized by the curvature. Fundamental reason for inapplicability of the continuum model to the description of the shape relaxation of crystalline atomic clusters has been revealed in (Combe et al,. 2000). Using the methods of computer simulation, it was established that, at temperatures far away from the melting point (more precisely, below the temperature of roughening), the shape relaxation rate is limited by the stage of attachment of transferred atoms to flat facets, which, in this process, are being rounded. Since the presence of steps on small flat facets is extremely unlikely, the attachment of transferred atoms to these facets is hampered. The stage of the rounding of facets starts with the appearance of critical nuclei on flat facets. At low temperatures, the appearance of critical nuclei is unlikely and, for this reason, the shape relaxation of facets is retarded.

In the same paper (Combe et al,. 2000), it was established that the law *τT* ~ *N*4/3 of the continuum model is obtained in the model under consideration at elevated temperatures, when granules practically have no flat facets. With decreasing temperature, the relation between *τT* and ~ *N* changes and can be well approximated by the relationship *τ*~*Nm*, where *m*(*T*) monotonically increases from 4/3 to 5 with decreasing *T*.

The parameters of the model considered in (Combe et al,. 2000) regretfully are more appropriate for the description of aluminum, rather than silver, clusters. However, the qualitative results obtained for *N*≤104, in our opinion, allow one to apply them to the system under study.

For the values of *τ* ≈102s and N≈104, close to ours, from the data of (Combe et al,. 2000), we have *m*≈4.2 and, correspondingly, *T*= 450 K. As the temperature decreases by 50 K, the shape relaxation time increases by a factor of three. These data are close to those obtained in our experiments.

The model contains a single energy parameter, 0.4 eV, which, by its meaning, is the activation energy of the surface diffusion over the [100] facet of a cubic body-centered crystal. With allowance made for a slight difference between this energy and the appropriate quantity for silver (0.33–0.34 eV) and the difference between the surfaces of the body-centered lattice of the model and face-centered lattice of silver, the proximity of the model shape relaxation time to the observed value allows us to conclude that our observations can be qualitatively explained by the model of (Combe et al,. 2000). Thus, we believe that, in the above experiments, we observed a delay of the self-diffusion mass transfer over the surfaces of nanoparticles at the stage of roughening of their facets.

#### **7.3 Mechanism of the shape relaxation of metal nanoparticles**

The agreement between our experimental data and the model proposed in (Combe et al,. 2000) is only a qualitative result. For the strong dependence of the shape relaxation time of granules on their sizes, the order-of-magnitude errors are inevitable in experimental determination of the rate of changes of the islands' morphology and other characteristics of islands. At the same time, it is evident that, in spite of the inhomogeneity in the films of nanoparticles, their surfaces, and the parameters of self-diffusion of atoms over them, the temperature dependence of the mass transfer rate is characterized by the activation energies of various processes providing shape relaxation of granules. Due to the mosaic structure of

This model provides the relationship (26), which we used to estimate the shape relaxation

The continuum model is evidently too rough to describe the phenomenon under study. This is obvious from the fact that the difference between flat facets cannot be characterized by the curvature. Fundamental reason for inapplicability of the continuum model to the description of the shape relaxation of crystalline atomic clusters has been revealed in (Combe et al,. 2000). Using the methods of computer simulation, it was established that, at temperatures far away from the melting point (more precisely, below the temperature of roughening), the shape relaxation rate is limited by the stage of attachment of transferred atoms to flat facets, which, in this process, are being rounded. Since the presence of steps on small flat facets is extremely unlikely, the attachment of transferred atoms to these facets is hampered. The stage of the rounding of facets starts with the appearance of critical nuclei on flat facets. At low temperatures, the appearance of critical nuclei is unlikely and, for this reason, the shape

In the same paper (Combe et al,. 2000), it was established that the law *τT* ~ *N*4/3 of the continuum model is obtained in the model under consideration at elevated temperatures, when granules practically have no flat facets. With decreasing temperature, the relation between *τT* and ~ *N* changes and can be well approximated by the relationship *τ*~*Nm*, where

The parameters of the model considered in (Combe et al,. 2000) regretfully are more appropriate for the description of aluminum, rather than silver, clusters. However, the qualitative results obtained for *N*≤104, in our opinion, allow one to apply them to the system

For the values of *τ* ≈102s and N≈104, close to ours, from the data of (Combe et al,. 2000), we have *m*≈4.2 and, correspondingly, *T*= 450 K. As the temperature decreases by 50 K, the shape relaxation time increases by a factor of three. These data are close to those obtained in

The model contains a single energy parameter, 0.4 eV, which, by its meaning, is the activation energy of the surface diffusion over the [100] facet of a cubic body-centered crystal. With allowance made for a slight difference between this energy and the appropriate quantity for silver (0.33–0.34 eV) and the difference between the surfaces of the body-centered lattice of the model and face-centered lattice of silver, the proximity of the model shape relaxation time to the observed value allows us to conclude that our observations can be qualitatively explained by the model of (Combe et al,. 2000). Thus, we believe that, in the above experiments, we observed a delay of the self-diffusion mass

transfer over the surfaces of nanoparticles at the stage of roughening of their facets.

The agreement between our experimental data and the model proposed in (Combe et al,. 2000) is only a qualitative result. For the strong dependence of the shape relaxation time of granules on their sizes, the order-of-magnitude errors are inevitable in experimental determination of the rate of changes of the islands' morphology and other characteristics of islands. At the same time, it is evident that, in spite of the inhomogeneity in the films of nanoparticles, their surfaces, and the parameters of self-diffusion of atoms over them, the temperature dependence of the mass transfer rate is characterized by the activation energies of various processes providing shape relaxation of granules. Due to the mosaic structure of

**7.3 Mechanism of the shape relaxation of metal nanoparticles** 

*m*(*T*) monotonically increases from 4/3 to 5 with decreasing *T*.

time.

relaxation of facets is retarded.

under study.

our experiments.

the granule surfaces, the self-diffusion depends not only on their material, but also on their shape. This results, in particular, in the characteristic nonexponential kinetics of the shape relaxation demonstrated by long intervals of absence of any changes (Combe et al,. 2000). The proposed optical method makes it possible to reveal these features.

It follows from the proposed mechanism of the shape relaxation of nanometer-sized granules that the time of relaxation to the equilibrium shape is determined not only by the temperature at which the relaxation occurs, but also by the state from which they relax. In particular, a non-equilibrium granule obtained by cooling a hot equilibrium granule to a certain temperature will relax faster than a granule obtained by heating an equilibrium cold granule to the same temperature. The difference between the two relaxation times results from the fact that, in the first case, the initial surface of the granule is disordered, and the facets on the surface will form faster than in the second case when a slow stage of formation of the critical nuclei is needed for roughening of the initial flat facets. The proposed reason for slowing down the shape relaxation of granular particles with flat facets presents new opportunities for controlling their shape by external perturbations of particles at the stage of their relaxation.
