Extended Smoluchowski Model for the Formation ... - ACS Publications

Article pubs.acs.org/JPCC

Extended Smoluchowski Model for the Formation of Size-Selected Silver Nanoclusters Generated via Modulated Pulsed Power Magnetron Sputtering Chuhang Zhang,*,†,‡ Hironori Tsunoyama,‡,§ Yuanxin Feng,† and Atsushi Nakajima*,‡,§,∥ †

School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, People’s Republic of China JST, ERATO, Nakajima Designer Nanocluster Assembly Project, 3-2-1 Sakado, Takatsu-ku, Kawasaki 213-0012, Japan § Department of Chemistry, Faculty of Science and Technology, and ∥Keio Institute of Pure and Applied Science (KiPAS), Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan ‡

S Supporting Information *

ABSTRACT: Size-selected silver nanoclusters (NCs) were generated using a modulated pulsed power magnetron sputtering (MPP-MSP) system coupled with a mass spectrometer, and their formation mechanism was investigated by applying an extended Smoluchowski model. Attractive interactions between NC cations and anions were incorporated into the conventional Smoluchowski equations to explain the origin of the size tuning of Ag NCs prepared via MPPMSP. Bunches overlapped as the repetition rate (f) and peak power (Pp) increased, resulting in the initial formation of large ionic nanoclusters, ultimately causing a reduction in the size of the NC ions due to neutralization via NC cation−anion charge recombination and the formation of aggregates.

1. INTRODUCTION The study of nanoclusters (NCs) has been extended to research fields including photocatalysis, biochemistry, and electrocatalysis.1−4 In particular, for photocatalysis, noble metal NCs are considered to be promising cocatalysts for enhancing efficient water splitting and inducing visible-light responses.5−7 For instance, efficient size-dependent oxygen and hydrogen generation via the decoration of Au NCs on a perovskite semiconductor crystal has been reported. This behavior was attributed to the size-specific physical and chemical properties of the NCs.8,9 Therefore, it is of significant importance to synthesize NCs with well-defined sizes, which requires a complete understanding of the their formation mechanisms. NCs with well-defined sizes can be generated using a magnetron sputtering source (MSP), and their size-dependent properties can be investigated using mass spectrometry.10 To couple an MSP with a mass spectrometer, a higher NC ion current is generally preferable, but under such higher density conditions, aggregation of the NCs (neutral, cationic, and anionic) contributes to NC formation in addition to sequential atom aggregation. Previously, we reported an advanced NC preparation method based on a modulated pulsed power magnetron sputtering (MPP-MSP) technique.11,12 Ag NC ions with intensities up to 10 nA were generated, and their size distributions were modulated by changing the repetition rate ( f) and peak power (Pp), as well as the shape of the pulse waveform. In addition, © XXXX American Chemical Society

the mechanism of formation of the size-selected Ag NC ions was qualitatively investigated using a retarding potential method. However, owing to the complexity of NC formation in the gas phase, the detailed mechanism for tuning the size of NC ions remains unknown. The Smoluchowski rate equation has been shown to describe the irreversible condensation of NCs.13,14 Huang and coworkers further considered NC ion-neutral interactions in the vapor phase, and obtained good agreement with the results of a laser vaporization experiment.15 However, sputtered materials in the plasma plume of a pulsed magnetron sputtering system comprise high-density neutral, cationic, and anionic species (up to 1020 m−3)16 owing to their high peak power. While the peak power density on the target of a conventional direct current (DC) MSP is less than 50 W/cm2, that of an MPP-MSP can approach several kW/cm2.17 NC cation−anion interactions should thus be explicitly taken into account to accurately describe the cluster formation mechanism. In the present study, therefore, the Smoluchowski equations were extended by adding a term for cation−anion interactions to investigate the formation mechanism of NCs in the gas phase. The calculated results were compared to data obtained using mass spectrometry for Ag NCs generated via MPP-MSP, and the Received: October 27, 2015 Revised: February 12, 2016

A

DOI: 10.1021/acs.jpcc.5b10531 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

the cations and anions, that is, −e2/4πε0r, which further promotes collisions between cationic and anionic NCs. Therefore, the first term on the right side of eq 1 represents the formation of neutral NCs of size k via the aggregation of neutral NCs with sizes i and k−i. The second term indicates the annihilation of neutral NCs of size k via aggregation with other neutral NCs. The third term represents the annihilation of neutral NCs of size k via aggregation with cationic and anionic NCs. The last term denotes the formation of neutral NCs of size k via aggregation of NC cations and anions. Similarly, the kinetic processes for the formation of NC cations and anions are

size tuning and formation mechanism of the Ag NC ions were discussed.

