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Concentration and size depth profile of colloidal silver particles in glass surfaces produced by sodium-silver ion-exchange
Journal of Non-Crystalline Solids 151 (1992) 88-94 North-Holland
~
]OURNAL
OF
U
Concentration and size depth profile of colloidal silver particles in glass surfaces produced by sodium-silver ion-exchange A. B e r g e r Center for Research in Electro-Optics and Lasers, University of Central Florida, 12424 Research Parkway, Suite 400, Orlando, FL 32826, USA
Received 12 May 1992 Revised manuscript received 7 July 1992
The sodium-silver ion exchange of a commercial soda-lime silicate glass followed by an annealing process yield small silver particles in a surface layer. Microspectrophotometryspectra and transmission electron microscopyshow that the mean particle size increases with the penetration depth, whereas the volume concentration is constant. A model on the basis of a homogeneous nucleation including ion exchange, diffusion, reduction and particle formation is given. The measured depth dependence is described with the model by a non-linear system of differential equations.
I. Introduction Staining of alkali-silicate glasses by exchanging silver ions from pastes or liquids with alkali ions of the glass yields a yellow coloured surface layer, which is caused by small silver particles [1,6]. Following Weyl [1] it is usually assumed that the formation of these particles is a result of the complex combination of ion exchange, diffusion of silver ions, their reduction to atoms and aggregation to particles. Whereas ion exchange and silver ion diffusion in glasses are well understood (see, for example, refs. [2,3]), there are many unsolved questions concerning reduction and particle formation. Especially the microscopic processes resulting in the reduction of the silver ions to atoms and their aggregation to crystalline particles are unknown. A detailed model describing the formation of the small silver particles has not yet been given.
Correspondence to: Dr A. Berger, Center for Research in Electro-Optics and Lasers, University of Central Florida, 12424 Research Parkway,Suite 400, Orlando, FL 32826, USA. Tel: + 1-407 658 3991. Telefax: + 1-407 658 3955.
Dependent on the glass composition in commercial soda-lime glasses, iron (concentration << 1 wt%) generally can be found in the Fe 2+ as well as in the Fe 3÷ state [4]. Using optical spectroscopy a relation of F e 2 + / F e 3÷= 0.43 was estimated for the investigated glass [5]. This ratio is important for the reduction of silver ions (in a non-reducing atmosphere). The following thermoreducing reaction is to be expected [1]: Fe 2 + ~ Fe3++ e A g + + e---* Ag °. According to this assumption the maximum number of silver atoms has to be equal to or less than the number of Fe 2+. In ref. [6], a method was presented, from which size as well as volume concentration of silver particles can be determined from optical extinction spectra. From a measurement of optical spectra with a microspectrophotometer as a function of penetration depth, an estimate of particle size and concentration as a function of depth from the exchanged surface can be made.
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A. Berger / Colloidal silver particles in glass surfaces
This paper presents experimental results of size and volume concentration of silver particles as a function of penetration depth after a low temperature ion exchange ( T = 400°C) followed by an annealing process near the transition temperature of the glass, Tg. Size and concentration were determined using the 'optical' method of ref. [6] and by transmission electron microscopy. The results are explained by a model on the basis of a homogeneous nucleation. The system of non-linear differential equations explains the formation of silver particles by ion exchange, diffusion, reduction and aggregation. A quantitative description of the complex process is given.
2. Experiments and results For all the investigations, samples of a commercial soda-lime glass (for composition, see table 1) with a thickness of 2.2 mm were used. The samples were subjected to an ion exchange in a salt liquid of 2 wt% AgNO3-98 wt% NaNO 3 at 400°C for 2 h and 2 wt% AgNO3-98 wt% KNO 3 at 400°C for 30 min, respectively. Electron-probe microanalysis detected a silver containing surface layer with a thickness of 55 and 25 txm, respectively, after the ion exchange [7]. In the case of the AgNO3-NaNO 3 salt liquid, 80% of the sodium ions are exchanged by silver ions at the surface. The weight change of the sample resulting from ion exchange gave a total amount of silver of M = 1.85 txm -2, if the silver is assumed to be in an infinitesimal thin surface layer (appropriate to eq. (20)). From optical spectra, I conclude that nearly all the silver is in the ionic state, because there is no absorption caused by silver atoms or particles. Silver particles cannot be found by electron microscopy. The ion exchanged samples were subsequently annealed at 550, 600 or 650°C. By cutting, grinding and polishing, cross-sectional samples with a
89
thickness of 25 Ixm were produced. On these samples, optical extinction spectra of the silver particles could be measured with a microspectrophotometer [9] at different depths from the surface exposed to the salt melt. The beam from the microspectrophotometer had a width of 10 I~m.
