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Journal of Thermal Analysis and Calorimetry

, Volume 133, Issue 1, pp 429–433 | Cite as

Thermodynamic model and high-temperature Raman spectra of 25Na2O·75B2O3 glassforming melts

  • Branislav Hruška
  • Armenak A. Osipov
  • Leyla M. Osipova
  • Mária Chromčíková
  • Andrea Černá
  • Marek Liška
Article
  • 106 Downloads

Abstract

The temperature dependence of Raman spectra of 25Na2O·75B2O3 melts recorded in the temperature range 321–1142 °C was studied by statistical methods (principal component analysis, PCA, and multivariate curve resolution, MCR) and by the thermodynamic model of Shakhmatkin and Vedishcheva. The PCA of thermally corrected spectra resulted in two independent components. Four components with not negligible abundance were found by the thermodynamic model in the studied temperature range (Na2O·B2O3, Na2O·2B2O3, Na2O·3B2O3, and Na2O·4B2O3). The temperature dependence of Na2O·2B2O3 equilibrium molar amount reaches maximum at ≈670 °C with very low molar amount (0.064 mol). The strong positive correlation was found between the equilibrium molar amounts of Na2O·B2O3 and Na2O·4B2O3. The Malfait spectral decomposition was performed by considering Na2O·3B2O3 and Na2O·4B2O3 as independent system components. The obtained partial Raman spectra correspond to the linear combination of particular Raman spectra of system components, i.e. Na2O·3B2O3 with small admixture of Na2O·2B2O3 for the first partial Raman spectrum and Na2O·4B2O3 with significant admixture of Na2O·B2O3 and with small admixture of Na2O·2B2O3 for the second partial Raman spectrum. The MCR analysis performed for two components resulted in loadings that were practically identical with partial Raman spectra obtained by Malfait decomposition. Both the MCR and the Malfait decomposition reproduced the experimental Raman spectra very well—on the level of experimental error. This result is considered as an example that the correctness of thermodynamic model can be validated by the analysis of temperature dependence of Raman spectra obtained for single glass composition.

Keywords

Thermodynamic model Raman spectra Borate glass melts MCR 

Introduction

The thermodynamic model of Shakhmatkin and Vedishcheva enables the non-empirical interpretation of the composition–structure–property relationships for glasses and melts of various multicomponent glassforming systems [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. However, the validity of the model has to be confirmed by comparison of its results with available structural data for glasses and melts with various compositions [6, 7, 8, 9]. The main aim of the present work is to manifest that the statistical analysis of temperature series of Raman spectra can be used as valuable tool for validation of particular thermodynamic model even in the case when only one glass composition is studied.

Method

Thermodynamic model of Shakhmatkin and Vedishcheva

Mainly in the field of silicate glasses the thermodynamic model of Shakhmatkin and Vedishcheva was successfully applied in previous years [1, 2, 3, 4, 5, 6, 7, 8, 9]. This model considers glasses and melts as an ideal solution formed from salt-like products of equilibrium chemical reactions between the simple chemical entities (oxides, halogenides, chalcogenides, etc.) and from the original (un-reacted) entities. These salt-like products (also called associates, groupings, or species) have the same stoichiometry as the crystalline compounds, which exist in the equilibrium phase diagram of the considered system. The model does not use adjustable parameters; only the molar Gibbs energies of pure crystalline compounds and the analytical composition of the system considered are used as input parameters. The minimization of the system’s Gibbs energy constrained by the overall system composition has to be performed with respect to the molar amount of each system species to reach the equilibrium system composition [11]. When the crystalline state data are used, the model can be simply applied to most multicomponent glasses including the non-oxide ones. The contemporary databases of thermodynamic properties containing the molar Gibbs energies of various species (like the FACT database [12, 13]) enable the routine construction of the Shakhmatkin and Vedishcheva model for many important multicomponent systems.