2. EXPERIMENTAL AND THEORETICAL METHODS Materials and Methods. The experiment setup (MPPMSP and quadrupole mass spectrometer, QMS) and methods are described elsewhere.12 A 99.9% purity Ag disk (diameter = 2 in) purchased from Rare Metallic Co. Ltd. was used as the sputtering target. The typical Ar and He flows were 100 and 400 sccm, respectively, leading to a background pressure of approximately 20 Pa. The length of the condensation cell was 290 mm, and the total aggregation time (drifting time) for the NCs was estimated to be approximately 0.05 s.12 By controlling just two parameters, that is, repetition rate ( f) and peak power (Pp), the size distribution and intensity of the NCs were manipulated. As reported previously, the pulse waveform was characterized by the duration and slope of the macro-pulse comprising the generated micropulses,12 and the duration of the macro-pulse was approximately 1.5 ms in the present study. Typical values for the peak voltage, discharge current, and peak power fell between −270 and −500 V, 0.7 and 4.0 A, and 180 and 2000 W, respectively. The average power was estimated by the product of Pp and the duty cycle, which is proportional to f, for example, 10 Hz represents a 1.5% duty cycle. In the present study, the peak power Pp was used as a parameter to tune sputtering conditions, which were determined from the product of the peak voltage and the peak current. Note also that all of the ion optics were optimized in order to reduce the massdependency of the transmission of the present QMS to the NCs. Theory. For the simulation model, the NC cation−anion interactions were considered, and the formation of neutral NCs was assumed to follow the kinetic processes:

mi+ + nk − i → mk+ , mi− + nk − i → mk−

and the kinetic equations for NC cations and anions are thus expressed as k−1

k−1

(3)

Note that the initial number density of atomic cations = σn0, where n0 is the initial number density of sputtered species (neutral, cationic, and anionic) and σ represents the ionization fraction. Previously we demonstrated that the ratio between the number densities of NC anions and cations for our MPP-MSP is approximately 1.6. Therefore, the initial number density of atomic anions is m−1 = 1.6σn0 and that of Ag atoms is n1 = (1 − 2.6σ)n0. By comparing the simulated results with experimentally obtained mass spectral data, the parameters n0, σ, u, and a were optimized. The details of the parameter optimization are presented in the Supporting Information.

3. RESULTS AND DISCUSSION Figure 1a shows the mass spectrum of Ag NC anions, wherein only small NC anions were observed and the atomic anion was the most populous species. A previous time-resolved analysis revealed that at a low sputtering repetition rate ( f ≤ 10 Hz), the sputtered materials are divided into discrete ion “bunches” rather than continuous plasma. At a low repetition rate and low Pp (≤200 W), on the contrary, the material density in bunches is relatively small.12 In the latter case, NCs are thought to form via the so-called sequential growth, wherein the NCs grow up by capturing single atomic species and NC−NC aggregation can be neglected.11 Therefore, the kinetic equations for NC growth can be simplified as follows:

∞

∞

i

∞

m+1

dnk 1 = K i , k − i ∑ nink − i − Kk , i ∑ nk ni dt 2 i=1 i=1

i=1

∞

dmk− = Kk′− i , i ∑ mk−− ini − Kk′ , i ∑ mk−ni − Kk″. i ∑ mk−mi+ dt i=1 i=1 i=1

The Smoluchowski equation can therefore be extended to

− Kk′ , i ∑ nk (mi+ + mi−) + K i″, k − i ∑ mi+mk−− i

∞

(2)

ni + nk − i → nk , mi+ + mk−− i → nk k−1

∞

dmk+ = Kk′− i , i ∑ mk+− ini − Kk′ , i ∑ mk+ni − Kk″, i ∑ mk+mi− dt i=1 i=1 i=1

(1)

