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Structure of partially reduced xPbO (1 − x)SiO2 glasses: combined EXAFS and MD study
Journal of Non-Crystalline Solids 351 (2005) 380–393 www.elsevier.com/locate/jnoncrysol
Structure of partially reduced xPbO (1 x)SiO2 glasses: combined EXAFS and MD study Agnieszka Witkowska a
a,b,*
, Jarosław Rybicki
a,b
, Andrea Di Cicco
c
Department of Solid State Physics, Faculty of Technical Physics and Applied Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland b TASK Computer Centre, Narutowicza 11/12, 80-952 Gdansk, Poland c INFM Dipartimento di Fisica, Universita` di Camerino, via Madonna delle Carceri, I-62032 Camerino (MC), Italy Received 6 July 2003; received in revised form 2 November 2004
Abstract We have studied the structure of partially reduced lead-silicate glasses using combined EXAFS (extended X-ray absorption fine structure) and MD (molecular dynamics) methods. The analysis was performed for glasses of x[(1 p)Pb pPbO] (1 x)SiO2 composition, x = 0.3, 0.5, 0.7, where parameter (1 p) describes the degree of reduction, i.e. the content of the granular metallic phase, appearing as the result of the reduction process (e.g. annealing in hydrogen atmosphere). In the EXAFS experiment (1 p) was expressed via the time of reduction realized at 400 C (1.5 h, 24 h, 70 h), whereas in the MD simulations it was determined precisely by using proper numbers of particles (corresponding to (1 p) = 0.0, 0.25, 0.5, 0.75 and 1.0). In the paper we describe in detail the local structure around lead atoms and its changes in the function of glass composition and reduction degree. The tendency for agglomeration of Pb0 into clusters, the formation of the granular metallic phase, and continuity of silica and lead oxide subnetworks are discussed. A good agreement between EXAFS-extractcd and MD-extracted parameters of the short-range structure encouraged us to preform a medium-range order analysis, based on the MD simulations only. Moreover, combining the EXAFS and MD methods we could correlate the reduction time (technological parameter) with the degree of reduction (1 p) and the actual state of the granular structure. The latter relation may be useful for controlled production of reduced glasses of pre-requcstcd physical properties. 2005 Elsevier B.V. All rights reserved.
1. Introduction Lead-silicate glasses, due to their specific and very interesting properties, find a lot of industrial applications. As special materials, they are used in the production of plate image amplifiers and scintillators [1]. *
Corresponding author. Present address: Department of Solid State Physics, Faculty of Technical Physics and Applied Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland. E-mail address: [email protected] (A. Witkowska). 0022-3093/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.01.036
Lead-silicate glasses submitted to the reduction process (e.g. in hydrogen atmosphere), undergo dramatic changes of optical properties [2] and electrical surface conductivity, and reveal a very high secondary emission coefficient, thus finding applications in the production of electron channel multipliers [3]. The atomic structure of lead-silicate glasses has been investigated for sixty years, using various experimental techniques, including IR spectroscopy [4], Raman spectroscopy [4–6], NMR [5,7,8], XPS [9], X-ray [10,11], neutron diffraction methods [12,13], and EXAFS [8,14]. Computer simulations of their structure have also
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been performed [14–18]. However, as far as the authors are aware, no experimental work on the atomic-level structure of partially reduced glasses has been reported in the literature, other than our work [19] presenting preliminary results concerning lead-silicate glass of one composition only, x = 0.5. Some simulation results on completely reduced glasses (xPbO (1 x)SiO2) have been described in [20,21]. Recently, we have performed EXAFS (extended X-ray absorption fine structure) and MD (molecular dynamics) investigations of the structure of partially reduced lead-silicate glasses as a function of PbO content in the initially unreduced glass. These new results are described in the present paper. X-ray absorption spectroscopy (XAS) is a powerful structural technique that allows us to investigate the neighbourhood of a photoabsorber atom embedded in a condensed medium. The main and very important advantages of the XAS method in the case of multi-component systems are its atomic selectivity and very high sensitivity. In our work we have used the GNXAS package for EXAFS data analysis [22–24]. Molecular dynamics (MD) is a widely used and very convenient simulation method for investigation of material structures on the atomic level (e.g. [25–35]). The MD method gives trajectories of all the atoms—and thus the structure—in the short- and medium-range, can be analysed in detail, and its results can be compared with experimental data. Moreover, MD simulations offer insight into structural features inaccessible to experiment. A combined usage of EXAFS and MD methods, as has been shown in [14,36,37], is a source of detailed information on the atomic structure in disordered materials. The paper is organised as follows. In Section 2 we discuss technical issues related to sample preparation, the EXAFS measurement and the MD simulation technique. In Section 3 we present an EXAFS analysis of the short range structure around Pb atoms. Our MD simulations are presented in Section 4. In Section 5 the granular structure of reduced glasses is characterised using the combined EXAFS and MD methods. Section 6 contains concluding remarks.
2. Measurement and simulation techniques 2.1. Sample preparation and XAS measurements xPbO (1 x)SiO2, 0.3 6 x 6 0.7 glasses were prepared as follows. A mixture of powdered SiO2 and PbO was placed in a platinum crucible and heated during 3–4 h in an electric furnace from room temperature (RT) to melting temperature, ranging from 1100 C for x = 0.7 to 1350 C for x = 0.3. The melts were kept slightly above their melting temperature for several minutes and stirred mechanically in order to homogenize the
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samples. Then the melts were rapidly cooled by pouring them onto a brass plate. The resulting glasses had uniform colour and were free of ceramic inclusions. No Bragg peaks have been identified by X-ray diffraction. The powdered samples (grain diameter of several lm) were then annealed in hot hydrogen atmosphere to permit their reduction. The reduction process was realised at the temperature of Tred = 400 C, for periods of tred = 1.5, 24, 70 h [38]. Next, the reduced glass powder was mixed with BN in optimal proportions, and pellets for XAS measurements were formed. The pelletsÕ mass, diameter and thickness amounted to 200 mg, 12 mm and 0.5–1 mm, respectively. Their homogeneity was being checked using an optical microscope. The spectra of the lead-silicate glasses at the Pb L3edge were recorded at the BM29 beam-line of the European Synchrotron Radiation Facility (Grenoble), using a double-crystal monochromator equipped with Si(3 1 1) crystals. The spectrum of solid Pb at the Pb L3-edge was collected at the D42 beam-line of the LURE laboratories (Orsay, Paris), equipped with an Si(331) channel-cut monochromator. The room temperature measurements were performed in the energy range 12 900–14 000 eV (the Pb L3-edge energy amounts to 13 035 eV). The sampling steps were equal to 3 eV and 0.4 eV in the 12 900–13 000 eV and 13 000–13 100 eV energy ranges, respectively. In the range between 13 100 eV and 14 000 eV the sampling step was increased from 0.4 to 2 eV. This scanning procedure yielded high quality data of both pre- and post-edge background analyses used for normalisation of the spectra. More details on the experimental setup can be found in [39]. Experimental data have been analysed within an advanced technique, using theoretical calculations of the X-ray absorption cross-section in the framework of the GNXAS method [22–24]. The method is based on the comparison in the energy space between the experimental signal and the theoretical one, attempting to optimise the relevant structural parameter values. The post-edge background was modelled with three splines of degree 3, 4, and 4 on the subsequent energy intervals: 13 080–13 300 eV, 13 300–13 500 eV, 13 500– 13 900 eV. The model absorption signal contained a contribution accounting for possible multielectron excitation channels (2p3/24f) and (2p3/24d). The importance of these contributions for the Pb L3-edge has been discussed in [40]. 2.2. MD simulations Our MD simulations were performed in the constant volume regime (NVE ensemble), using the MDSIM code [41]. The atoms were assumed to interact by the twobody Born–Mayer–Huggins potential, containing the Born–Mayer repulsive contribution, and Coulomb interactions. Null charge was assigned to neutral lead
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atoms, Pb0, while the remaining interaction parameters remained the same as for lead ions, Pb+2. Thus, the Pb0–X interactions, X = Pb0, Pb+2, Si+4 and O2, contain only repulsive contributions, while Pb0–Pb0 and Pb0–O2 interactions are strictly packing in character. Moreover, the polarizability of lead atoms and ions has been neglected, and no charge transfer has been allowed for. Nevertheless, despite the simplicity of the applied force-field with the potential parameter taken from [15], the simulation reproduces quite well the experimental interatomic distances and co-ordination numbers, even the shape of the first peak of the partial pair distribution functions (see e.g. [14,16,17,42]). Therefore, the results of our simulations can be safely used as the first guess of GNXAS local structure refinement procedures for partially reduced glasses. In this paper we report on simulations of partially reduced lead-silicate glasses of x[(1 p)Pb pPbO] (1 x)SiO2 composition, x = 0.3, 0.5, 0.7, and (1 p) = 0.0, 0.25, 0.5, 0.75 and 1.0. Numbers of atoms and edge lengths of the cubic simulation boxes are given in Table 1. The sample was initially prepared in a well-equilibrated molten state at 10 000 K, then cooled at the average rate of 2 · 1013 K/s down to 300 K, passing equilibrium states at 8000 K, 6000 K, 5000 K, 4000 K, 3000 K, 2500 K, 2000 K, 1500 K, 1000 K, and 600 K. Temperature scaling was applied whenever the rolling average of temperature (calculated over last 100 time steps) exceeded the (T DT, T + DT) interval. At each intermediate temperature the system was equilibrated during 30 000 fs time steps, using DT = 100 K for T P 1000 K, DT = 20 K for T = 600 K and DT = 10 K for T = 300 K. Equilibrated systems were sampled during 10 000 time steps Dt, Dt = 1015 s. Structural information on short-range correlations was obtained in a conventional way, mainly from the
partial pair and angular distribution functions (pPDFs and ADFs). As it has been shown in a number of papers, pPDFs can be decomposed in disordered systems into a short-range peak of well-defined shape and a long-range tail [43,44]. A simple Gaussian shape of the short-range peak of a pPDF is usually insufficient to describe accurately the short-range ordering of highly disordered systems. A useful parameterisation of the first partial PDF peak, as shown in [44,45], has the form: gAB ðrÞ ¼
1 pAB ðrÞ ; qB 4pr2
ð1Þ
where qB is the atomic density of B-neighbours, and pAB(r) is the bond length probability density of finding atoms B within shell r and r + dr around atoms A, described by a CF distribution. The corresponding formula, valid for (r R)b > 2r, reads as follows [23,46–48]: 4 1 2N 4 2ðr RÞ b2 þ pAB ðrÞ ¼ rb rjbjCð4=b2 Þ b2 4 2ðr RÞ exp 2 þ : ð2Þ rb b Here, R is the average first-shell A–B distance, r2 is the distance variance (a Debye–Waller-like parameter), b is the asymmetry (skewness) parameter, N is the co-ordination number, and C(x) is the Euler C-function. In order to describe the second and further co-ordination shells, i.e. to describe the medium-range order, instead of analysing PDFs and ADFs, one should use more advanced methods of structural analysis. One of the possible approaches is to analyse properly constructed clusters of edge and/or face-sharing Voronoı¨ polyhedra [49–51]. This method, although very efficient in detecting crystalline regions of various symmetries [52], works well for close-packed systems. In open
Table 1 Numbers of atoms used in the simulations and the corresponding edge lengths of the simulation box Pb+2
Si+4
O2
˚] L [A
0 150 300 450 600
600 450 300 150 0
1400 1400 1400 1400 1400
3400 3250 3100 2950 2800
42.731 42.574 42.415 42.256 42.095
0.0 0.25 0.5 0.75 1.0
0 250 500 750 1000
1000 750 500 250 0
1000 1000 1000 1000 1000
3000 2750 2500 2250 2000
42.870 42.668 42.463 42.256 42.048
0.0 0.25 0.5 0.75 1.0
0 350 700 1050 1400
1400 1050 700 350 0
600 600 600 600 600
2600 2250 1900 1550 1200
44.292 44.054 43.813 43.569 43.322
X
1p
0.3
0.0 0.25 0.5 0.75 1.0
0.5
0.7
Pb0
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systems, serious problems appear in the construction of Voronoı¨ network and in procedures eliminating short edges and small faces. In such cases, cation–anion ring analysis seems to be an ideal tool for characterising the medium-range order. The medium-range order was studied mainly via cation–anion ring analysis performed with a new, highly efficient, redundancy-aware algorithm. The algorithm is based on the communication network simulation proposed for an efficient and exact solution of the Ring Perception Problem [53]. In our calculations, a new approach was used [54,55], viz. the pre-filtering technique. The structure of the determined basal rings was investigated using the ANELLI programme package [56–58].
