Kinetics of growth of nanosized gold clusters in glass

J O U R N A L OF

ELSEVIER

Journal of Non-CrystallineSolids 203 (1996) 202-205

Kinetics of growth of nanosized gold clusters in glass Pratima Rao, Robert Doremus * Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA

Abstract

Gold particles were grown in a potassium borosilicate glass. An analysis of the growth kinetics suggests that the particles were of uniform size, spherical, and grew by diffusion of gold atoms from the glass matrix to the particles. Knowledge of the growth kinetics allows one to calculate the size of particles when they are so small that they can be observed only with difficulty in the electron microscope.

1. Introduction

Glass is an excellent medium for growing small metallic particles and studying their optical properties. The particles can be nucleated thermally or by radiation, and then grown at a higher temperature, so that the number of growing particles remains constant. Thus the distribution of particle sizes remains small. This small distribution is of value in studying optical properties of the particles, because these properties depend on particle size. The mechanism of growth of the particles can often be deduced from the kinetics of growth. The kinetics allow one to calculate the particle sizes even for clusters so small that they are difficult to observe in the electron microscope. In this work we grew small gold particles in a potassium borosilicate glass and measured their optical absorption as a function of time at wavelengths less than 0.45 ~ m where the optical absorption is directly proportional to the volume of the particles

* Corresponding author. Tel.: + 1-518 276 6372; fax: + 1-518 276 8554; e-mail: [email protected].

[1-3], so the rate of particle growth can be calculated from measurements of the optical absorption at these wavelengths at different times at constant temperature. We found that the kinetics of growth were consistent with a constant number of particles, and that the rate of growth was determined by the diffusion of gold in the glass to the growing particles.

2. G r o w t h equations In this section, equations for the diffusion-controlled growth of a fixed number of spherical particles, all of the same size, will be given. The flux J of material diffusing to a spherical particle of radius R is [4,5]

j

(C i - Ce)D R

(l)

in which D is the diffusion coefficient, C i is the uniform initial concentration of diffusing substance, and C e is the equilibrium concentration of diffusing material (gold) in the glass. We assume that C e is negligibly small compared to C i. In Eq. (1) it is assumed that the particle size at time zero is zero.

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P. Rao, R. Doremus/ Journal of Non-Crystalline Solids 203 (1996) 202-205

The flux to the particle is related to the rate of growth of the particle by the equation dR J =/9-dt

(2)

in which p is the density of gold in concentration units. If Eqs. (1) and (2) are equated and the result integrated from R = 0, t = 0 to R, t: R 2 = 2C i D t / p .

(3)

This equation shows that the radius of the particle grows proportional to t ~/2, so the volume of the particle is proportional to t 3/2. The above equations are valid for an isolated particle growing in an infinite matrix. In a glass containing many particles the concentration of gold between the particles decreases as the particles grow. To account for this competition for dissolved gold, Eq. (1) must be modified to [6,7] J = CiD(1 - W ) / R

(4)

in which W is the fraction of total precipitation:

(5)

w = NVp/<

in which N is the number of growing particles per unit volume and V = 4/3"rrR 3 is the volume of an individual particle. Eq. (4) is valid only if the particles are much farther apart than their size, so that the concentration of gold far from the particles is uniform, although decreasing with time. There is a rigorous and complicated solution to Eqs. (2) and (4); however, up to about W = 0.8, the following equation is a satisfactory approximation (Ref. [8], p. 23): W= 1 - exp(-kt3/2).

(6)

At small times this equation becomes W = kt 3/2 ;

(7)

so, from Eqs. (3) and (5), k

3

- -

"

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different form. However, an electromicrographic study of gold and silver particles grown in glass in a similar method to the present study showed them to be spheres [10]. With the slow growth at high temperatures in this work, one would expect the particles to grow with their lowest energy shape, which is spherically symmetrical (Ref. [11]; Ref. [8], p. 39). Furthermore, the position of the optical absorption maximum for the largest particles in this study was exactly at the wavelength (0.525 Ixm) expected for spherical particles [2]. If the particles are not spherical, the wavelength of maximum absorption is shifted to longer wavelengths [12,13]. For all these reasons, we feel justified in treating the particles as spherically symmetrical.