Note that the potential evaporation of “hot atoms” from the NCs was neglected based on the assumption that the NCs are sufficiently cooled by the buffer gas. All electrons are assumed to be attached to the atoms, and only atomic species (atomic anions, cations, and atoms) are assumed to exist when the model starts. Here, nk, m+k , and m−k denote the number densities of the neutral, cationic, and anionic NCs, respectively, each with number of atoms k. The parameter Ki,j represents the reaction rate constant between neutral NCs with sizes i and j, and is given by K i , j = i + j /ij × (i1/3 + j1/3 )2 × K1,1. For Ag atoms, K1,1 is approximately 10−17 m3/s.18 Correspondingly, K′i,j and K″i,j stand for the rate constants between neutral−ionic and cationic−anionic NCs. Here we propose that K′i,j = uKi,j, with the constant u greater than 1 because long-range ionneutral interactions have an extra potential of −αe2/8πε0r4 (a; polarizability), which accelerates the collisions between ionic and neutral NCs. Meanwhile, the rate constant for cationic−anionic NC interactions is expressed as Ki,j″ = aKi,j, where a is greater than u due to the attractive potential between

dnk = Kk − 1,1nk − 1n1 − Kk ,1nk n1 − Kk′ ,1nk (m1+ + m1−) dt + Kk″− 1,1mk+− 1m1−

(4)

dmk+ = Kk′− 1,1mk+− 1n1 − Kk′ ,1mk+n1 − Kk″,1mk+m1− dt

(5)

dmk− dt

= Kk′− 1,1mk−− 1n1 − Kk′ ,1mk−n1 − Kk″,1mk−m1+

(6)

Using the deduction theorem, the number density of NC anions of size k can be determined as a function of t: B


Article


Figure 1. (a) Mass spectrum of Ag NC anions generated via MPPMSP. Repetition rate = 10 Hz, Pp = 180 W. (b) Relationship between the NC number density and size obtained using kinetic eq 7. Parameter set: n0 = 3 × 1017 m−3, σ = 3%, t = 0.05 s, u = 2, and a = 4.

mk−

⎛ c1 ⎞k − 1 − = ⎜ ⎟ m1 (1 − exp(−c 2t ))k − 1 ⎝ c2 ⎠

Figure 2. (a) Mass spectra of Ag NC anions generated with repetition rates of 10 Hz (blue), 70 Hz (green), 100 Hz (red), and 150 Hz (black) at Pp = 200 W. (b) Size distribution of NC anions obtained using the simulation model. Initial sputtered material number density n0 = 4.0 × 1017 m−3 (blue), 1.7 × 1018 m−3 (green), 3.5 × 1018 m−3 (red), and 6.5 × 1018 m−3 (black). Other parameter set: σ = 4%, t = 0.05 s, u = 2, and a = 4.

(7)

Here, c1 = Kk−1,1 ′ n1 and c2 = Kk,1 ′ n1 + Kk,1 ″ m+1 . To simulate NC formation under low f and Pp conditions, the initial sputtered material number density n0 and ionization fraction σ were carefully estimated using values of 3 × 1017 m−3 and 3% for f = 10 Hz and Pp = 180 W, respectively, considering previous experimental findings.12,16,20 Meanwhile, the values for u and a were optimized step-by-step based on comparisons of the calculated results with the mass spectral data (see Supporting Information). The relationship between the number density and size of the NC anions is shown in Figure 1b, and a similar size distribution for the calculated and experiment data was obtained (see Figure 1a). The agreement between the experimental and simulated results suggests that the growth of NCs during sputtering at low f and Pp values mainly occurs via sequential growth. As the repetition rate f increases, both the size and total intensity of the NC anions increases, as shown in Figure 2a. Note that the intensities of the NCs were determined from the corresponding mass spectra using a MATLAB program. It can be clearly seen in the figure that the NC size with the highest intensity shifted from atomic anions to 26-mers as f increased from 10 to 150 Hz. Meanwhile, the size distribution of the NC anions broadened as the value of f increased. This f effect on the size distribution and intensity of NC anions is thought to be due to the overlap of ion “bunches,” which in turn enhances the local density of sputtered species.12 This overlapping process can be simulated by increasing the initial number density of the sputtered materials (n0), while maintaining a constant ionization fraction (σ) independent of f.