3. Results of the XAS experiment 3.1. XANES spectra analysis In Fig. 1 we present the X-ray absorption spectra collected at the Pb L3-edge for a series of modified lead-silicate glasses. In order to show better the influence of modification time on the spectraÕs shape, the spectra of unmodified glass (0 h) [14] and of pure crystalline Pbfcc [59] have been added. It is apparent that for each composition (x = 0.3, 0.5, 0.7) annealing in hydrogen atmosphere changes the absorption spectra. At subsequent stages of reduction the spectra become more and more similar to the spectrum of pure lead. The rate of change strongly depends on the lead oxide content. For x = 0.3 and 0.5, the first evidence of glass modification become apparent after about 70 h and 24 h, respectively. The first peak becomes less intensive, and its maximum shifts towards lower energies by 1.5 eV in comparison with unmodified glass (the 0 h curve in the figure), and the second peak also become flatter and
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wider. In the x = 0.5 glass, 70 h reduction modifies the photoabsorber neighbourhood completely; the L3 Pb spectrum of the glass resembles the spectrum of pure crystalline Pb, with its characteristic pre-peak and two peaks in the 13 080–13 130 eV range. In the case of the 0.7PbO 0.3SiO2 glass, 24 h reduction suffices to obtain the Pb-like spectrum, and further reduction does not change its shape much. The above observations relative to the XANES region allow us to formulate the following conclusions: • A metallic lead phase appears in the glasses as a result of annealing in hydrogen atmosphere. • In the glasses with a low lead oxide content, even after long reduction times, the ratio of the number of lead atoms remaining in the bulk of the metallic phase to the number of other lead atoms (i.e. remaining in the lead oxide subsystem or laying on the surface of the metallic phase) remains very small. • For x = 0.5 and x = 0.7 glasses, most Pb atoms gain a full fcc Pb neighbourhood after less than 70 h and 24 h of reduction, respectively. It follows from the above analysis that the XAFS signal from modified lead-silicate glasses can be considered as a superposition of signals from Pb atoms of neighbourhoods similar to those found in a glass and from Pb atoms remaining in the metallic phase (fcc-like coordination): amod ðEÞ ¼ p aunmod ðEÞ þ ð1 pÞ aPbfcc ðEÞ;
ð3Þ
where parameter p measures the contribution due to the unmodified phase. The results of fitting the model spectra given by Eq. (3) to the experimental ones in the XANES region are shown in Fig. 2, where the best fit values of p are also shown. The fit is not always satisfactory. This may be due to the presence of PbOn with n < 4 or to
Fig. 1. X-ray absorption spectra in the XANES region for reduced lead-silicate glasses in the function of glass composition, x, and reduction time, tred (Tred = 400 C).
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Fig. 2. Comparison of the experimental XANES spectra of reduced x[(1 p)Pb pPbO] (1 x)SiO2 glasses, aexp(E) (—), with the model spectra given by Eq. (3) (- - -): (a) x = 0.3; (b) x = 0.5; (c) x = 0.7.
the fact that metallic granules embedded in a glass matrix are finite in size and that surface atoms have oxygen atoms in their neighbourhood. Since neither of these cases has been taken into account in model (3), (1 p) estimates the number of Pb atoms remaining in the bulk of granules only. However, (1 p) can still be considered a rough estimation of the level of glass modification. 3.2. EXAFS spectra analysis Basing on the XANES analysis results, a detailed investigation has been undertaken of the local neighbourhood of lead atoms in modified x[(1 p)Pb pPbO]
(1 x)SiO2 glasses. In the GNXAS analysis, two body configurations were taken into account. A theoretical total signal was modelled as a sum of contributions from configurations characteristic of pure, crystalline PbO (c1ð2Þ ) and of crystalline Pbfcc ðc2ð2Þ Þ. The result of fitting the theoretical signal to the experimental one in the wave vector space is presented in Fig. 3, where two reference signals (from unmodified glasses and from pure Pbfcc) are also shown. Apparently, the influence of the reduction is also distinct in the EXAFS region. The changes in intensity, shapes and positions of oscillations are obviously related to the variations of the nearest neighbourhood of the photoabsorber, while the changes
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385
Fig. 3. The results of fitting the model EXAFS signal to the experimental one in modified xPbO (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7, and in crystalline (Pbfcc). (Points) experimental signal; (solid line) model signal.
of structural parameters offer insight into the structural evolution during the reduction process. Two-body Pb–O and Pb–Pb spatial distributions have been modelled using CF functions (see Eq. (2)). The obtained parameter values for all modified lead-silicate glasses are listed in Table 2. The accuracy of the mean distance between the photoabsorber and the near˚ . Parameters r2 and N est neighbour amounts to 0.01 A are determined with an error of about 10%. The accuracy of b determination is about 30%. A 1.5 h reduction in hydrogen atmosphere at the temperature of 400 C of the glass of the smallest contents of PbO (x = 0.3) results in only a slight increase of the r2 parameter. Thus, the only effect after such a short reduction time is an increase of the Pb–O distancesÕ dispersion. Lead atoms still belong mainly to PbO4 units (N = 3.84). The same situation persists over successive
22.5 h of reduction (a continuous increase of r2). However, the measurements performed on the sample submitted to 70 h reduction reveal a significant decrease in the number of oxygen neighbours (to about 2.5), which might suggest the appearance of PbO3 groups. Simultaneously, the Pb–Pb spatial correlation changes. The average Pb–Pb distance decreases, while the Pb–Pb coordination number increases. Both of these parameters undergo a jump in respect to their values in the sample after 24 h of reduction. The dispersion of the Pb–Pb distances increases with increasing time of reduction (up to ˚ 2 in the 70 h sample). 0.067 A In the 0.5PbO 0.5SiO2 glass, a 1.5 h reduction at T = 400 C also results only in an increase of r2. Further reduction causes greater and greater distance dispersion r2 and asymmetry of the distance distribution. The average Pb–O co-ordination number decreases to about 2.2
Table 2 Structural parameters characterizing the nearest neighbourhood of lead atoms in reduced xPbO (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7, obtained from GNXAS analysis (Tred = 400 C) ˚] ˚] ˚ 2] x [mol] tred [h] Ro [A R [A r2 [A b N 0.3
0.5
0.7
Pbfcc
0 1.5 24 70 0 1.5 24 70 0 1.5 24 70
Pb–O
Pb–Pb
Pb–O
Pb–Pb
Pb–O
Pb–Pb
Pb–O
Pb–Pb
Pb–O
Pb–Pb
2.31 2.27 2.25 2.26 2.30 2.27 2.25 2.25 2.30 2.26 2.25 2.25
– 3.76 3.75 3.54 – 3.82 3.58 3.52 – 3.89 3.51 3.51
2.40 2.40 2.40 2.42 2.39 2.39 2.40 2.45 2.38 2.43 2.46 2.46
– 3.90 3.87 3.67 – 3.91 3.71 3.63 – 4.04 3.64 3.61
0.028 0.035 0.041 0.051 0.022 0.031 0.041 0.065 0.020 0.045 0.066 0.065
– 0.046 0.059 0.067 – 0.058 0.077 0.055 – 0.070 0.064 0.048
0.90 1.20 1.30 1.16 1.00 1.12 1.46 1.28 1.00 1.46 1.51 1.40
– 0.90 0.80 0.80 – 0.50 0.63 0.69 – 0.96 0.84 0.67
4.05 3.84 3.73 2.49 4.05 3.77 2.21 1.63 4.10 3.55 1.52 1.67
– 2.80 3.51 8.02 – 4.88 7.53 11.12 – 6.40 11.47 11.56
–
3.47
–
3.49
–
0.019
–
0.11
–
12.00
2
Ro is the most probable distance of the first-shell Pb–O and Pb–Pb correlations; R, r , b and N – as described in Section 2.2, see Eq. (2).