Two thin sections of the glass (thickness = 40 Ixm) were cut perpendicular to the stained surface. These two thin sections were glued between two aluminium supporting rings in such a way that their ion exchanged surfaces were contiguous along a ring diameter. The specimen was then thinned by ion beam etching until there was a hole in the centre with a radius greater than the depth of the coloured region. Afterwards the sample was covered with a thin carbon film by vacuum evaporation to avoid charging effects in the transmission electron microscope (TEM). Ion beam thinning shaped the edge of the hole to a wedge allowing electron transmission through the thinner parts. Due to the specimen geometry, TEM measurements along the edge of the hole yield the possibility of imaging the sample at different penetration depths [10]. Figure 1 shows two typical transmission electron micrographs of silver particles taken at different penetration depths. There are spherical particles which electron diffraction identifies as crystalline silver (see refs. [6,13]). The particles are predominantly monocrystalline, as was shown by high resolution electron microscopy [10]. In each volume element, a distribution of the particle sizes occurs, which can be fitted by a Gaussian distribution [6,13]. The mean particle size changes with the penetration depth, whereas their volume concentration is nearly constant, both in correspondence with fig. 3 and fig. 4, respectively. In fig. 2 a typical change of the optical extinction spectra with the penetration depth can be seen. Using the method described in ref. [6], the dependence of particle radius (fig. 3) and volume
Table 1 Glass composition (wt%) (measured by X-ray fluorescence analysis [8]) SiO 2
Na20
CaO
MgO
ml203
KzO
SO 3
Fe203
72.20%
14.20
6.50
4.42
1.49
0.71
0.39
0.13
A. Berger / Colloidal siluer particles in glass surfaces
90
a
b
5o
5O i
45
2Omin/6OO~ • 2h/600'C 17h/60~C 95h/600"C
40
i
45
lh/650~C 100h/650"C 100h/600'C 100h/550~
4O
]
c
"23o 0)
~o251
+
~ 20
o
~2°i
o &
'7
10
lO
,
,o
5
~^~o~ ~ qllF • ' ' I
o o
I
,
I
q v
v
o
p~v r
0
,
~ v
ee&
v
vv
L
,
V
i
,
J
,
300 600 900 1 200 ~enetrotion depth (/~m)
1 O0 200 500 400 penetration depth (p.m)
l~ig. 3. Particle radius in dependence on the pentration depth for sample (a) exchanged in 2 wt% AgNO3-98 wt% KNO 3 and (b) exchanged in 2 wt% AgNO3-98 wt% NaNO 3 after different annealing processes.
Fig. 1. TEM micrographs at penetration depths of (a) 24 i~m and (b) 286 I~m after 30 min ion exchange in 2 wt% AgNO 3-98 wt% KNO 3 at 400°C and 95 h annealing at 600°C.
concentration (fig. 4) on the penetration depth was determined. (Because of the Gaussian distribution this method yields mean partical sizes.) The diameters and volume concentrations estimated, thus, are in a good agreement with the electron microscopical mean partical sizes [6,13]. The results: can be summarized as follows. (1) After the ion exchange, only ionic silver occurs in a very small surface layer.
e
6Q
b 2O
70
I .........
/\ I t I !.
"~.50
--
t /I
I-
~
,.C.
-
.~ 40 .u
x= 30,u.m x=l 30p.m x=230#m x= 280,u.m x=3OO,u,m
--- -
1 / I I s 1 ~
-
-
x=31
20min/600*C 2h/600"c 17~/800'C 95h/600'c
16
, ioso i00h/650"C
I J
14 12
~t0
-
•
v0
• Oe•
v
10
~ +0
8
• • 00
6
6
~> 4
20
"
16
~12
8
~
1
18
~14
O,u.m
x=3,30/zm x=350p.m
----
2.0 i
18
ee
Ooo~ OoOo
4
0 v
2
2 0 0
,
550
i
370
,
i
390
,
I
410
,
i
430
,
i
450
,
i
470
wovelength (nm)
,
I
490
,
I
510
,
i
5.30
,
~
550
Fig. 2. Optical extinction spectra for different penetration depths, x, after an ion exchange of 30 min in 2 wt% AgNO 398 wt% KNO 3 at 400°C and 95 h annealing at 600°C.
,
i
0
,
i
,
i
,
100 200 500 400 penetrotion depth (#m)
0
.300 600 900 1 200 lenetrotion depth (,u.m)
Fig. 4. Volume concentration of crystalline silver in dependence on the pentration depth for sample (a) exchanged in 2 wt% AgNO3-98 wt% KNO 3 and (b) exchanged in 2 wt% AgNO 3-98 wt% NaNO 3 after different annealing processes.
91
A. Berger / Colloidal silver particles in glass surfaces
(2) After the annealing process near Tg, there are spherical, monocrystalline silver particles within a surface layer. (3) In each volume element of this surface layer exists a Gaussian distribution of particle sizes. (4) The mean particle size increases with growing penetration depth. (5) At the glass surface, the particle size increases with the duration of the annealing. (6) After very long annealing processes, the particles near the end of the surface layer have very large diameters, which strongly increase with growing penetration depth. (7) After annealing, the volume concentration of silver particles is nearly constant within the coloured surface layer and decreases at their end to zero.