Therefore, initial sputtered material number densities of 4.0 × 1017 m−3, 1.7 × 1018 m−3, 3.5 × 1018 m−3, and 6.5 × 1018 m−3, which correspond to repetition rates of 10, 70, 100, and 150 Hz, respectively, were used for the calculations. Note that by increasing n 0 , the NC growth may involve NC−NC aggregation, which is described by eqs 1−3. The simulated results are shown in Figure 2b, where the NC size with the highest number density evolved from atomic anions, to 14mers, 18-mers, and finally 26-mers. This observation is quite consistent with the experimental results shown in Figure 2a. In addition, the size distributions of the NCs obtained using the model were also quite similar to those of the corresponding mass spectra. The consistency between the calculated and the experimental results suggests that the origin of the size tuning of NCs as a function of repetition rate can be ascribed to the increased sputtered material density due to bunch overlapping, rather than the ionization fraction. The total number densities of the NC anions shown in Figure 2b were calculated to be 2.4 × 1016 m−3, 9.0 × 1016 m−3, 2.0 × 1017 m−3, and 2.4 × 1017 m−3 for the blue, green, red, and black curves in Figure 2b, respectively. However, the initial number densities of atomic anions were 2.6 × 1016 m−3, 1.0 × 1017 m−3, 2.2 × 1017 m−3, and 4.2 × 1017 m−3, which equal 1.6n0 × σ. The discrepancy between the total number density before and after aggregation clearly indicates that some NC ions were “lost” after aggregation and suggests that the NC cation−anion recombination occurred as the repetition rate increased. At high f values, the total number density of charged species is C


Article


that as Pp increases, both the quantity and the ionization fraction of the sputtered material in a single bunch are enhanced, leading to the formation of larger NCs. However, when Pp was greater than 500 W, the size of the NC anions decreased, as can be seen in Figure 4a. In addition,

enhanced, which may lead to interactions between cations and anions and thus neutralization via charge recombination, as indicated by the last term in eq 1. It has been demonstrated that both the initial number density and the ionization fraction of sputtered materials can be enhanced by increasing Pp during MPP-MSP,19,20 which is an essential advantage of MPP-MSP over DC-MSP. In this case, the size distribution of the NCs is expected to be manipulated by the magnitude of Pp. Thus, in order to study the pure effect of Pp on the size distribution of NCs, the repetition rate was fixed at 10 Hz to prevent bunch overlapping. As Pp increases, both the number density and ionization fraction of the sputtered materials in individual bunches are expected to increase. Figure 3a presents the mass spectra of Ag NC anions

Figure 4. (a) Relationship between the intensity and size of NC anions generated via MPP-MSP at Pp = 500 (blue), 700 (green), and 1100 W (red) and f = 10 Hz. (b) Size distributions of NC anions calculated using the simulation model with varied initial sputtered material number densities and ionization fractions (n0, σ): (1.0 × 1018 m−3, 11%) (blue), (1.5 × 1018 m−3, 16%) (green), and (2.2 × 1018 m−3, 21%) (red); other parameters: t = 0.05 s, u = 2, and a = 4.