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after 24 h and to about 1.6 after 70 h. This means that many Pb–On units present in the unmodified glass disappear during the reduction process. Simultaneously, RPb–Pb and Ro,Pb–Pb decrease, while the Pb–Pb average co-ordination number increases approaching the nearest neighbour distance and co-ordination number of fcc ˚ and 12.0, respectively). The r2 parameter lead (3.47 A of Pb–Pb correlation assumes its maximum value, ˚ 2, after 24 h of reduction, and decreases to 0.077 A ˚ 0.055 A2 at the end of the modification process. This suggests that the reduction process was completed after a time shorter than 70 h, and later an ordering of the Pb neutral atoms into crystalline granules took place. As follows from Table 2, the same scenario applies to the x = 0.7 glass.
4. Molecular dynamics simulations A series of molecular dynamics simulations of x[(1 p)Pb pPbO] (1 x)SiO2, x = 0.3, 0.5, 0.7 systems, with p = 0.0, 0.25, 0.5, 0.75, and 1.0 have been performed. First, a discussion of changes of the environments of Pb+2 and Si+4 ions will be presented (Section 4.1). Then the granular structure of neutral Pb atoms will be described (Section 4.2). 4.1. Neighbourhood of Pb+2 and Si+4 cations Fig. 4 shows the spatial Pb+2–O2 correlations versus x and (1 p). For each x, the increasing reduction degree results in variations of the gPbþ2 O2 ðrÞ shape. Please note that the changes for x = 0.7 have a slightly different character than those observed for x = 0.3 and 0.5. In the latter cases the disorder around Pb+2 ions increases with progressing reduction (the first PDF peak flattens and the most probable Pb+2–O2 distance, Ro, moves towards lower values of r), while in the x = 0.7 case the lead ion environment changes less significantly and in a different way (the height of the first PDF peak in˚ ). creases and its position remains fixed at Ro 2.32 A
These differences can be related to the role played by lead oxide in glasses of different composition [7–9]. For x 6 0.5, lead oxide is a glass modifier and does not form a continuous network. For greater x, PbO becomes a glass former and thus still forms a continuous network in partially reduced glasses (together with silica), in which neutral Pb clusters are located. Detailed structural data for the Pb+2–O2 correlation (Ro, R, r2, b and N) are presented in Table 3. The parameter values confirm the previous observations. It should be noted, however, that for all partially reduced x = 0.3 glasses and for the x = 0.5 glass at the reduction degrees equal to (1 p) = 0.5 and 0.75, as well as for the x = 0.7 glass at the reduction degree equal to (1 p) = 0.75, the asymmetry of the first peak cannot be described appropriately with the aid of a single value of b, and the second CF function must be used (gAB(r) = g1,AB(r) + g2,AB(r), parameters of the subsequent subshell are indicated by 1 and 2, see e.g. Table 3). In x 6 0.5 cases, increasing (1 p) is accompanied by an increase of the Pb+2–O2 disorder (distance dispersion and peak assymetry parameters increase gradually for both of the obtained subshells). For x = 0.3, the mean distance to the oxygen neighbours from both subshells increases. The related lead-oxygen co-ordination number for the closer subshell increases (up to about 3.0), whereas it decreases for the more distant subshell (down to about 1.8). For x = 0.5, the first subshell R1 ˚ , with a simultaneous change value decreases by 0.07 A of N1 (from 4.0 to 2.0). At the same time R2 and N2 values increase. Finally, the structural parameters are practically constant in the function of (1 p) for the x = 0.7 glass. The local Pb+2–O2 ordering remains constant at least up to the reduction degree (1 p) = 0.5. However, at (1 p) = 0.75 the correlation asymmetry increases, which means that, although the lead ions are always surrounded by 4 oxygen atoms, one of them is usually more ˚ ) from the Pb ion than the others. distant (by 0.50 A Fig. 5(a) illustrates how the Si+4–O2 pair correlation evolves with the increasing reduction degree. First of all,
Fig. 4. Pb+2–O2 PDF functions in x[(1 p)Pb pPbO] (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7, in the function of the reduction degree (1 p), obtained in MD simulations.
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Table 3 MD-extractcd structural parameters, describing the shape of the first Pb+2–O2 peaks in partially reduced lead-silicate glasses ˚] ˚] ˚ 2] ˚] ˚ 2] R1 [A r21 [A b1 N1 R2 [A r22 [A b2 x [mol] 1p Ro [A 0.3
0.5
0.7
N2
0.0 0.25 0.50 0.75
2.32 2.31 2.31 2.31
2.42 2.31 2.37 2.45
0.026 0.009 0.019 0.044
0.83 0.35 0.63 0.98
3.85 1.49 2.69 3.04
2.58 2.86 2.91
0.052 0.077 0.12
0.80 1.02 1.28
3.00 2.58 1.77
0.0 0.25 0.50 0.75
2.33 2.33 2.32 2.31
2.41 2.40 2.36 2.34
0.020 0.019 0.015 0.023
0.83 0.71 0.46 0.89
4.10 3.85 3.16 2.06
2.78 2.83
0.053 0.18
1.00 1.40
1.39 2.77
0.0 0.25 0.50 0.75
2.32 2.32 2.32 2.31
2.39 2.38 2.39 2.36
0.017 0.017 0.018 0.018
0.83 0.72 0.73 0.50
4.00 3.91 3.89 3.29
2.85
0.061
1.10
0.98
Ro is the most probable Pb+2–O2 distance; R1, r21 , b1 and N1 are the first subshellÕs structural parameters; R2, r22 , b2 and N2 are the second subshellÕs structural parameters.
the dispersion of Si+4–O2 distances decreases. Moreover, in the x = 0.7 glass, the most probable Si+4–O2 distance, Ro, increases with increasing (1 p) values. Simultaneously, for O2–O2 spatial correlations (Fig. 5(b)), the changes are most distinct for x = 0.7, and with increasing (1 p) both distance dispersion and distribution asymmetry increase. In order to follow the evolution of structural units in partially reduced glasses, distributions of the co-ordination numbers around Pb+2 and Si+4 ions have been calculated. The shape of dominating structural units has also been analysed.
It follows from the analysis that, independently of the initial contents of lead oxide in the glasses, the increase of reduction degree weakens the dominating role of basic structural units (i.e. of PbO4 and PbO5 groups for x 6 0.5, and of PbO4 groups for x = 0.7), increasing the fractions of PbO3 and PbO6 units. At lower PbO contents, say x 6 0.5, the fraction of PbO5 groups significantly decreases with increasing (1 p), while the fraction of PbO4 groups remains almost constant. Moreover, among PbO4 groups, independently of glass composition, the relative contribution of pyramids (where all four oxygen neighbours lie on the same side of a
Fig. 5. MD-simulated pair distribution functions: (a) Si+4–O2 and (b) O2–O2, in x[(1 p)Pb pPbO] (1 x)SiO2, x = 0.3, 0.5, 0.7 glasses in the function of the reduction degree, (1 p).
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Table 4 Occurrence of tetrahedron-like and pyramid-like PbO4 groups in partially reduced MD-simulated x[(1 p)Pb pPbO] (1 x)SiO2 glasses 1p
0.00 0.25 0.50 0.75
x = 0.3
x = 0.5
x = 0.7
Pyramid
Tetrahedron
Pyramid
Tetrahedron
Pyramid
Tetrahedron
[%] 27.23 34.00 60.00 57.14
[%] 72.77 66.00 40.00 42.86
[%] 19.03 26.20 30.65 54.88
[%] 80.97 73.80 69.35 45.12
[%] 6.39 12.87 19.88 28.15
[%] 93.61 87.13 80.12 71.85
plane containing the Pb ion) increases at the cost of a decreasing contribution of tetrahedral groups (see Table 4). Our analysis of the Pb+2–O2 . . . and the Si+4–O2 . . . rings reveals a distinct influence of the reduction process on the medium-range structure of the considered glasses. General trends, however, are weakly dependent on glass composition. PbO: At low lead oxide contents, connectivity of the Pb+2–O2 subnetwork decreases and more and more bifurcated linear chains appear even at early reduction stages (i.e. at (1 p) = 0.25). In the x = 0.5 glass, significant changes in the number and size of connected clusters and in the numbers of Pb+2 and O2 atoms not belonging to any ring appear only at the (1 p) = 0.75 reduction degree. Finally, even at the highest reduction degree over 85% of all adjacent Pb+2 and O2 atoms in the x = 0.7 glass form one connected cluster, and 70% of the atoms belong to rings. Moreover, the fraction of 2-member Pb+2–O2 rings increases with increasing reduction degree for all the considered compositions. In other words, the number of edge-sharing PbOn units increases. SiO2: The distribution of n-member Si+4–O2 . . . rings in the function of glass composition and reduction degree reveals a strong tendency of the SiO2 subsystem to form a relaxed (pure silicalike), continuous 3D network. The more reduced glasses, the greater the contribution of 5- and 6member rings and the lesser the contribution of rings shorter than 4 and longer than 7.