3. Theory After the ion exchange, there is only ionic silver within a very small surface layer. Two independent processes, ion exchange and reaction processes during annealing, will be discussed. Silver particles are formed by the annealing process. Subsequently, three different scales will be assumed. (1) Within scale 1, the diffusion of ionic silver takes place. It is assumed to be slow compared with the reaction processes. (2) The reaction processes are discussed on a smaller local scale. In small volume elements, the concentration of ionic silver and reduction agent are assumed to be constant. On this scale, mean values for particle size and concentration are used. (3) On a still smaller scale, a spatial varying concentration of ionic silver and reaction agent, i.e., within scale 2 an inhomogeneous distribution results in a size and concentration distribution of silver particles (as discussed for example in ref. [11]). In the following, only scale 2 will be discussed. Scale 1 is only of interest inasmuch as it gives the conditions of scale 2 (i.e., the concentration of ionic silver). Considering this, the simplest model
on the basis of a homogeneous nucleation including diffusion, reduction and particle formation is given by rate equations (1)-(4): 0i
0 (D0i]_
Ot = d x ~
-~x ]
nklinrn
Or - - -= - n k li"r n - k z i r N , Ot ON -- =kli"r",
-kzirN'
(1) (2) (3)
0t
0c Ot = V a [ n k l i n r "
+ kzirU]"
(4)
Equation (1) describes the change of the concentration of ionic silver i (with the exception of the volume concentration, all the concentrations used subsequently are numbers of ions, atoms or particles per volume unit). Starting from the silver ion concentration profile after the ion exchange, there is a diffusion process (term 1 in eq. (1)) as well as a decrease of the silver ion concentration by reducing processes (formation of stable nuclei of n silver atoms (term 2) and growth of nuclei or particles (term 3)). The concentration of the reducing agent, r, which is constant, r = r0, throughout the glass volume before the annealing process, is decreased by this reducing process in the same way (eq. (2)). The concentration of the stable nuclei, which is equal to the concentration of the particles, is given by N (eq. (3)), whereas c, the volume concentration of silver particles (i.e., particle volume per glass volume, which is equal the product of particle concentration and particle volume), is given by eq. (4). Va is the volume of a silver atom in the particle; k 1 and k 2 are the rates of nuclei formation and addition of silver atoms to nuclei or particles, respectively. If it is assumed that the concentration of ionic silver is much higher than the concentration of the reducing agent, and therefore the 'reduced' silver, the second and third terms in eq. (1) can be neglected. Equation (1) reduces then to 0t
-
0x
D
(5)
i.e., eq. (1) (scale 1) and eqs. (2)-(4) (scale 2) are separated.
92
A. Berger/ Colloidalsilverparticlesin glasssurfaces
Because of r = r 0 for t = 0 (in correspondence with the constant concentration of Fe 2+ in the glass before annealing) and r ~ 0 for t ~ oo (resulting from eq. (2)), r = r 0 exp( - C l t )
The equilibrium concentrations (i.e., on scale 2: t ~ oo) are r ( t --* oo) = 0,
(14)
C ( t ~ oo) = roVa,
(15)
(6)
can be assumed. Since c = 0 for t = 0 and c ~ c 0 for t ~ oo (in correspondence with the measured constant particle volume concentration), c is supposed to be c = c o i l - exp( - C z t ) ] .
N ( t ~ oo) = ~ kl--~nin-lrnk2
From c = NV, with V the particle volume, the particle radius, R, can be calculated as
(7)
Equations (2) and (4) then give c I = c 2 (c 1 and c 2 are constants) and Var o = c o. Since diffusion is assumed to be slower than reaction, the concentration of ionic silver is constant within the time of particle formation. With N = 0 for t = 0, from eqs. (3) and (6) it follows that klinr~ N= --[1 - exp(-Qnt)]. (8)
C1n
~ R =
-roC 1 exp(-clt ) = -nkli"r ~ exp(-clnt ) kai"r ~ - k z i r o exp( - C l t ) - -
cln
× [1 - exp( - c l n t ) ] .
(9)
For sufficiently large values of n and t, the terms containing e x p ( - c ~ n t ) in eq. (9) can be neglected. Equation (9) then reduces to klinr~ - r o C 1 e x p ( - C l t ) = - k 2 i r o exp(--Clt ) - -
cln
16"rr2k2Va2 9nkli,_lr~_
(17)
2 .
Since kl, k2, Va and r 0 are constant within the entire glass volume, the particle radius depends only on the concentration of ionic silver: R = A i (1-n)/6, (18)
~ 16-ffzk2Va2 A =
With this and eq. (6), eq. (2) gives
(16)
o •
(19)
9nklr~_ 2
In case of long annealing times, the total penetration depth, xi, of ionic silver after the ion exchange (25 and 55 Ixm, respectively) is very small compared with the width, x a, of the particle containing layers resulting from annealing (350 and 1000 Ixm, respectively), so that the simple model of diffusion from a plane source in a semi-infinite medium can be applied. With the assumption D = const., eq. (5) gives i
vr~D~ exp - ~
,
(20)
(10) With the assumption of constant silver ion concentration within the time of reaction and large values of n and t, the system of eqs. (2)-(4) gives
[ ~] r = r 0 exp
-klk2in+lr~ t 17 '
with M the total amount of ionic silver in an infinitesimal thin layer [12]. Combining eqs. (18) and (20), the dependence of particle radius on the penetration depth and annealing time is then given by
(11)
M R=A'
c
=
(
Var o 1 - exp
-
n
t
1
'
(12)
~
x 2
]/(l-n)/6
e x p [ - ~D-~-]1
.
(21)
The logarithmic form of eq. (21) is ln R = l n A
g=~nin-lr~{1-exp[-~klk2nin+lr~t]).
[
__(n-l)6 In[ ~ D t ] + ( n- - -l ) 24
x2 Dt '
(22)
(13)
i.e., In R c~ (x2/t).