the total intensities of the NC anions were calculated to be approximately 7700, 9900, and 16000 (relative intensity) for Pp = 500, 700, and 1100 W, respectively. The reduced size of the NC ions under these conditions may be due to the increased ionization fraction, because the average number of atoms in each NC ion is reduced. The simulation was then performed, where the initial sputtered number density n0 and ionization fraction σ were set as (n0, σ) = (1.0 × 1018 m−3, 11%), (1.5 × 1018 m−3, 16%), and (2.2 × 1018 m−3, 21%), corresponding to Pp values of 500, 700, and 1100 W, as shown in Figure 4a. The simulated results are presented in Figure 4b. Size shifts similar to those seen in Figure 4a were observed. In addition, the total number densities of the NC anions after aggregation were calculated for (n0, σ) = (1.0 × 1018 m−3, 11%), (1.5 × 1018 m−3, 16%), and (2.2 × 1018 m−3, 21%) to be 1.1 × 1017 m−3, 1.9 × 1017 m−3, and 3.1 × 1017 m−3, respectively, and are in good agreement with the corresponding experimental results (see Figure 4a). The anomalous dependent behavior of the total intensity and size of the NC ions on Pp is thus ascribed to two possibilities: (1) an enhanced cation−anion aggregation rate resulting from the richer NC ion densities at higher Pp and (2) a high plasma temperature at high Pp prohibiting the formation of large NCs.

Figure 3. (a) Relationship between the intensity and size of NC anions generated via MPP-MSP at Pp = 200 (blue), 300 (green), 400 (red), and 500 W (black) and a repetition rate of 10 Hz. (b) Calculated size distributions of NC anions for different initial sputtered material number densities and ionization fractions: n0 = 4 × 1017 m−3, σ = 4% (blue), n0 = 6.0 × 1017 m−3, σ = 7% (green), n0 = 8.0 × 1017 m−3, σ = 9% (red), and n0 = 1018 m−3, σ = 11% (black); other parameters: t = 0.05 s, u = 2, and a = 4.

generated at different values of Pp. It can be seen that the NC size with the greatest number density increased from atomic anions to 24-mers as the value of Pp increased from 200 to 500 W. To simulate the effect of Pp on the NC size distribution, the initial sputtered number density n0 and ionization fraction σ were set at (n0, σ) = (4.0 × 1017 m−3, 4%), (6.0 × 1017 m−3, 7%), (8.0 × 1017 m−3, 9%), and (1018 m−3, 11%), as estimated from refs 16 and 20. Note that the data below an average power of 50 W were fitted with a straight line in order to deduce the ionization fraction.20 The calculated results are shown in Figure 3b. A similar size shift in the NC size distribution was clearly observed: the NC size with the highest number density increased from atomic anions to 26-mers. This result suggests D


Article

The Journal of Physical Chemistry C The two possibilities were examined by measuring the size distributions of NC anions, cations, and neutrals under the same sputtering conditions. To detect the corresponding NC neutrals sufficiently, a higher repetition rate of f = 130 Hz was used because the neutrals were measured with electron impact ionization. As Pp increased from 200 to 800 W, the average size of the NC anions increased from approximately 20-mers to 45mers; simultaneously, larger NC cations (larger than 10-mers) were emphasized (see Figure S3 in Supporting Information). For the corresponding neutrals, NCs with the size of 30-mers to 130-mers appeared above Pp = 400 W and the average size gradually increased as Pp was increased to 800 W (see Figure S4 in Supporting Information). In fact, the extended Smoluchowski model in which all the parameters were adjusted for f = 130 Hz consistently reproduced the experimental size distributions of NC anions and cations; NC cations are usually smaller than NC anions, and the size of NC cations is shifted to larger size with increasing Pp. Furthermore, the Pp dependence of the corresponding NC neutrals could be simulated (see Figures S5 and S6 in Supporting Information). Note that a further increase of Pp from 800 to 1000 W not only resulted in a reduction of NC size and number density for NC anions and cations, but also in a reduction of number density for the NC neutrals. The agreement between experiments and simulations suggests that the size reduction of NC ions is attributable to the accelerated NC anion−cation aggregation in this power range of 500 W < Pp < 800 W for f = 130 Hz. Although the dramatic reduction of intensity above Pp = 800 W could not be explained by the extended Smoluchowski model, the size reduction for NC ions appears to result from NC growth prohibition due to the buffer-gas temperature increase caused by the high plasma temperature. Given the above calculated and experimental results, a growth mechanism for NCs generated via MPP-MSP was proposed, consisting of three regimes. Regime I (growth): At low sputtered power (f ≤ 10 Hz and Pp ≤ 200 W), sequential growth is the dominant process for NC growth because the sputtered material number density and ionization fraction in each ion bunch are low. As f increases, ion bunches generated by MPP-MSP begin to overlap, resulting in the enhancement of the material number density in the overlapping regions. Consequently, the NC size increases with f. Comparatively, as Pp increases, both sputtered material number density and ionization fraction increase, which causes an increase in the NC size and number density. Regime II (anion−cation neutralization dominant): As Pp further increases, the formation of large NCs is suppressed, because the NC cation−anion aggregation rate is accelerated at richer NC ion concentrations. This leads to a reduction in the total intensity of NC anions after neutralization of the NC cations via aggregation. Regime III (temperature effect dominant): As Pp further increases, depending on the repetition rate, the formation of large NCs is prohibited because of nonthermalized sputtered material initiated by a higher Pp, leading to a reduction of the number density of NC ions and neutrals.