3. Experimental methods The base glass composition contained 63.9 wt% SiO 2, 9.6 wt% B20 3, and 26.5 wt% K20. This composition was chosen because it dissolved a larger amount of gold than many soda-lime and aluminosilicate compositions. For the study of kinetics, 0.2 wt% of gold as HAuCI 4 • 3HeO was added to the base composition. Twenty gram batches of glass were melted in alumina crucibles at 1400°C to 1500°C, and then quenched rapidly by pouring onto a metal plate. Strains were annealed by holding the glass for few minutes at 400°C and cooling slowly. Samples were then cut, and polished to give a smooth surface. The glass was irradiated with a high intensity mercury UV lamp for 15 to 20 min to nucleate particles. Then samples were heated in hydrogen at 550°C or 600°C to grow the particles. The hydrogen served to reduce the gold ions to atoms in the glass. Optical absorption spectra were measured with a spectrophotometer (Perkin-Elmer 330). The samples were about 2 mm thick.

(8)

4. Experimental results If the particles are not spherical, but can be described as rotational ellipsoids, Ham [9] has shown that the exponent in Eqs. (6) and (7) is still 3 / 2 if the particles do not change their shape as they grow. For ellipsoids, Eq. (8) has a similar but slightly

A sample of glass was heated four hours at 550°C to remove radiation damage and then at 600°C for 4, 8, 12, 16, 20, and 24 h. It is estimated that the growth is twice as fast at 600°C as at 550°C, based

P. Rao, R. Doremus / Journal of Non-Crystalline Solids 203 (1996)202-205

204

Table 1 Optical absorption of gold in glass and values at different times

Time (h)

Optical absorption at 0.44 p.m

W

6 10 14 18 22 26

0.281 0.374 0.606 0.731 1.07 1.17

0.181 0.241 0.391 0.472 0.690 0.755

upon an activation energy of about 84 k J / m o l [1]. Thus two hours was added to the times at 600°C to account for the heating at 550°C. Optical absorption coefficients at 0.44 p~m at each heating time are given in Table 1. A value of 0.25 was subtracted from the measured absorption values. This absorption was measured at longer wavelengths (0.7 and 0.8 p.m) after the radiation damage was annealed out. It results from reflection losses from the two surfaces (0.08) and scattering from polishing scratches and background instrumental noise (0.17). From the data, a value of 1.55 for the corrected absorption after complete growth of the particles was estimated. Then measured values of W, the fraction of total growth, are the absorption divided by 1.55, and are given in Table 1.

5. D i s c u s s i o n

The experimental data are compared with Eq. (6) in Fig. 1, with a value of 0.010 h - 3 / 2 for k. The comparison is within the experimental scatter of the data. We conclude that the measured absorption values are proportional to the total volume of the gold particles and that Eq. (6) is a reasonable fit to the experimental results. Thus we also conclude that the particles were nucleated together, so that the total number of particles was constant throughout growth, and the particles are close to the same size, giving a small size distribution. Furthermore, the exponent of 3 / 2 on time suggests the diffusion-controlled growth of a constant number of spheres. It is possible to estimate the number density of particles N and the diffusion coefficient D from the particle size at a particular time or N. From the optical absorption spectra one can roughly estimate

the particle diameter at W = 0.5 as 2 nm. This estimate comes from a comparison of optical absorption spectra measured in this study [3] with those in an earlier study [2]; the particle sizes for curves in Ref. [2] were corrected as a result of the electron microscopic study in Ref. [10]. The ratio of the peak height (absorption) at 0.525 Ixm to the absorption at 0.44 ~ m changes linearly with particle size up to a particle diameter of about 10 nm. (Ref. [2]; Ref. [14], Fig. 3). Therefore, the size can be estimated from this ratio. With a particle size of 2 nm at W = 0.5, the volume per particle is about 4 × 10 -21 cm 3. With W = 0.5, p = 19.3 g / c m 3, and C i =0.002 × 2.5 =0.005 g / c m 3, N is 3 × 1016 particles/cm 3. The measured value of k is about 4.53 × 10 -8 s- 15. The equation for the diffusion coefficient D is