4.2. Local ordering around neutral Pb atoms Fig. 6 presents snapshots of atomic configurations at the last simulation step for the 0.3[(1 p)Pb pPbO] 0.7SiO2 glass. Even this simple visualisation shows a strong and progressive tendency to agglomeration of neutral Pb atoms. Fig. 7 shows the structure around neutral Pb atoms viewed by partial pair distribution functions, where an increase of the reduction degree is followed by changes
in the Pb0–Pb0 spatial correlation function. Decomposition of gPb0 Pb0 ðrÞs into CF functions (Fig. 7 and Table 5) reveals that for the x = 0.3 glass at any stage of reduction and for the x = 0.5 glass of (1 p) 6 0.5 the Pb0– Pb0 correlation can be described by two co-ordination subshells, with the average interatomic distances of ˚ and 3.60 A ˚. 3.05 A The first Pb0–Pb0 co-ordination subshell is related to the existence of small lead clusters (dimers, trimers and tetramers). In our simulations the interatomic distance ˚ . This value is in such clusters is equal to about 3.1 A compatible with the radius of the lead atom, calculated ˚ [60] or experimentally numerically to be rat = 1.54 A ˚ [61,62]. Also investigations on determined to be 1.47 A Pb dimers, trimers and tetramers on Ge, Si or Cu sur˚ or faces have yielded Pb0–Pb0 distances of 3.16 A ˚ [63,64]. As follows from Fig. 7 such small Pb0 3.30 A clusters are dominant in x = 0.3 and 0.5 glasses at lower reduction degrees (e.g. over 65% in a glass with x = 0.3 and (1 p) = 0.25).
5. Granular structure viewed by coupled EXAFS/MD methods The EXAFS method cannot distinguish between neutral Pb0 atoms and Pb+2 ions. This means that Pb–O correlations extracted from an EXAFS analysis of partially reduced glasses contain contributions from oxygen neighbours of lead ions built in the PbOSiO2 network (Pb+2–O2) and from neutral Pb0 atoms not completely surrounded by other Pb0 atoms, thus remaining in contact with oxygen ions built in the network (Pb0–O2 contribution). However, as follows from Fig. 8(a), the contribution of Pb0–O2 pairs to the EXAFS Pb–O signal is expected to be negligible (cf. RPb–O and r2 values from Table 2), and the EXAFS-extracted Pb–O correlation can be considered a Pb+2–O2 correlation. Similarly, the EXAFS-extracted Pb–Pb spatial correlation contains contributions from Pb0–Pb0, Pb0–Pb+2, Pb+2–Pb0 and Pb+2–Pb+2 pairs. For the above spatial correlations we have the following parameters: ˚ , N = 12.0; • Pb0–Pb0 in pure crystalline Pb: R0 = 3.49 A 0 +2 +2 0 • Pb –Pb and Pb –Pb in our simulated glasses: ˚ , N 1.0 (Fig. 8(b)); Ro = 3.30–3.80 A
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Fig. 6. Atomic configurations at the last simulation step for the x = 0.3 glass, (a) unreduced glass: small balls – Si+4 and O2 ions, large balls – Pb+2; (b)–(d) partially reduced glasses, only Pb0 atoms are shown; (e) and (f) completely reduced glass: small balls – Si+4 and O2 ions, large balls – Pb0.
˚, • Pb+2–Pb+2 in crystalline red PbO: Ro,1 = 3.69 A ˚ N1 = 4.0 i Ro,2 = 3.85 A, N2 = 4.0; in partially ˚ and N reduced MD-simulated glasses: Ro = 3.44 A varies from 2.0 to 4.5 depending on glass composition.
These distributions overlap, which was taken into account when the MD and EXAFS data were compared. In Fig. 9(a) we present the result of an exemplary comparison, namely of the Pb–Pb co-ordination numbers obtained from MD simulations and EXAFS
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Fig. 7. Partial Pb0–Pb0 pair distribution functions in simulated x[(1 p)Pb pPbO] (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7 in the function of the reduction degree, (1 p). (r) MD simulation, (—) total model function, (- - -) CF components (subshells).
Table 5 Structural parameters describing the first Pb0–Pb0 peaks in MD-simulated reduced glasses ˚] ˚] ˚ 2] x [mol] 1p Ro [A R1 [A r21 [A b1 N1
˚] R2 [A
˚ 2] r22 [A
b2
N2
0.3
0.25 0.50 0.75 1.0
3.00 3.19 3.15 3.23
3.02 3.14 3.09 3.09
0.030 0.036 0.038 0.051
0.46 0.39 0.82 1.01
2.07 2.37 1.84 0.98
3.59 3.57 3.52 3.52
0.130 0.120 0.140 0.150
0.80 0.79 1.17 1.07
1.59 4.51 6.98 8.61
0.5
0.25 0.50 0.75 1.0
3.18 3.29 3.30 3.32
3.18 2.90 3.61 3.63
0.051 0.005 0.190 0.195
0.70 0.15 1.15 1.11
1.79 0.18 9.93 10.33
3.65 3.61
0.190 0.200
1.03 1.21
3.41 8.61
0.7
0.25 0.50 0.75 1.0
3.52 3.50 3.47 3.45
4.06 3.92 3.93 3.92
0.473 0.351 0.360 0.356
1.22 1.11 1.23 1.26
7.86 9.41 10.61 11.08
3.47
3.49
0.019
0.11
12.00
Pbfcc 0
0
Ro the most probable Pb –Pb distance; structural parameters.
R2, r21 ,
b1 and N1 are the first subshellÕs structural parameters; R1, r22 , b2 and N2 are the second subshellÕs
analysis ((1 p) values for EXAFS data are taken from XANES analysis). Analogical comparisons have been performed for other structural parameters (RPb–O, NPb–O and RPb–Pb) and similarly good agreement has been obtained. Basing on this agreement, we have tried to correlate directly the reduction time with the reduction degree (see Fig. 9(b)). The former is exactly controlled during the experiment, while the latter is precisely known in the simulations, and can be simultaneously estimated from Eq. (3) (XANES analysis). A pair-wise comparison of the corresponding characteristics obtained from the XANES analysis and those
obtained from the coupled EXAFS/MD method leads to the conclusion that both methods give very similar estimations of the reduction degree (p parameters do not differ by more than 5%). It is also interesting to relate the technological parameters (glass composition, duration of the reduction process) to the average size of metallic clusters appearing in the structure. To this end, we calculated the N (average co-ordination number) to N0 (ideal fcc co-ordination number, i.e. 12) ratios for fcc spherical clusters of various radii as reference data (the right-hand panels of Fig. 10, ÔModel PbfccÕ). Then, assuming the spherical shape and
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Fig. 8. MD-simulated pair distribution functions: (a) Pb0–O2 and (b) Pb0–Pb+2, in x[(1 p)Pb pPbO] (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7, in the function of the reduction degree, (1 p).
Fig. 9. (a) A comparison of the co-ordination numbers obtained from the EXAFS analysis and MD simulations in the function of x and (1 p) (all Pb–Pb correlations). (b) Dependence between p and tred values for all the compositions and stages of reduction, obtained using the combined EXAFS and MD method. For the sake of comparison, the XANES-extracted data are also presented (- - -).
fcc structure of granules in the glasses, the average size of granules in subsequent samples could be estimated. The corresponding construction is shown in Fig. 10. The Pb–Pb co-ordination number determined for modified glasses and normalised to N0 = 12 is shown on the vertical axes. On the horizontal axes of the left-hand figures, the reduction time and the reduction degree are shown, according to the method. The right-hand figures have the granule radius as an independent argument. The smallest fcc cluster (13 atoms) has the radius of ˚ , while the average Pb–Pb co-ordination number 3.5 A amounts to N = 5.462. It has thus been assumed that values of N/N0 lower than 5.462/12 0.45 correspond to dispersed lead atoms, not forming any granules. Basing on the dependences shown in Figs. 9 and 10, it is easy to note that such a situation takes place for the x = 0.3 glasses reduced for not longer than 50 h ((1 p) <
0.45) and for the x = 0.5 glasses reduced for less than 6 h ((1 p) < 0.15). One can read the average Pb granule radii appearing in our modified glasses, both real and simulated, from Fig. 10. A comparison of all the results reveals a distinct dependence between the initial glass composition, the reduction parameter (p or tred) and the average granule size. Moreover, the end of the reduction does not mean that big granules have been formed. Only a sufficiently long annealing time leads to the appearance of the final granular structure.
6. Concluding remarks In this paper we have presented the results of EXAFS and MD analyses of partially reduced lead-silicate glasses. The obtained experimental and numerical
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Fig. 10. Estimation of the average Pb cluster size: (a) from EXAFS experiment, (b) from MD simulation, N has been taken from CF approximation of g(r) for the Pb0–Pb0 pair.
results are compatible with each other and provide detailed information on the atomic-level structure around lead atoms, which can belong either to the oxide subsystem or to the metallic one. From our analysis of the glass structure it follows that:
7. Structural changes of the PbOSiO2 glass matrix are quite different for x 6 0.5 and 0.7 glasses, due to a different role played by PbO in low-PbO content systems (viz. glass modifier) and PbO-rich ones (viz. glass former).
1. Independently of the glass composition, the reduction process leads to the appearance of a metallic lead phase, in particular to the appearance of a lead nanogranular structure embedded in the glass matrix. 2. XANES and coupled EXAFS/MD analyses of partially reduced glasses give good estimations of the reduction degree and the average granule size. The estimated granule radius equals several nanometers. 3. With increasing lead oxide contents in the virgin glass, the susceptibility to reduction and agglomeration of neutral lead atoms increases. 4. No separation of the lead oxide and silica oxide phases has been observed in the matrix of partially reduced glasses. 5. A strong tendency of the SiO2 subsystem to form a relaxed (pure silica-like), continuous 3D network during the reduction should be noted. 6. The greater the reduction degree, the greater the contribution of PbO4 pyramids and PbO3 and PbO6 groups.
Good agreement between the EXAFS-extracted and MD-simulated structural parameters obtained for the considered set of lead-silicate glasses has been the basis for correlating certain parameters, which can be precisely (without error) determined from the experiment (reduction time) and from the simulation (the simulated glass composition). Such correlations will certainly be useful for the elaboration of a technology for the production of materials with pre-requested structural parameters.
Acknowledgments We gratefully acknowledge the support of the European Synchrotron Radiation Facility in providing synchrotron radiation facilities for the HS1663 experiment and would like to thank the BM29 staff members. We would also like to thank Dr Angela Trapananti and Dr Emiliano Principi for their assistance in the XAS measurements.