influence the initial sputtered material number density and ionization fraction. Under low f (≤10 Hz) and Pp (≤200 W) conditions, sequential growth is the main mechanism for NC growth due to the small material number density and ionization fraction in the ion bunches. At a relatively high Pp (≥500 W) at f = 10 Hz, on the contrary, NC cation−anion aggregation is the dominant process for NC growth, which in turn reduces the intensities of larger NC anions due to their neutralization via charge recombination with NC cations. Depending on the repetition rate, the prohibition of large NCs is enhanced because of the high thermal energy of sputtered materials initiated by high Pp, which results in the intensity reduction of NCs.

■

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b10531. The details of the parameter optimization, including n0, σ, u, and a are presented. Figure S1: Optimization of (n0, u) by comparing the model and experimental results. Figure S2: Step by step optimization of a for silver NC anions generated at Pp = 500 W and f = 10 Hz. Figure S3: Mass spectra of NC ions under varied Pp = 200, 400, 600, 800, and 1000 W at f = 130 Hz. Figure S4: Mass spectra of NC neutrals under Pp = 200, 400, 600, 800, and 1000 W at f = 130 Hz. Figure S5: Calculated size distributions of NC ions under varied (n0, σ). Figure S6: Calculated size distributions of NC neutrals under varied (n0, σ; PDF).

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Fax: +86-571-8507-0707. *E-mail: [email protected]. Fax: +81-45-566-1697. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is partly supported by JSPS KAKENHI of Grant-inAids for Scientific Research (A) Grant Number 15H02002. Financial supports from the National Natural Science Foundation of China (Grant Nos. 11547217 and 11374082). Scientific Research Foundation of Zhejiang University of Science and Technology (Grant No. F702108E04) is also acknowledged.

■

REFERENCES

(1) Burda, C.; Chen, X. B.; Narayanan, R.; El-sayed, M. A. Chemistry and properties of nanocrystals of different shapes. Chem. Rev. 2005, 105, 1025−1102. (2) Kunz, S.; Schweinberger, F. F.; Habibpour, V.; Röttgen, M.; Harding, C.; Arenz, M.; Heiz, U. Temperature dependent CO oxidation mechanisms on size-selected clusters. J. Phys. Chem. C 2010, 114, 1651−1654. (3) Wender, H.; Migovski, P.; Feil, A. F.; Teixeira, S. R.; Dupont, J. Sputtering deposition of nanoparticles onto liquid substrates: Recent advances and future trends. Coord. Chem. Rev. 2013, 257, 2468−2483. (4) Marzun, G.; Nakamura, J.; Zhang, X. R.; Barcikowski, S.; Wagener, P. Size control and supporting of palladium nanoparticles made by laser ablation in saline solution as a facile route to heterogeneous catalysts. Appl. Surf. Sci. 2015, 348, 75−84.