3k 12/3( p )1/3 D = ( -ff-~ ] --~, .

(9)

From the known values of k, N, p, and C i one can calculate a D of about 4 × 10 -16 c m 2 / s . This value is a rough estimate because of the uncertainty in the exact particle size. This value is much smaller than the diffusion coefficient of gold of about 3 × 10-~z cmZ/s at 600°C in a glass of composition (wt%) 71.5%SIO z, 23%NazO, 4%AIzO 3, and l%ZnO, plus minor amounts of other oxides. Perhaps the difference in compositions is the reason for the different D values.

0.800 0.700 0.600

~: 0.5oo 0.400

=

0.300 0.200

O.lO0 0.~0 0

5

10

15 Time

20

25

30

(hrs.)

Fig. 1. Volume fraction of gold in particles as a function of time. Points are from optical absorption at 0.44 p,m. Line is from Eq. (6) with n = 3 / 2 .

P. Rao, R. Doremus / Journal of Non-Crystalline Solids 203 (1996) 202-205

These kinetic parameters provide for an estimate of particle size at early times. For example, if the particle diameter is 2 nm when W = 0.5, as assumed above, then at W = 0.18, it is 1.4 nm.

6. Conclusions Diffusion-controlled growth of a constant number of spheres was determined as a result of reasonable agreement between the measured absorption values and Eq. (6). A constant number of particles which had nucleated together resulted in a small size distribution. The conclusion that optical absorption at 0.44 Ixm is proportional to the total particle volume agrees with results of earlier studies of the kinetics [1,2] and optical absorption [2] of small growing gold particles. The kinetic results fit Eq. (6) down to a particle diameter of 1 nm at least [10], using the optical absorption at 0.436 Ixm as a measure of particle volume. This size is deduced from electron micrographs of the gold particles [10]. The shape of the optical absorption curve broadens at wavelengths greater than about 0.45 I~m as the particle becomes

205

smaller. However, below 0.45 Ixm, the absorption coefficient remains unchanged as the particles become smaller, until they become extremely small.

References [1] R.H. Doremus, in: Symposium on Nucleation and Crystallization in Glasses and Melts (American Ceramic Society, Columbus, OH, 1962) p. 119. [2] R.H. Doremus, J. Chem. Phys. 40 (1964) 2389. [3] R.H. Doremus and P. Rao, to be published. [4] C. Zener, J. Appl. Phys. 20 (1949) 950. [5] F.C. Frank, Proc. R. Soc. 201A (1950) 586. [6] C.A. Wert and C. Zener, J. Appl. Phys. 21 (1950) 5. [7] V.K. La Mer and R.H. Dinegar, J. Am. Chem. Soc. 72 (1950) 4847. [8] R.H. Doremus, Rates of Phase Transformations (Academic Press, San Diego, CA, 1985). [9] F.S. Ham, J. Phys. Chem. Solids 6 (1958) 335. [10] R.H. Doremus and A.M. Turkalo, J. Mater. Sci. 11 (1976) 903. [11] B.E. Sundquist, Acta Metall. 12 (1964) 588. [12] E. David, Z. Phys. 114 (1939) 389. [13] G.F. Boren and D.R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983) ch. 5, p. 146. [14] U. Kreibig and L. Genzel, Surf. Sci. 156 (1985) 678.