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The MD simulations have been performed at the TASK Computer Centre (Gdansk, Poland).
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Structure of partially reduced xPbO (1 x)SiO2 glasses: combined EXAFS and MD study Agnieszka Witkowska a
a,b,*
, Jarosław Rybicki
a,b
, Andrea Di Cicco
c
Department of Solid State Physics, Faculty of Technical Physics and Applied Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland b TASK Computer Centre, Narutowicza 11/12, 80-952 Gdansk, Poland c INFM Dipartimento di Fisica, Universita` di Camerino, via Madonna delle Carceri, I-62032 Camerino (MC), Italy Received 6 July 2003; received in revised form 2 November 2004
Abstract We have studied the structure of partially reduced lead-silicate glasses using combined EXAFS (extended X-ray absorption fine structure) and MD (molecular dynamics) methods. The analysis was performed for glasses of x[(1 p)Pb pPbO] (1 x)SiO2 composition, x = 0.3, 0.5, 0.7, where parameter (1 p) describes the degree of reduction, i.e. the content of the granular metallic phase, appearing as the result of the reduction process (e.g. annealing in hydrogen atmosphere). In the EXAFS experiment (1 p) was expressed via the time of reduction realized at 400 C (1.5 h, 24 h, 70 h), whereas in the MD simulations it was determined precisely by using proper numbers of particles (corresponding to (1 p) = 0.0, 0.25, 0.5, 0.75 and 1.0). In the paper we describe in detail the local structure around lead atoms and its changes in the function of glass composition and reduction degree. The tendency for agglomeration of Pb0 into clusters, the formation of the granular metallic phase, and continuity of silica and lead oxide subnetworks are discussed. A good agreement between EXAFS-extractcd and MD-extracted parameters of the short-range structure encouraged us to preform a medium-range order analysis, based on the MD simulations only. Moreover, combining the EXAFS and MD methods we could correlate the reduction time (technological parameter) with the degree of reduction (1 p) and the actual state of the granular structure. The latter relation may be useful for controlled production of reduced glasses of pre-requcstcd physical properties. 2005 Elsevier B.V. All rights reserved.
1. Introduction Lead-silicate glasses, due to their specific and very interesting properties, find a lot of industrial applications. As special materials, they are used in the production of plate image amplifiers and scintillators [1]. *
Corresponding author. Present address: Department of Solid State Physics, Faculty of Technical Physics and Applied Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland. E-mail address: [email protected] (A. Witkowska). 0022-3093/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2005.01.036
Lead-silicate glasses submitted to the reduction process (e.g. in hydrogen atmosphere), undergo dramatic changes of optical properties [2] and electrical surface conductivity, and reveal a very high secondary emission coefficient, thus finding applications in the production of electron channel multipliers [3]. The atomic structure of lead-silicate glasses has been investigated for sixty years, using various experimental techniques, including IR spectroscopy [4], Raman spectroscopy [4–6], NMR [5,7,8], XPS [9], X-ray [10,11], neutron diffraction methods [12,13], and EXAFS [8,14]. Computer simulations of their structure have also
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been performed [14–18]. However, as far as the authors are aware, no experimental work on the atomic-level structure of partially reduced glasses has been reported in the literature, other than our work [19] presenting preliminary results concerning lead-silicate glass of one composition only, x = 0.5. Some simulation results on completely reduced glasses (xPbO (1 x)SiO2) have been described in [20,21]. Recently, we have performed EXAFS (extended X-ray absorption fine structure) and MD (molecular dynamics) investigations of the structure of partially reduced lead-silicate glasses as a function of PbO content in the initially unreduced glass. These new results are described in the present paper. X-ray absorption spectroscopy (XAS) is a powerful structural technique that allows us to investigate the neighbourhood of a photoabsorber atom embedded in a condensed medium. The main and very important advantages of the XAS method in the case of multi-component systems are its atomic selectivity and very high sensitivity. In our work we have used the GNXAS package for EXAFS data analysis [22–24]. Molecular dynamics (MD) is a widely used and very convenient simulation method for investigation of material structures on the atomic level (e.g. [25–35]). The MD method gives trajectories of all the atoms—and thus the structure—in the short- and medium-range, can be analysed in detail, and its results can be compared with experimental data. Moreover, MD simulations offer insight into structural features inaccessible to experiment. A combined usage of EXAFS and MD methods, as has been shown in [14,36,37], is a source of detailed information on the atomic structure in disordered materials. The paper is organised as follows. In Section 2 we discuss technical issues related to sample preparation, the EXAFS measurement and the MD simulation technique. In Section 3 we present an EXAFS analysis of the short range structure around Pb atoms. Our MD simulations are presented in Section 4. In Section 5 the granular structure of reduced glasses is characterised using the combined EXAFS and MD methods. Section 6 contains concluding remarks.
2. Measurement and simulation techniques 2.1. Sample preparation and XAS measurements xPbO (1 x)SiO2, 0.3 6 x 6 0.7 glasses were prepared as follows. A mixture of powdered SiO2 and PbO was placed in a platinum crucible and heated during 3–4 h in an electric furnace from room temperature (RT) to melting temperature, ranging from 1100 C for x = 0.7 to 1350 C for x = 0.3. The melts were kept slightly above their melting temperature for several minutes and stirred mechanically in order to homogenize the
381
samples. Then the melts were rapidly cooled by pouring them onto a brass plate. The resulting glasses had uniform colour and were free of ceramic inclusions. No Bragg peaks have been identified by X-ray diffraction. The powdered samples (grain diameter of several lm) were then annealed in hot hydrogen atmosphere to permit their reduction. The reduction process was realised at the temperature of Tred = 400 C, for periods of tred = 1.5, 24, 70 h [38]. Next, the reduced glass powder was mixed with BN in optimal proportions, and pellets for XAS measurements were formed. The pelletsÕ mass, diameter and thickness amounted to 200 mg, 12 mm and 0.5–1 mm, respectively. Their homogeneity was being checked using an optical microscope. The spectra of the lead-silicate glasses at the Pb L3edge were recorded at the BM29 beam-line of the European Synchrotron Radiation Facility (Grenoble), using a double-crystal monochromator equipped with Si(3 1 1) crystals. The spectrum of solid Pb at the Pb L3-edge was collected at the D42 beam-line of the LURE laboratories (Orsay, Paris), equipped with an Si(331) channel-cut monochromator. The room temperature measurements were performed in the energy range 12 900–14 000 eV (the Pb L3-edge energy amounts to 13 035 eV). The sampling steps were equal to 3 eV and 0.4 eV in the 12 900–13 000 eV and 13 000–13 100 eV energy ranges, respectively. In the range between 13 100 eV and 14 000 eV the sampling step was increased from 0.4 to 2 eV. This scanning procedure yielded high quality data of both pre- and post-edge background analyses used for normalisation of the spectra. More details on the experimental setup can be found in [39]. Experimental data have been analysed within an advanced technique, using theoretical calculations of the X-ray absorption cross-section in the framework of the GNXAS method [22–24]. The method is based on the comparison in the energy space between the experimental signal and the theoretical one, attempting to optimise the relevant structural parameter values. The post-edge background was modelled with three splines of degree 3, 4, and 4 on the subsequent energy intervals: 13 080–13 300 eV, 13 300–13 500 eV, 13 500– 13 900 eV. The model absorption signal contained a contribution accounting for possible multielectron excitation channels (2p3/24f) and (2p3/24d). The importance of these contributions for the Pb L3-edge has been discussed in [40]. 2.2. MD simulations Our MD simulations were performed in the constant volume regime (NVE ensemble), using the MDSIM code [41]. The atoms were assumed to interact by the twobody Born–Mayer–Huggins potential, containing the Born–Mayer repulsive contribution, and Coulomb interactions. Null charge was assigned to neutral lead
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atoms, Pb0, while the remaining interaction parameters remained the same as for lead ions, Pb+2. Thus, the Pb0–X interactions, X = Pb0, Pb+2, Si+4 and O2, contain only repulsive contributions, while Pb0–Pb0 and Pb0–O2 interactions are strictly packing in character. Moreover, the polarizability of lead atoms and ions has been neglected, and no charge transfer has been allowed for. Nevertheless, despite the simplicity of the applied force-field with the potential parameter taken from [15], the simulation reproduces quite well the experimental interatomic distances and co-ordination numbers, even the shape of the first peak of the partial pair distribution functions (see e.g. [14,16,17,42]). Therefore, the results of our simulations can be safely used as the first guess of GNXAS local structure refinement procedures for partially reduced glasses. In this paper we report on simulations of partially reduced lead-silicate glasses of x[(1 p)Pb pPbO] (1 x)SiO2 composition, x = 0.3, 0.5, 0.7, and (1 p) = 0.0, 0.25, 0.5, 0.75 and 1.0. Numbers of atoms and edge lengths of the cubic simulation boxes are given in Table 1. The sample was initially prepared in a well-equilibrated molten state at 10 000 K, then cooled at the average rate of 2 · 1013 K/s down to 300 K, passing equilibrium states at 8000 K, 6000 K, 5000 K, 4000 K, 3000 K, 2500 K, 2000 K, 1500 K, 1000 K, and 600 K. Temperature scaling was applied whenever the rolling average of temperature (calculated over last 100 time steps) exceeded the (T DT, T + DT) interval. At each intermediate temperature the system was equilibrated during 30 000 fs time steps, using DT = 100 K for T P 1000 K, DT = 20 K for T = 600 K and DT = 10 K for T = 300 K. Equilibrated systems were sampled during 10 000 time steps Dt, Dt = 1015 s. Structural information on short-range correlations was obtained in a conventional way, mainly from the
partial pair and angular distribution functions (pPDFs and ADFs). As it has been shown in a number of papers, pPDFs can be decomposed in disordered systems into a short-range peak of well-defined shape and a long-range tail [43,44]. A simple Gaussian shape of the short-range peak of a pPDF is usually insufficient to describe accurately the short-range ordering of highly disordered systems. A useful parameterisation of the first partial PDF peak, as shown in [44,45], has the form: gAB ðrÞ ¼
1 pAB ðrÞ ; qB 4pr2
ð1Þ
where qB is the atomic density of B-neighbours, and pAB(r) is the bond length probability density of finding atoms B within shell r and r + dr around atoms A, described by a CF distribution. The corresponding formula, valid for (r R)b > 2r, reads as follows [23,46–48]: 4 1 2N 4 2ðr RÞ b2 þ pAB ðrÞ ¼ rb rjbjCð4=b2 Þ b2 4 2ðr RÞ exp 2 þ : ð2Þ rb b Here, R is the average first-shell A–B distance, r2 is the distance variance (a Debye–Waller-like parameter), b is the asymmetry (skewness) parameter, N is the co-ordination number, and C(x) is the Euler C-function. In order to describe the second and further co-ordination shells, i.e. to describe the medium-range order, instead of analysing PDFs and ADFs, one should use more advanced methods of structural analysis. One of the possible approaches is to analyse properly constructed clusters of edge and/or face-sharing Voronoı¨ polyhedra [49–51]. This method, although very efficient in detecting crystalline regions of various symmetries [52], works well for close-packed systems. In open
Table 1 Numbers of atoms used in the simulations and the corresponding edge lengths of the simulation box Pb+2
Si+4
O2
˚] L [A
0 150 300 450 600
600 450 300 150 0
1400 1400 1400 1400 1400
3400 3250 3100 2950 2800
42.731 42.574 42.415 42.256 42.095
0.0 0.25 0.5 0.75 1.0
0 250 500 750 1000
1000 750 500 250 0
1000 1000 1000 1000 1000
3000 2750 2500 2250 2000
42.870 42.668 42.463 42.256 42.048
0.0 0.25 0.5 0.75 1.0
0 350 700 1050 1400
1400 1050 700 350 0
600 600 600 600 600
2600 2250 1900 1550 1200
44.292 44.054 43.813 43.569 43.322
X
1p
0.3
0.0 0.25 0.5 0.75 1.0
0.5
0.7
Pb0
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systems, serious problems appear in the construction of Voronoı¨ network and in procedures eliminating short edges and small faces. In such cases, cation–anion ring analysis seems to be an ideal tool for characterising the medium-range order. The medium-range order was studied mainly via cation–anion ring analysis performed with a new, highly efficient, redundancy-aware algorithm. The algorithm is based on the communication network simulation proposed for an efficient and exact solution of the Ring Perception Problem [53]. In our calculations, a new approach was used [54,55], viz. the pre-filtering technique. The structure of the determined basal rings was investigated using the ANELLI programme package [56–58].