A. Berger / Colloidal silver particles in glass surfaces
4. Discussion
As an example, the experimental data of the samples ion exchanged in 2 wt% AgNO3-98 wt% NaNO 3 will be compared with the theoretical description. The results for the samples exchanged in the A g N O 3 - K N O 3 liquid are similar. For the theoretical discussion, a diffusion coefficient was used, which was extrapolated from the temperature range 300-480°C by an Arrhenius function ( D O= 5.6 x 10 +3 ixm2/s; E , = 1.1 x 10-19 j) [7]). According to the weight change, the total amount of silver in the glass assumed to be in an infinitesimal thin surface layer is M = 1.85 ~m 2. The model (1)-(4) with solution (14)-(16) and (18) gave the following qualitative results: (i) high concentration of ionic silver (near surface, short times) results in many stable nuclei and small size particles; (ii) small concentration of ionic silver (great depths, long times) results in few stable nuclei and large size particles; (iii) the volume concentration of particles is constant within the silver containing layer; (iv) for a given annealing time the particle size increase with the depth.
a 50
2?
35
2.5
U
~
, h/650"C
45
le0nleO0.c 100~/ss0.c A=5 e . 1 0 - " ; . - i 8
40
A~5 6.10-11; n = l 8 5 --
35
v
A = I S.10-'; .-I 0.8
^=1 S-,O-';n=
eo
3O
20 o
/
15 z V ,
05~ 0
10
~Z , 0 o . / e s - ~
]
1 ×2/t
4
2 5 in/~mZ/s
93
The logarithmic plot (fig. 5(a)) shows that for long annealing times all values calculated from experiments can be fitted to a linear function, which is in agreement with eq. (22) and which allows an estimate of n and A. The value of n, the number of silver atoms building a stable nucleus, increases from 6.0 to 18.5 in the temperature range from 550 to 650°C. The value of A reduces from 1.5 x 10 -5 i~m(3-n)/2 to 5.6 x 10 -11 ixm(3-")/2, i.e., the ratio k2/k 1 decreases (eq. (19)). Since, however, the concentration of the particles, N, depends on the product k,i"r", the number of particles per volume decreases with increasing temperature. With r o = rFe2+, then c(t --->oo) = 1.2 x 10 -4, which is in a good agreement with the experimental data (see fig. 4). A plot of particle radius versus penetration depth (fig. 5(b)) shows that there is agreement between experimental and theoretical values for long annealing times. In the logarithmic plot, the data for shorter annealing times also fits a linear function. These have, however, a different slope. As fig. 5(b) shows, experimental and theoretical values differ in the depth dependence of the particle size. This difference is, however, not a general failure of the model, but is to be expected, because the assumption x i << x a is not valid for small annealing times. To solve the diffusion eq. (5), a constant diffusion coefficient was assumed. For long annealing times resulting in small silver ion concentrations, this assumption is good. For small annealing times, i.e., larger silver ion concentrations, however, the concentration dependence of the diffusion coefficient has to be taken into account. To give a correct description for short annealing times, it seems to be necessary to carry out numerical calculations for the diffusion of the silver ions taking into account the measured silver depth profile after ion exchange and a concentration dependent diffusion coefficient.
5 0 300
600 900 X in/zm
1200
Fig. 5. Comparison of experimental data (ion exchange in 2 wt% AgNO3-98 wt% NaNO 3 at 400°C for 2 h, different annealing processes) with the theoretical description according eqs. (21) and (22), respectively. (A is in Ixm(3-n)/2).
5. Conclusions
The proposed model of particle formation on the basis of a homogeneous nucleation in which ion exchange, silver ion diffusion, reduction and
94
A. Berger / Colloidal silver particles in glass surfaces
agglomeration are the parameters gives a qualitative explanation of the dependence of particle size and concentration on the penetration depth. For long annealing times at temperatures in the range 550-650°C, even a quantitative description of the dependence is possible. The number of silver atoms building a stable nucleus was estimated to be in the order of 6-18 in the temperature range from 550 to 650°C. When this work was done, the author was with the Martin-Luther-Universit~it, Halle-Wittenberg, Fachbereich Physik, Germany. He thanks Dr. Herbert Hofmeister from the Max-Planck-Institut fiir Mikrostrukturphysik Halle for carrying out the electron microscopical investigations.
References [1] W.A. Weyl, Coloured Glasses (Dawsons, London, 1959). [2] G.H. Frischat, Ionic Diffusion In Oxide Glasses (TransTech, Aedermannsdorf, 1975). [3] H. Garfinkel, in: Membranes, ed. G. Eisenman, Vol. 1 (Dekker, New York, 1972) p. 179. [4] C.R. Bamford, Colour Generation and Control in Glass (Elsevier, Amsterdam, 1977). [5] D. Ganka, dissertation (A), Martin-Luther Universitiit Halle, in preparation. [6] K.-J. Berg, A. Berger and H. Hofmeister, Z. Phys. D20 (1991) 309. [7] M. Petzold, unpublished work. [8] S. Thiel, Praktikumsbeleg (Torgau, 1989). [9] K.H. Brauer and F. Frfhlich, Exp. Techn. Phys. 6 (1958) 216. [10] M. Dubiel, H. Hofmeister, K.-J. Berg and A. Berger, Ver6ffentlichungen zur 12. Tagung Elektronen mikroskopie, Dresden, 1988, p. 424. [11] S.C. Jain, A.E. Hughes, J. Mater. Sci. 13 (1978) 1611. [12] J. Crank, The Mathematics of Diffusion (Oxford University Press, 1956). [13] A. Berger, thesis, Martin-Luther Universitiit Halle (1988).