4. CONCLUSIONS Size-selected Ag NCs were generated via MPP-MSP, and their formation mechanism was investigated using an extended Smoluchowski model wherein NC cation−anion aggregation was taken into account. It was found that size tuning of Ag NCs prepared using MPP-MSP occurs because variations in f affect the overlapping of ion bunches, whereas variations in Pp E


Article

The Journal of Physical Chemistry C (5) Yang, F.; Takahashi, Y.; Sakai, N.; Tatsuma, T. Visible light driven photocatalysts with oxidative energy storage abilities. J. Mater. Chem. 2011, 21, 2288−2293. (6) Schweinberger, F. F.; Berr, M. J.; Döblinger, M.; Sanwald, K. E.; Wolff, C.; Crampton, A. S.; Ridge, C. J.; Jäckel, F.; Feldmann, J.; Tschurl, M. Cluster size effects in the photocatalytic hydrogen evolution reaction. J. Am. Chem. Soc. 2013, 135, 13262−13265. (7) Qian, K.; Sweeny, B. C.; Johnston-Peck, A. C.; Niu, W. X.; Graham, J. O.; DuChene, J. S.; Qiu, J. J.; Wang, Y. C.; Engelhard, M. H.; Su, D. Surface plasmon-driven water reduction: Gold nanoparticle size matters. J. Am. Chem. Soc. 2014, 136, 9842−9845. (8) Shi, X.; Ueno, K.; Oshikiri, T.; Misawa, H. Improvement of pasmon-enhanced photocurrent generation by interference of TiO2 thin film. J. Phys. Chem. C 2013, 117, 24733−24739. (9) Shi, X.; Ueno, K.; Oshikiri, T.; Misawa, H. Plasmon-enhanced photocurrent generation and water oxidation with a gold nanoislandloaded Titanium dioxide photoelectrode. J. Phys. Chem. C 2013, 117, 2494−2499. (10) Messerli, S.; Schintke, S.; Morgenstern, K.; Sanchez, A.; Heiz, U.; Schneider, W. D. Imaging size-selected silicon clusters with a lowtemperature scanning tunneling microscope. Surf. Sci. 2000, 465, 331− 338. (11) Tsunoyama, H.; Zhang, C. H.; Akatsuka, H.; Sekiya, H.; Nagase, T.; Nakajima, A. Development of high-flux ion source for size-selected nanocluster ions based on high-power impulse magnetron sputtering. Chem. Lett. 2013, 42, 857−859. (12) Zhang, C. H.; Tsunoyama, H.; Akatsuka, H.; Sekiya, H.; Nagase, T.; Nakajima, A. Advanced nanocluster ion source based on highpower impulse magnetron sputtering and time-resolved measurements of nanocluster formation. J. Phys. Chem. A 2013, 117, 10211−10217. (13) von Smoluchowski, M. Drei Vorträge fiber diffusion, brownsche bewegung und koagulation von kolloidteilche. Z. Phys. 1916, 17, 557− 585. (14) von Smoluchowski, M. Versuch einer mathematischen Theorie der koagulationskinetik kolloider Lösungen. Z. Phys. Chem. 1917, 92, 129−168. (15) Huang, R. B.; Zhang, Q.; Chen, H.; Zheng, L. S. Application of the Smoluchowski equation to the formation kinetics of cluster ions. Eur. Phys. J. D 1999, 9, 253−256. (16) Helmersson, U.; Lattemann, M.; Bohlmark, J.; Ehiasarian, A. P.; Gudmundsson, J. T. Ionized physical vapor deposition (IPVD): A review of technology and applications. Thin Solid Films 2006, 513, 1− 24. (17) Kouznetsov, V.; Macák, K.; Schneider, J. M.; Helmersson, U.; Petrov, I. A novel pulsed magnetron sputter technique utilizing very high target power densities. Surf. Coat. Technol. 1999, 122, 290−293. (18) Smirnov, B. M.; Strizhev, A. J. Kinetics of clustering processes in expanding vapor. Phys. Scr. 1994, 49, 615−618. (19) Poolcharuansin, P.; Bowes, M.; Petty, T. J.; Bradley, J. W. Ionized metal flux fraction measurements in HiPIMS discharges. J. Phys. D: Appl. Phys. 2012, 45, 322001(1−5). (20) Hala, M.; Capek, J.; Zabeida, J.; Klemberg-Sapieha, J. E.; Martinu, L. Pulse management in high power pulsed magnetron sputtering of niobium. Surf. Coat. Technol. 2012, 206, 4186−4193.

F


Extended Smoluchowski Model for the Formation ... - ACS Publications

Recommend Documents