3. Results of the XAS experiment 3.1. XANES spectra analysis In Fig. 1 we present the X-ray absorption spectra collected at the Pb L3-edge for a series of modified lead-silicate glasses. In order to show better the influence of modification time on the spectraÕs shape, the spectra of unmodified glass (0 h) [14] and of pure crystalline Pbfcc [59] have been added. It is apparent that for each composition (x = 0.3, 0.5, 0.7) annealing in hydrogen atmosphere changes the absorption spectra. At subsequent stages of reduction the spectra become more and more similar to the spectrum of pure lead. The rate of change strongly depends on the lead oxide content. For x = 0.3 and 0.5, the first evidence of glass modification become apparent after about 70 h and 24 h, respectively. The first peak becomes less intensive, and its maximum shifts towards lower energies by 1.5 eV in comparison with unmodified glass (the 0 h curve in the figure), and the second peak also become flatter and
383
wider. In the x = 0.5 glass, 70 h reduction modifies the photoabsorber neighbourhood completely; the L3 Pb spectrum of the glass resembles the spectrum of pure crystalline Pb, with its characteristic pre-peak and two peaks in the 13 080–13 130 eV range. In the case of the 0.7PbO 0.3SiO2 glass, 24 h reduction suffices to obtain the Pb-like spectrum, and further reduction does not change its shape much. The above observations relative to the XANES region allow us to formulate the following conclusions: • A metallic lead phase appears in the glasses as a result of annealing in hydrogen atmosphere. • In the glasses with a low lead oxide content, even after long reduction times, the ratio of the number of lead atoms remaining in the bulk of the metallic phase to the number of other lead atoms (i.e. remaining in the lead oxide subsystem or laying on the surface of the metallic phase) remains very small. • For x = 0.5 and x = 0.7 glasses, most Pb atoms gain a full fcc Pb neighbourhood after less than 70 h and 24 h of reduction, respectively. It follows from the above analysis that the XAFS signal from modified lead-silicate glasses can be considered as a superposition of signals from Pb atoms of neighbourhoods similar to those found in a glass and from Pb atoms remaining in the metallic phase (fcc-like coordination): amod ðEÞ ¼ p aunmod ðEÞ þ ð1 pÞ aPbfcc ðEÞ;
ð3Þ
where parameter p measures the contribution due to the unmodified phase. The results of fitting the model spectra given by Eq. (3) to the experimental ones in the XANES region are shown in Fig. 2, where the best fit values of p are also shown. The fit is not always satisfactory. This may be due to the presence of PbOn with n < 4 or to
Fig. 1. X-ray absorption spectra in the XANES region for reduced lead-silicate glasses in the function of glass composition, x, and reduction time, tred (Tred = 400 C).
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Fig. 2. Comparison of the experimental XANES spectra of reduced x[(1 p)Pb pPbO] (1 x)SiO2 glasses, aexp(E) (—), with the model spectra given by Eq. (3) (- - -): (a) x = 0.3; (b) x = 0.5; (c) x = 0.7.
the fact that metallic granules embedded in a glass matrix are finite in size and that surface atoms have oxygen atoms in their neighbourhood. Since neither of these cases has been taken into account in model (3), (1 p) estimates the number of Pb atoms remaining in the bulk of granules only. However, (1 p) can still be considered a rough estimation of the level of glass modification. 3.2. EXAFS spectra analysis Basing on the XANES analysis results, a detailed investigation has been undertaken of the local neighbourhood of lead atoms in modified x[(1 p)Pb pPbO]
(1 x)SiO2 glasses. In the GNXAS analysis, two body configurations were taken into account. A theoretical total signal was modelled as a sum of contributions from configurations characteristic of pure, crystalline PbO (c1ð2Þ ) and of crystalline Pbfcc ðc2ð2Þ Þ. The result of fitting the theoretical signal to the experimental one in the wave vector space is presented in Fig. 3, where two reference signals (from unmodified glasses and from pure Pbfcc) are also shown. Apparently, the influence of the reduction is also distinct in the EXAFS region. The changes in intensity, shapes and positions of oscillations are obviously related to the variations of the nearest neighbourhood of the photoabsorber, while the changes
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385
Fig. 3. The results of fitting the model EXAFS signal to the experimental one in modified xPbO (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7, and in crystalline (Pbfcc). (Points) experimental signal; (solid line) model signal.
of structural parameters offer insight into the structural evolution during the reduction process. Two-body Pb–O and Pb–Pb spatial distributions have been modelled using CF functions (see Eq. (2)). The obtained parameter values for all modified lead-silicate glasses are listed in Table 2. The accuracy of the mean distance between the photoabsorber and the near˚ . Parameters r2 and N est neighbour amounts to 0.01 A are determined with an error of about 10%. The accuracy of b determination is about 30%. A 1.5 h reduction in hydrogen atmosphere at the temperature of 400 C of the glass of the smallest contents of PbO (x = 0.3) results in only a slight increase of the r2 parameter. Thus, the only effect after such a short reduction time is an increase of the Pb–O distancesÕ dispersion. Lead atoms still belong mainly to PbO4 units (N = 3.84). The same situation persists over successive
22.5 h of reduction (a continuous increase of r2). However, the measurements performed on the sample submitted to 70 h reduction reveal a significant decrease in the number of oxygen neighbours (to about 2.5), which might suggest the appearance of PbO3 groups. Simultaneously, the Pb–Pb spatial correlation changes. The average Pb–Pb distance decreases, while the Pb–Pb coordination number increases. Both of these parameters undergo a jump in respect to their values in the sample after 24 h of reduction. The dispersion of the Pb–Pb distances increases with increasing time of reduction (up to ˚ 2 in the 70 h sample). 0.067 A In the 0.5PbO 0.5SiO2 glass, a 1.5 h reduction at T = 400 C also results only in an increase of r2. Further reduction causes greater and greater distance dispersion r2 and asymmetry of the distance distribution. The average Pb–O co-ordination number decreases to about 2.2
Table 2 Structural parameters characterizing the nearest neighbourhood of lead atoms in reduced xPbO (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7, obtained from GNXAS analysis (Tred = 400 C) ˚] ˚] ˚ 2] x [mol] tred [h] Ro [A R [A r2 [A b N 0.3
0.5
0.7
Pbfcc
0 1.5 24 70 0 1.5 24 70 0 1.5 24 70
Pb–O
Pb–Pb
Pb–O
Pb–Pb
Pb–O
Pb–Pb
Pb–O
Pb–Pb
Pb–O
Pb–Pb
2.31 2.27 2.25 2.26 2.30 2.27 2.25 2.25 2.30 2.26 2.25 2.25
– 3.76 3.75 3.54 – 3.82 3.58 3.52 – 3.89 3.51 3.51
2.40 2.40 2.40 2.42 2.39 2.39 2.40 2.45 2.38 2.43 2.46 2.46
– 3.90 3.87 3.67 – 3.91 3.71 3.63 – 4.04 3.64 3.61
0.028 0.035 0.041 0.051 0.022 0.031 0.041 0.065 0.020 0.045 0.066 0.065
– 0.046 0.059 0.067 – 0.058 0.077 0.055 – 0.070 0.064 0.048
0.90 1.20 1.30 1.16 1.00 1.12 1.46 1.28 1.00 1.46 1.51 1.40
– 0.90 0.80 0.80 – 0.50 0.63 0.69 – 0.96 0.84 0.67
4.05 3.84 3.73 2.49 4.05 3.77 2.21 1.63 4.10 3.55 1.52 1.67
– 2.80 3.51 8.02 – 4.88 7.53 11.12 – 6.40 11.47 11.56
–
3.47
–
3.49
–
0.019
–
0.11
–
12.00
2
Ro is the most probable distance of the first-shell Pb–O and Pb–Pb correlations; R, r , b and N – as described in Section 2.2, see Eq. (2).