~
]OURNAL
OF
U
Concentration and size depth profile of colloidal silver particles in glass surfaces produced by sodium-silver ion-exchange A. B e r g e r Center for Research in Electro-Optics and Lasers, University of Central Florida, 12424 Research Parkway, Suite 400, Orlando, FL 32826, USA
Received 12 May 1992 Revised manuscript received 7 July 1992
The sodium-silver ion exchange of a commercial soda-lime silicate glass followed by an annealing process yield small silver particles in a surface layer. Microspectrophotometryspectra and transmission electron microscopyshow that the mean particle size increases with the penetration depth, whereas the volume concentration is constant. A model on the basis of a homogeneous nucleation including ion exchange, diffusion, reduction and particle formation is given. The measured depth dependence is described with the model by a non-linear system of differential equations.
I. Introduction Staining of alkali-silicate glasses by exchanging silver ions from pastes or liquids with alkali ions of the glass yields a yellow coloured surface layer, which is caused by small silver particles [1,6]. Following Weyl [1] it is usually assumed that the formation of these particles is a result of the complex combination of ion exchange, diffusion of silver ions, their reduction to atoms and aggregation to particles. Whereas ion exchange and silver ion diffusion in glasses are well understood (see, for example, refs. [2,3]), there are many unsolved questions concerning reduction and particle formation. Especially the microscopic processes resulting in the reduction of the silver ions to atoms and their aggregation to crystalline particles are unknown. A detailed model describing the formation of the small silver particles has not yet been given.
Correspondence to: Dr A. Berger, Center for Research in Electro-Optics and Lasers, University of Central Florida, 12424 Research Parkway,Suite 400, Orlando, FL 32826, USA. Tel: + 1-407 658 3991. Telefax: + 1-407 658 3955.
Dependent on the glass composition in commercial soda-lime glasses, iron (concentration << 1 wt%) generally can be found in the Fe 2+ as well as in the Fe 3÷ state [4]. Using optical spectroscopy a relation of F e 2 + / F e 3÷= 0.43 was estimated for the investigated glass [5]. This ratio is important for the reduction of silver ions (in a non-reducing atmosphere). The following thermoreducing reaction is to be expected [1]: Fe 2 + ~ Fe3++ e A g + + e---* Ag °. According to this assumption the maximum number of silver atoms has to be equal to or less than the number of Fe 2+. In ref. [6], a method was presented, from which size as well as volume concentration of silver particles can be determined from optical extinction spectra. From a measurement of optical spectra with a microspectrophotometer as a function of penetration depth, an estimate of particle size and concentration as a function of depth from the exchanged surface can be made.
0022-3093/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A. Berger / Colloidal silver particles in glass surfaces
This paper presents experimental results of size and volume concentration of silver particles as a function of penetration depth after a low temperature ion exchange ( T = 400°C) followed by an annealing process near the transition temperature of the glass, Tg. Size and concentration were determined using the 'optical' method of ref. [6] and by transmission electron microscopy. The results are explained by a model on the basis of a homogeneous nucleation. The system of non-linear differential equations explains the formation of silver particles by ion exchange, diffusion, reduction and aggregation. A quantitative description of the complex process is given.
2. Experiments and results For all the investigations, samples of a commercial soda-lime glass (for composition, see table 1) with a thickness of 2.2 mm were used. The samples were subjected to an ion exchange in a salt liquid of 2 wt% AgNO3-98 wt% NaNO 3 at 400°C for 2 h and 2 wt% AgNO3-98 wt% KNO 3 at 400°C for 30 min, respectively. Electron-probe microanalysis detected a silver containing surface layer with a thickness of 55 and 25 txm, respectively, after the ion exchange [7]. In the case of the AgNO3-NaNO 3 salt liquid, 80% of the sodium ions are exchanged by silver ions at the surface. The weight change of the sample resulting from ion exchange gave a total amount of silver of M = 1.85 txm -2, if the silver is assumed to be in an infinitesimal thin surface layer (appropriate to eq. (20)). From optical spectra, I conclude that nearly all the silver is in the ionic state, because there is no absorption caused by silver atoms or particles. Silver particles cannot be found by electron microscopy. The ion exchanged samples were subsequently annealed at 550, 600 or 650°C. By cutting, grinding and polishing, cross-sectional samples with a
89
thickness of 25 Ixm were produced. On these samples, optical extinction spectra of the silver particles could be measured with a microspectrophotometer [9] at different depths from the surface exposed to the salt melt. The beam from the microspectrophotometer had a width of 10 I~m.
Two thin sections of the glass (thickness = 40 Ixm) were cut perpendicular to the stained surface. These two thin sections were glued between two aluminium supporting rings in such a way that their ion exchanged surfaces were contiguous along a ring diameter. The specimen was then thinned by ion beam etching until there was a hole in the centre with a radius greater than the depth of the coloured region. Afterwards the sample was covered with a thin carbon film by vacuum evaporation to avoid charging effects in the transmission electron microscope (TEM). Ion beam thinning shaped the edge of the hole to a wedge allowing electron transmission through the thinner parts. Due to the specimen geometry, TEM measurements along the edge of the hole yield the possibility of imaging the sample at different penetration depths [10]. Figure 1 shows two typical transmission electron micrographs of silver particles taken at different penetration depths. There are spherical particles which electron diffraction identifies as crystalline silver (see refs. [6,13]). The particles are predominantly monocrystalline, as was shown by high resolution electron microscopy [10]. In each volume element, a distribution of the particle sizes occurs, which can be fitted by a Gaussian distribution [6,13]. The mean particle size changes with the penetration depth, whereas their volume concentration is nearly constant, both in correspondence with fig. 3 and fig. 4, respectively. In fig. 2 a typical change of the optical extinction spectra with the penetration depth can be seen. Using the method described in ref. [6], the dependence of particle radius (fig. 3) and volume
Table 1 Glass composition (wt%) (measured by X-ray fluorescence analysis [8]) SiO 2
Na20
CaO
MgO
ml203
KzO
SO 3
Fe203
72.20%
14.20
6.50
4.42
1.49
0.71
0.39
0.13
A. Berger / Colloidal siluer particles in glass surfaces
90
a
b
5o
5O i
45
2Omin/6OO~ • 2h/600'C 17h/60~C 95h/600"C
40
i
45
lh/650~C 100h/650"C 100h/600'C 100h/550~
4O
]
c
"23o 0)
~o251
+
~ 20
o
~2°i
o &
'7
10
lO
,
,o
5
~^~o~ ~ qllF • ' ' I
o o
I
,
I
q v
v
o
p~v r
0
,
~ v
ee&
v
vv
L
,
V
i
,
J
,
300 600 900 1 200 ~enetrotion depth (/~m)
1 O0 200 500 400 penetration depth (p.m)
l~ig. 3. Particle radius in dependence on the pentration depth for sample (a) exchanged in 2 wt% AgNO3-98 wt% KNO 3 and (b) exchanged in 2 wt% AgNO3-98 wt% NaNO 3 after different annealing processes.
Fig. 1. TEM micrographs at penetration depths of (a) 24 i~m and (b) 286 I~m after 30 min ion exchange in 2 wt% AgNO 3-98 wt% KNO 3 at 400°C and 95 h annealing at 600°C.
concentration (fig. 4) on the penetration depth was determined. (Because of the Gaussian distribution this method yields mean partical sizes.) The diameters and volume concentrations estimated, thus, are in a good agreement with the electron microscopical mean partical sizes [6,13]. The results: can be summarized as follows. (1) After the ion exchange, only ionic silver occurs in a very small surface layer.
e
6Q
b 2O
70
I .........
/\ I t I !.
"~.50
--
t /I
I-
~
,.C.
-
.~ 40 .u
x= 30,u.m x=l 30p.m x=230#m x= 280,u.m x=3OO,u,m
--- -
1 / I I s 1 ~
-
-
x=31
20min/600*C 2h/600"c 17~/800'C 95h/600'c
16
, ioso i00h/650"C
I J
14 12
~t0
-
•
v0
• Oe•
v
10
~ +0
8
• • 00
6
6
~> 4
20
"
16
~12
8
~
1
18
~14
O,u.m
x=3,30/zm x=350p.m
----
2.0 i
18
ee
Ooo~ OoOo
4
0 v
2
2 0 0
,
550
i
370
,
i
390
,
I
410
,
i
430
,
i
450
,
i
470
wovelength (nm)
,
I
490
,
I
510
,
i
5.30
,
~
550
Fig. 2. Optical extinction spectra for different penetration depths, x, after an ion exchange of 30 min in 2 wt% AgNO 398 wt% KNO 3 at 400°C and 95 h annealing at 600°C.
,
i
0
,
i
,
i
,
100 200 500 400 penetrotion depth (#m)
0
.300 600 900 1 200 lenetrotion depth (,u.m)
Fig. 4. Volume concentration of crystalline silver in dependence on the pentration depth for sample (a) exchanged in 2 wt% AgNO3-98 wt% KNO 3 and (b) exchanged in 2 wt% AgNO 3-98 wt% NaNO 3 after different annealing processes.
91
A. Berger / Colloidal silver particles in glass surfaces
(2) After the annealing process near Tg, there are spherical, monocrystalline silver particles within a surface layer. (3) In each volume element of this surface layer exists a Gaussian distribution of particle sizes. (4) The mean particle size increases with growing penetration depth. (5) At the glass surface, the particle size increases with the duration of the annealing. (6) After very long annealing processes, the particles near the end of the surface layer have very large diameters, which strongly increase with growing penetration depth. (7) After annealing, the volume concentration of silver particles is nearly constant within the coloured surface layer and decreases at their end to zero.