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after 24 h and to about 1.6 after 70 h. This means that many Pb–On units present in the unmodified glass disappear during the reduction process. Simultaneously, RPb–Pb and Ro,Pb–Pb decrease, while the Pb–Pb average co-ordination number increases approaching the nearest neighbour distance and co-ordination number of fcc ˚ and 12.0, respectively). The r2 parameter lead (3.47 A of Pb–Pb correlation assumes its maximum value, ˚ 2, after 24 h of reduction, and decreases to 0.077 A ˚ 0.055 A2 at the end of the modification process. This suggests that the reduction process was completed after a time shorter than 70 h, and later an ordering of the Pb neutral atoms into crystalline granules took place. As follows from Table 2, the same scenario applies to the x = 0.7 glass.
4. Molecular dynamics simulations A series of molecular dynamics simulations of x[(1 p)Pb pPbO] (1 x)SiO2, x = 0.3, 0.5, 0.7 systems, with p = 0.0, 0.25, 0.5, 0.75, and 1.0 have been performed. First, a discussion of changes of the environments of Pb+2 and Si+4 ions will be presented (Section 4.1). Then the granular structure of neutral Pb atoms will be described (Section 4.2). 4.1. Neighbourhood of Pb+2 and Si+4 cations Fig. 4 shows the spatial Pb+2–O2 correlations versus x and (1 p). For each x, the increasing reduction degree results in variations of the gPbþ2 O2 ðrÞ shape. Please note that the changes for x = 0.7 have a slightly different character than those observed for x = 0.3 and 0.5. In the latter cases the disorder around Pb+2 ions increases with progressing reduction (the first PDF peak flattens and the most probable Pb+2–O2 distance, Ro, moves towards lower values of r), while in the x = 0.7 case the lead ion environment changes less significantly and in a different way (the height of the first PDF peak in˚ ). creases and its position remains fixed at Ro 2.32 A
These differences can be related to the role played by lead oxide in glasses of different composition [7–9]. For x 6 0.5, lead oxide is a glass modifier and does not form a continuous network. For greater x, PbO becomes a glass former and thus still forms a continuous network in partially reduced glasses (together with silica), in which neutral Pb clusters are located. Detailed structural data for the Pb+2–O2 correlation (Ro, R, r2, b and N) are presented in Table 3. The parameter values confirm the previous observations. It should be noted, however, that for all partially reduced x = 0.3 glasses and for the x = 0.5 glass at the reduction degrees equal to (1 p) = 0.5 and 0.75, as well as for the x = 0.7 glass at the reduction degree equal to (1 p) = 0.75, the asymmetry of the first peak cannot be described appropriately with the aid of a single value of b, and the second CF function must be used (gAB(r) = g1,AB(r) + g2,AB(r), parameters of the subsequent subshell are indicated by 1 and 2, see e.g. Table 3). In x 6 0.5 cases, increasing (1 p) is accompanied by an increase of the Pb+2–O2 disorder (distance dispersion and peak assymetry parameters increase gradually for both of the obtained subshells). For x = 0.3, the mean distance to the oxygen neighbours from both subshells increases. The related lead-oxygen co-ordination number for the closer subshell increases (up to about 3.0), whereas it decreases for the more distant subshell (down to about 1.8). For x = 0.5, the first subshell R1 ˚ , with a simultaneous change value decreases by 0.07 A of N1 (from 4.0 to 2.0). At the same time R2 and N2 values increase. Finally, the structural parameters are practically constant in the function of (1 p) for the x = 0.7 glass. The local Pb+2–O2 ordering remains constant at least up to the reduction degree (1 p) = 0.5. However, at (1 p) = 0.75 the correlation asymmetry increases, which means that, although the lead ions are always surrounded by 4 oxygen atoms, one of them is usually more ˚ ) from the Pb ion than the others. distant (by 0.50 A Fig. 5(a) illustrates how the Si+4–O2 pair correlation evolves with the increasing reduction degree. First of all,
Fig. 4. Pb+2–O2 PDF functions in x[(1 p)Pb pPbO] (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7, in the function of the reduction degree (1 p), obtained in MD simulations.
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Table 3 MD-extractcd structural parameters, describing the shape of the first Pb+2–O2 peaks in partially reduced lead-silicate glasses ˚] ˚] ˚ 2] ˚] ˚ 2] R1 [A r21 [A b1 N1 R2 [A r22 [A b2 x [mol] 1p Ro [A 0.3
0.5
0.7
N2
0.0 0.25 0.50 0.75
2.32 2.31 2.31 2.31
2.42 2.31 2.37 2.45
0.026 0.009 0.019 0.044
0.83 0.35 0.63 0.98
3.85 1.49 2.69 3.04
2.58 2.86 2.91
0.052 0.077 0.12
0.80 1.02 1.28
3.00 2.58 1.77
0.0 0.25 0.50 0.75
2.33 2.33 2.32 2.31
2.41 2.40 2.36 2.34
0.020 0.019 0.015 0.023
0.83 0.71 0.46 0.89
4.10 3.85 3.16 2.06
2.78 2.83
0.053 0.18
1.00 1.40
1.39 2.77
0.0 0.25 0.50 0.75
2.32 2.32 2.32 2.31
2.39 2.38 2.39 2.36
0.017 0.017 0.018 0.018
0.83 0.72 0.73 0.50
4.00 3.91 3.89 3.29
2.85
0.061
1.10
0.98
Ro is the most probable Pb+2–O2 distance; R1, r21 , b1 and N1 are the first subshellÕs structural parameters; R2, r22 , b2 and N2 are the second subshellÕs structural parameters.
the dispersion of Si+4–O2 distances decreases. Moreover, in the x = 0.7 glass, the most probable Si+4–O2 distance, Ro, increases with increasing (1 p) values. Simultaneously, for O2–O2 spatial correlations (Fig. 5(b)), the changes are most distinct for x = 0.7, and with increasing (1 p) both distance dispersion and distribution asymmetry increase. In order to follow the evolution of structural units in partially reduced glasses, distributions of the co-ordination numbers around Pb+2 and Si+4 ions have been calculated. The shape of dominating structural units has also been analysed.
It follows from the analysis that, independently of the initial contents of lead oxide in the glasses, the increase of reduction degree weakens the dominating role of basic structural units (i.e. of PbO4 and PbO5 groups for x 6 0.5, and of PbO4 groups for x = 0.7), increasing the fractions of PbO3 and PbO6 units. At lower PbO contents, say x 6 0.5, the fraction of PbO5 groups significantly decreases with increasing (1 p), while the fraction of PbO4 groups remains almost constant. Moreover, among PbO4 groups, independently of glass composition, the relative contribution of pyramids (where all four oxygen neighbours lie on the same side of a
Fig. 5. MD-simulated pair distribution functions: (a) Si+4–O2 and (b) O2–O2, in x[(1 p)Pb pPbO] (1 x)SiO2, x = 0.3, 0.5, 0.7 glasses in the function of the reduction degree, (1 p).
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Table 4 Occurrence of tetrahedron-like and pyramid-like PbO4 groups in partially reduced MD-simulated x[(1 p)Pb pPbO] (1 x)SiO2 glasses 1p
0.00 0.25 0.50 0.75
x = 0.3
x = 0.5
x = 0.7
Pyramid
Tetrahedron
Pyramid
Tetrahedron
Pyramid
Tetrahedron
[%] 27.23 34.00 60.00 57.14
[%] 72.77 66.00 40.00 42.86
[%] 19.03 26.20 30.65 54.88
[%] 80.97 73.80 69.35 45.12
[%] 6.39 12.87 19.88 28.15
[%] 93.61 87.13 80.12 71.85
plane containing the Pb ion) increases at the cost of a decreasing contribution of tetrahedral groups (see Table 4). Our analysis of the Pb+2–O2 . . . and the Si+4–O2 . . . rings reveals a distinct influence of the reduction process on the medium-range structure of the considered glasses. General trends, however, are weakly dependent on glass composition. PbO: At low lead oxide contents, connectivity of the Pb+2–O2 subnetwork decreases and more and more bifurcated linear chains appear even at early reduction stages (i.e. at (1 p) = 0.25). In the x = 0.5 glass, significant changes in the number and size of connected clusters and in the numbers of Pb+2 and O2 atoms not belonging to any ring appear only at the (1 p) = 0.75 reduction degree. Finally, even at the highest reduction degree over 85% of all adjacent Pb+2 and O2 atoms in the x = 0.7 glass form one connected cluster, and 70% of the atoms belong to rings. Moreover, the fraction of 2-member Pb+2–O2 rings increases with increasing reduction degree for all the considered compositions. In other words, the number of edge-sharing PbOn units increases. SiO2: The distribution of n-member Si+4–O2 . . . rings in the function of glass composition and reduction degree reveals a strong tendency of the SiO2 subsystem to form a relaxed (pure silicalike), continuous 3D network. The more reduced glasses, the greater the contribution of 5- and 6member rings and the lesser the contribution of rings shorter than 4 and longer than 7.