3. Theory After the ion exchange, there is only ionic silver within a very small surface layer. Two independent processes, ion exchange and reaction processes during annealing, will be discussed. Silver particles are formed by the annealing process. Subsequently, three different scales will be assumed. (1) Within scale 1, the diffusion of ionic silver takes place. It is assumed to be slow compared with the reaction processes. (2) The reaction processes are discussed on a smaller local scale. In small volume elements, the concentration of ionic silver and reduction agent are assumed to be constant. On this scale, mean values for particle size and concentration are used. (3) On a still smaller scale, a spatial varying concentration of ionic silver and reaction agent, i.e., within scale 2 an inhomogeneous distribution results in a size and concentration distribution of silver particles (as discussed for example in ref. [11]). In the following, only scale 2 will be discussed. Scale 1 is only of interest inasmuch as it gives the conditions of scale 2 (i.e., the concentration of ionic silver). Considering this, the simplest model
on the basis of a homogeneous nucleation including diffusion, reduction and particle formation is given by rate equations (1)-(4): 0i
0 (D0i]_
Ot = d x ~
-~x ]
nklinrn
Or - - -= - n k li"r n - k z i r N , Ot ON -- =kli"r",
-kzirN'
(1) (2) (3)
0t
0c Ot = V a [ n k l i n r "
+ kzirU]"
(4)
Equation (1) describes the change of the concentration of ionic silver i (with the exception of the volume concentration, all the concentrations used subsequently are numbers of ions, atoms or particles per volume unit). Starting from the silver ion concentration profile after the ion exchange, there is a diffusion process (term 1 in eq. (1)) as well as a decrease of the silver ion concentration by reducing processes (formation of stable nuclei of n silver atoms (term 2) and growth of nuclei or particles (term 3)). The concentration of the reducing agent, r, which is constant, r = r0, throughout the glass volume before the annealing process, is decreased by this reducing process in the same way (eq. (2)). The concentration of the stable nuclei, which is equal to the concentration of the particles, is given by N (eq. (3)), whereas c, the volume concentration of silver particles (i.e., particle volume per glass volume, which is equal the product of particle concentration and particle volume), is given by eq. (4). Va is the volume of a silver atom in the particle; k 1 and k 2 are the rates of nuclei formation and addition of silver atoms to nuclei or particles, respectively. If it is assumed that the concentration of ionic silver is much higher than the concentration of the reducing agent, and therefore the 'reduced' silver, the second and third terms in eq. (1) can be neglected. Equation (1) reduces then to 0t
-
0x
D
(5)
i.e., eq. (1) (scale 1) and eqs. (2)-(4) (scale 2) are separated.
92
A. Berger/ Colloidalsilverparticlesin glasssurfaces
Because of r = r 0 for t = 0 (in correspondence with the constant concentration of Fe 2+ in the glass before annealing) and r ~ 0 for t ~ oo (resulting from eq. (2)), r = r 0 exp( - C l t )
The equilibrium concentrations (i.e., on scale 2: t ~ oo) are r ( t --* oo) = 0,
(14)
C ( t ~ oo) = roVa,
(15)
(6)
can be assumed. Since c = 0 for t = 0 and c ~ c 0 for t ~ oo (in correspondence with the measured constant particle volume concentration), c is supposed to be c = c o i l - exp( - C z t ) ] .
N ( t ~ oo) = ~ kl--~nin-lrnk2
From c = NV, with V the particle volume, the particle radius, R, can be calculated as
(7)
Equations (2) and (4) then give c I = c 2 (c 1 and c 2 are constants) and Var o = c o. Since diffusion is assumed to be slower than reaction, the concentration of ionic silver is constant within the time of particle formation. With N = 0 for t = 0, from eqs. (3) and (6) it follows that klinr~ N= --[1 - exp(-Qnt)]. (8)
C1n
~ R =
-roC 1 exp(-clt ) = -nkli"r ~ exp(-clnt ) kai"r ~ - k z i r o exp( - C l t ) - -
cln
× [1 - exp( - c l n t ) ] .
(9)
For sufficiently large values of n and t, the terms containing e x p ( - c ~ n t ) in eq. (9) can be neglected. Equation (9) then reduces to klinr~ - r o C 1 e x p ( - C l t ) = - k 2 i r o exp(--Clt ) - -
cln
16"rr2k2Va2 9nkli,_lr~_
(17)
2 .
Since kl, k2, Va and r 0 are constant within the entire glass volume, the particle radius depends only on the concentration of ionic silver: R = A i (1-n)/6, (18)
~ 16-ffzk2Va2 A =
With this and eq. (6), eq. (2) gives
(16)
o •
(19)
9nklr~_ 2
In case of long annealing times, the total penetration depth, xi, of ionic silver after the ion exchange (25 and 55 Ixm, respectively) is very small compared with the width, x a, of the particle containing layers resulting from annealing (350 and 1000 Ixm, respectively), so that the simple model of diffusion from a plane source in a semi-infinite medium can be applied. With the assumption D = const., eq. (5) gives i
vr~D~ exp - ~
,
(20)
(10) With the assumption of constant silver ion concentration within the time of reaction and large values of n and t, the system of eqs. (2)-(4) gives
[ ~] r = r 0 exp
-klk2in+lr~ t 17 '
with M the total amount of ionic silver in an infinitesimal thin layer [12]. Combining eqs. (18) and (20), the dependence of particle radius on the penetration depth and annealing time is then given by
(11)
M R=A'
c
=
(
Var o 1 - exp
-
n
t
1
'
(12)
~
x 2
]/(l-n)/6
e x p [ - ~D-~-]1
.
(21)
The logarithmic form of eq. (21) is ln R = l n A
g=~nin-lr~{1-exp[-~klk2nin+lr~t]).
[
__(n-l)6 In[ ~ D t ] + ( n- - -l ) 24
x2 Dt '
(22)
(13)
i.e., In R c~ (x2/t).
A. Berger / Colloidal silver particles in glass surfaces
4. Discussion
As an example, the experimental data of the samples ion exchanged in 2 wt% AgNO3-98 wt% NaNO 3 will be compared with the theoretical description. The results for the samples exchanged in the A g N O 3 - K N O 3 liquid are similar. For the theoretical discussion, a diffusion coefficient was used, which was extrapolated from the temperature range 300-480°C by an Arrhenius function ( D O= 5.6 x 10 +3 ixm2/s; E , = 1.1 x 10-19 j) [7]). According to the weight change, the total amount of silver in the glass assumed to be in an infinitesimal thin surface layer is M = 1.85 ~m 2. The model (1)-(4) with solution (14)-(16) and (18) gave the following qualitative results: (i) high concentration of ionic silver (near surface, short times) results in many stable nuclei and small size particles; (ii) small concentration of ionic silver (great depths, long times) results in few stable nuclei and large size particles; (iii) the volume concentration of particles is constant within the silver containing layer; (iv) for a given annealing time the particle size increase with the depth.
a 50
2?
35
2.5
U
~
, h/650"C
45
le0nleO0.c 100~/ss0.c A=5 e . 1 0 - " ; . - i 8
40
A~5 6.10-11; n = l 8 5 --
35
v
A = I S.10-'; .-I 0.8
^=1 S-,O-';n=
eo
3O
20 o
/
15 z V ,
05~ 0
10
~Z , 0 o . / e s - ~
]
1 ×2/t
4
2 5 in/~mZ/s
93
The logarithmic plot (fig. 5(a)) shows that for long annealing times all values calculated from experiments can be fitted to a linear function, which is in agreement with eq. (22) and which allows an estimate of n and A. The value of n, the number of silver atoms building a stable nucleus, increases from 6.0 to 18.5 in the temperature range from 550 to 650°C. The value of A reduces from 1.5 x 10 -5 i~m(3-n)/2 to 5.6 x 10 -11 ixm(3-")/2, i.e., the ratio k2/k 1 decreases (eq. (19)). Since, however, the concentration of the particles, N, depends on the product k,i"r", the number of particles per volume decreases with increasing temperature. With r o = rFe2+, then c(t --->oo) = 1.2 x 10 -4, which is in a good agreement with the experimental data (see fig. 4). A plot of particle radius versus penetration depth (fig. 5(b)) shows that there is agreement between experimental and theoretical values for long annealing times. In the logarithmic plot, the data for shorter annealing times also fits a linear function. These have, however, a different slope. As fig. 5(b) shows, experimental and theoretical values differ in the depth dependence of the particle size. This difference is, however, not a general failure of the model, but is to be expected, because the assumption x i << x a is not valid for small annealing times. To solve the diffusion eq. (5), a constant diffusion coefficient was assumed. For long annealing times resulting in small silver ion concentrations, this assumption is good. For small annealing times, i.e., larger silver ion concentrations, however, the concentration dependence of the diffusion coefficient has to be taken into account. To give a correct description for short annealing times, it seems to be necessary to carry out numerical calculations for the diffusion of the silver ions taking into account the measured silver depth profile after ion exchange and a concentration dependent diffusion coefficient.
5 0 300
600 900 X in/zm
1200
Fig. 5. Comparison of experimental data (ion exchange in 2 wt% AgNO3-98 wt% NaNO 3 at 400°C for 2 h, different annealing processes) with the theoretical description according eqs. (21) and (22), respectively. (A is in Ixm(3-n)/2).
5. Conclusions
The proposed model of particle formation on the basis of a homogeneous nucleation in which ion exchange, silver ion diffusion, reduction and
94
A. Berger / Colloidal silver particles in glass surfaces
agglomeration are the parameters gives a qualitative explanation of the dependence of particle size and concentration on the penetration depth. For long annealing times at temperatures in the range 550-650°C, even a quantitative description of the dependence is possible. The number of silver atoms building a stable nucleus was estimated to be in the order of 6-18 in the temperature range from 550 to 650°C. When this work was done, the author was with the Martin-Luther-Universit~it, Halle-Wittenberg, Fachbereich Physik, Germany. He thanks Dr. Herbert Hofmeister from the Max-Planck-Institut fiir Mikrostrukturphysik Halle for carrying out the electron microscopical investigations.
References [1] W.A. Weyl, Coloured Glasses (Dawsons, London, 1959). [2] G.H. Frischat, Ionic Diffusion In Oxide Glasses (TransTech, Aedermannsdorf, 1975). [3] H. Garfinkel, in: Membranes, ed. G. Eisenman, Vol. 1 (Dekker, New York, 1972) p. 179. [4] C.R. Bamford, Colour Generation and Control in Glass (Elsevier, Amsterdam, 1977). [5] D. Ganka, dissertation (A), Martin-Luther Universitiit Halle, in preparation. [6] K.-J. Berg, A. Berger and H. Hofmeister, Z. Phys. D20 (1991) 309. [7] M. Petzold, unpublished work. [8] S. Thiel, Praktikumsbeleg (Torgau, 1989). [9] K.H. Brauer and F. Frfhlich, Exp. Techn. Phys. 6 (1958) 216. [10] M. Dubiel, H. Hofmeister, K.-J. Berg and A. Berger, Ver6ffentlichungen zur 12. Tagung Elektronen mikroskopie, Dresden, 1988, p. 424. [11] S.C. Jain, A.E. Hughes, J. Mater. Sci. 13 (1978) 1611. [12] J. Crank, The Mathematics of Diffusion (Oxford University Press, 1956). [13] A. Berger, thesis, Martin-Luther Universitiit Halle (1988).