4.2. Local ordering around neutral Pb atoms Fig. 6 presents snapshots of atomic configurations at the last simulation step for the 0.3[(1 p)Pb pPbO] 0.7SiO2 glass. Even this simple visualisation shows a strong and progressive tendency to agglomeration of neutral Pb atoms. Fig. 7 shows the structure around neutral Pb atoms viewed by partial pair distribution functions, where an increase of the reduction degree is followed by changes
in the Pb0–Pb0 spatial correlation function. Decomposition of gPb0 Pb0 ðrÞs into CF functions (Fig. 7 and Table 5) reveals that for the x = 0.3 glass at any stage of reduction and for the x = 0.5 glass of (1 p) 6 0.5 the Pb0– Pb0 correlation can be described by two co-ordination subshells, with the average interatomic distances of ˚ and 3.60 A ˚. 3.05 A The first Pb0–Pb0 co-ordination subshell is related to the existence of small lead clusters (dimers, trimers and tetramers). In our simulations the interatomic distance ˚ . This value is in such clusters is equal to about 3.1 A compatible with the radius of the lead atom, calculated ˚ [60] or experimentally numerically to be rat = 1.54 A ˚ [61,62]. Also investigations on determined to be 1.47 A Pb dimers, trimers and tetramers on Ge, Si or Cu sur˚ or faces have yielded Pb0–Pb0 distances of 3.16 A ˚ [63,64]. As follows from Fig. 7 such small Pb0 3.30 A clusters are dominant in x = 0.3 and 0.5 glasses at lower reduction degrees (e.g. over 65% in a glass with x = 0.3 and (1 p) = 0.25).
5. Granular structure viewed by coupled EXAFS/MD methods The EXAFS method cannot distinguish between neutral Pb0 atoms and Pb+2 ions. This means that Pb–O correlations extracted from an EXAFS analysis of partially reduced glasses contain contributions from oxygen neighbours of lead ions built in the PbOSiO2 network (Pb+2–O2) and from neutral Pb0 atoms not completely surrounded by other Pb0 atoms, thus remaining in contact with oxygen ions built in the network (Pb0–O2 contribution). However, as follows from Fig. 8(a), the contribution of Pb0–O2 pairs to the EXAFS Pb–O signal is expected to be negligible (cf. RPb–O and r2 values from Table 2), and the EXAFS-extracted Pb–O correlation can be considered a Pb+2–O2 correlation. Similarly, the EXAFS-extracted Pb–Pb spatial correlation contains contributions from Pb0–Pb0, Pb0–Pb+2, Pb+2–Pb0 and Pb+2–Pb+2 pairs. For the above spatial correlations we have the following parameters: ˚ , N = 12.0; • Pb0–Pb0 in pure crystalline Pb: R0 = 3.49 A 0 +2 +2 0 • Pb –Pb and Pb –Pb in our simulated glasses: ˚ , N 1.0 (Fig. 8(b)); Ro = 3.30–3.80 A
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Fig. 6. Atomic configurations at the last simulation step for the x = 0.3 glass, (a) unreduced glass: small balls – Si+4 and O2 ions, large balls – Pb+2; (b)–(d) partially reduced glasses, only Pb0 atoms are shown; (e) and (f) completely reduced glass: small balls – Si+4 and O2 ions, large balls – Pb0.
˚, • Pb+2–Pb+2 in crystalline red PbO: Ro,1 = 3.69 A ˚ N1 = 4.0 i Ro,2 = 3.85 A, N2 = 4.0; in partially ˚ and N reduced MD-simulated glasses: Ro = 3.44 A varies from 2.0 to 4.5 depending on glass composition.
These distributions overlap, which was taken into account when the MD and EXAFS data were compared. In Fig. 9(a) we present the result of an exemplary comparison, namely of the Pb–Pb co-ordination numbers obtained from MD simulations and EXAFS
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Fig. 7. Partial Pb0–Pb0 pair distribution functions in simulated x[(1 p)Pb pPbO] (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7 in the function of the reduction degree, (1 p). (r) MD simulation, (—) total model function, (- - -) CF components (subshells).
Table 5 Structural parameters describing the first Pb0–Pb0 peaks in MD-simulated reduced glasses ˚] ˚] ˚ 2] x [mol] 1p Ro [A R1 [A r21 [A b1 N1
˚] R2 [A
˚ 2] r22 [A
b2
N2
0.3
0.25 0.50 0.75 1.0
3.00 3.19 3.15 3.23
3.02 3.14 3.09 3.09
0.030 0.036 0.038 0.051
0.46 0.39 0.82 1.01
2.07 2.37 1.84 0.98
3.59 3.57 3.52 3.52
0.130 0.120 0.140 0.150
0.80 0.79 1.17 1.07
1.59 4.51 6.98 8.61
0.5
0.25 0.50 0.75 1.0
3.18 3.29 3.30 3.32
3.18 2.90 3.61 3.63
0.051 0.005 0.190 0.195
0.70 0.15 1.15 1.11
1.79 0.18 9.93 10.33
3.65 3.61
0.190 0.200
1.03 1.21
3.41 8.61
0.7
0.25 0.50 0.75 1.0
3.52 3.50 3.47 3.45
4.06 3.92 3.93 3.92
0.473 0.351 0.360 0.356
1.22 1.11 1.23 1.26
7.86 9.41 10.61 11.08
3.47
3.49
0.019
0.11
12.00
Pbfcc 0
0
Ro the most probable Pb –Pb distance; structural parameters.
R2, r21 ,
b1 and N1 are the first subshellÕs structural parameters; R1, r22 , b2 and N2 are the second subshellÕs
analysis ((1 p) values for EXAFS data are taken from XANES analysis). Analogical comparisons have been performed for other structural parameters (RPb–O, NPb–O and RPb–Pb) and similarly good agreement has been obtained. Basing on this agreement, we have tried to correlate directly the reduction time with the reduction degree (see Fig. 9(b)). The former is exactly controlled during the experiment, while the latter is precisely known in the simulations, and can be simultaneously estimated from Eq. (3) (XANES analysis). A pair-wise comparison of the corresponding characteristics obtained from the XANES analysis and those
obtained from the coupled EXAFS/MD method leads to the conclusion that both methods give very similar estimations of the reduction degree (p parameters do not differ by more than 5%). It is also interesting to relate the technological parameters (glass composition, duration of the reduction process) to the average size of metallic clusters appearing in the structure. To this end, we calculated the N (average co-ordination number) to N0 (ideal fcc co-ordination number, i.e. 12) ratios for fcc spherical clusters of various radii as reference data (the right-hand panels of Fig. 10, ÔModel PbfccÕ). Then, assuming the spherical shape and
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Fig. 8. MD-simulated pair distribution functions: (a) Pb0–O2 and (b) Pb0–Pb+2, in x[(1 p)Pb pPbO] (1 x)SiO2 glasses, x = 0.3, 0.5, 0.7, in the function of the reduction degree, (1 p).
Fig. 9. (a) A comparison of the co-ordination numbers obtained from the EXAFS analysis and MD simulations in the function of x and (1 p) (all Pb–Pb correlations). (b) Dependence between p and tred values for all the compositions and stages of reduction, obtained using the combined EXAFS and MD method. For the sake of comparison, the XANES-extracted data are also presented (- - -).
fcc structure of granules in the glasses, the average size of granules in subsequent samples could be estimated. The corresponding construction is shown in Fig. 10. The Pb–Pb co-ordination number determined for modified glasses and normalised to N0 = 12 is shown on the vertical axes. On the horizontal axes of the left-hand figures, the reduction time and the reduction degree are shown, according to the method. The right-hand figures have the granule radius as an independent argument. The smallest fcc cluster (13 atoms) has the radius of ˚ , while the average Pb–Pb co-ordination number 3.5 A amounts to N = 5.462. It has thus been assumed that values of N/N0 lower than 5.462/12 0.45 correspond to dispersed lead atoms, not forming any granules. Basing on the dependences shown in Figs. 9 and 10, it is easy to note that such a situation takes place for the x = 0.3 glasses reduced for not longer than 50 h ((1 p) <
0.45) and for the x = 0.5 glasses reduced for less than 6 h ((1 p) < 0.15). One can read the average Pb granule radii appearing in our modified glasses, both real and simulated, from Fig. 10. A comparison of all the results reveals a distinct dependence between the initial glass composition, the reduction parameter (p or tred) and the average granule size. Moreover, the end of the reduction does not mean that big granules have been formed. Only a sufficiently long annealing time leads to the appearance of the final granular structure.
6. Concluding remarks In this paper we have presented the results of EXAFS and MD analyses of partially reduced lead-silicate glasses. The obtained experimental and numerical
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Fig. 10. Estimation of the average Pb cluster size: (a) from EXAFS experiment, (b) from MD simulation, N has been taken from CF approximation of g(r) for the Pb0–Pb0 pair.
results are compatible with each other and provide detailed information on the atomic-level structure around lead atoms, which can belong either to the oxide subsystem or to the metallic one. From our analysis of the glass structure it follows that:
7. Structural changes of the PbOSiO2 glass matrix are quite different for x 6 0.5 and 0.7 glasses, due to a different role played by PbO in low-PbO content systems (viz. glass modifier) and PbO-rich ones (viz. glass former).
1. Independently of the glass composition, the reduction process leads to the appearance of a metallic lead phase, in particular to the appearance of a lead nanogranular structure embedded in the glass matrix. 2. XANES and coupled EXAFS/MD analyses of partially reduced glasses give good estimations of the reduction degree and the average granule size. The estimated granule radius equals several nanometers. 3. With increasing lead oxide contents in the virgin glass, the susceptibility to reduction and agglomeration of neutral lead atoms increases. 4. No separation of the lead oxide and silica oxide phases has been observed in the matrix of partially reduced glasses. 5. A strong tendency of the SiO2 subsystem to form a relaxed (pure silica-like), continuous 3D network during the reduction should be noted. 6. The greater the reduction degree, the greater the contribution of PbO4 pyramids and PbO3 and PbO6 groups.
Good agreement between the EXAFS-extracted and MD-simulated structural parameters obtained for the considered set of lead-silicate glasses has been the basis for correlating certain parameters, which can be precisely (without error) determined from the experiment (reduction time) and from the simulation (the simulated glass composition). Such correlations will certainly be useful for the elaboration of a technology for the production of materials with pre-requested structural parameters.
Acknowledgments We gratefully acknowledge the support of the European Synchrotron Radiation Facility in providing synchrotron radiation facilities for the HS1663 experiment and would like to thank the BM29 staff members. We would also like to thank Dr Angela Trapananti and Dr Emiliano Principi for their assistance in the XAS measurements.
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The MD simulations have been performed at the TASK Computer Centre (Gdansk, Poland).
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