Decomposition of Raman spectra based on glass composition obtained from thermodynamic model

The method of numerical analysis of Raman spectra was proposed by Malfait and co-workers [14, 15, 16]. The basic assumption of this approach is that the Raman spectra are a sum of partial Raman spectra (generated by individual species) multiplied by the abundance of the species. Thus, a set of Raman spectra obtained for series of glasses with different compositions spans a linear vector space with the dimensionality given by the number of independent species (i.e. species that independently vary their abundance) with different partial Raman spectra (PRS). Each experimental spectrum is recorded with an arbitrary scaling, i.e. it is known with the exception of a multiplication factor. Principal component analysis (PCA) [17, 18] is then used to find the number of independent components. It is worth noting that using PCA method is very advantageous also in the field of thermal analysis [19].The approach of Malfait and Zakaznova-Herzog can be mathematically formulated as:
$$ {\mathbf{AP}} = {\mathbf{EC}} $$
(1)
where the matrix formalism is used where A(N w/N s) is the matrix of Raman spectra with N w rows corresponding to N w wavenumbers and N s columns corresponding to N s samples/spectra, P(N s/N s) is the square diagonal matrix with the unknown coefficients each multiplying one particular Raman spectrum, E(N w/N c) is the matrix of PRS stored column wise (N c equals to the number of independent components), and C(N c/N s) is the matrix with abundances of N c individual components in N s samples. Let us suppose that the C matrix is known from the solution of the thermodynamic model. Then, the nonlinear least squares problem can be formulated:
$$ F\left( {{\mathbf{P,E}}} \right) = \sum\limits_{j}^{{N{\text{s}}}} {\sum\limits_{i}^{{N{\text{w}}}} {\left[ {A_{\text{i},\text{j}} P_{\text{j},\text{j}} - \sum\limits_{k}^{{N{\text{c}}}} {{\text{E}}_{\text{i},\text{.k}} C_{\text{k},\text{j}} } } \right]^{2} = \hbox{min} } } . $$
(2)
Without loss of generality, we can set P 1,1 = 1.

Multivariate curve resolution

The Multivariate Curve Resolution (MCR) method [20, 21] decomposes the set of experimental Raman spectra in the spectra of quasi-pure components (so-called loadings) and relative abundances of these components (so-called scores). It is important to emphasize that MCR does not use any information of the system composition. The MCR result has to be compared with the result of Malfait spectra decomposition (based on the thermodynamic model).

Experimental

Samples were prepared from boron oxide (special purity grade) and sodium carbonate (reagent grade). These batch components were dried in a furnace at 120 °C for 2 h. Then, they were weighed and mixed in the required ratio. The batch (15 g) was melted in a platinum crucible at 1100 °C for 1 h. The obtained melt was poured into both a small platinum crucible and a steel mould, and then, they were cooled in air at room temperature. The cooling rate in the steel mould was higher than in the platinum crucible. This allows glass samples to be obtained as cylinders with heights of 8 mm and diameters of 5 mm [22]. The small platinum crucible was placed into a compact electrical furnace to record Raman spectra at different temperatures. The temperature of the sample was controlled within ±1 °C. A detailed description of the high-temperature setup can be found in [23, 24].

The Raman spectra were excited by an LTI-701 solid-state pulsed laser (λ = 532 nm, <P> = 1 W) operating at a modulation frequency of 8.7 kHz. The pulse duration of the acousto-optic switch was equal to 2 μs. An FEU-79 photomultiplier was used for detection of the Raman signal. The spectral width of the slit was equal to 6 cm−1 in all experiments. All spectra were corrected using a Böse–Einstein population factor [22].

Results and discussion

The set of nine baseline subtracted and thermally corrected Raman spectra of 25Na2O·75B2O3 glassforming melt measured at temperatures (1142, 1060, 1022, 954, 880, 756, 670, 479, 321) °C was analysed (Fig. 1). The spectra were recorded with the wavelength step of 2 cm−1 in the range (350–1650) cm−1.
Fig. 1

Normalized and thermally corrected Raman spectra

The principal component analysis (PCA) was performed by the MATLAB software [18]. For two components, the real error of 1.8% approaches the experimental error (Fig. 2). The minimum of indicator function [17, 18] was observed for two components, and for two components, the value of Malinowski significance level (Fig. 3) falls below 5%. Thus, the PCA analysis resulted in two independent components.
Fig. 2

PCA—real error

Fig. 3

PCA—Malinowski significance level (%)

The multivariate curve analysis (MCR) [20, 21] performed for two independent components resulted in the Raman spectra (so-called loadings) and relative abundances (so-called scores) of both components (Figs. 4, 5). The MCR method reproduced the experimental data with high accuracy—within the experimental spectral noise level. As an example, the comparison between experimental and MCR calculated spectra is for four randomly selected temperatures plotted in Fig. 6.
Fig. 4

Comparison of normalized PRS with normalized loadings

Fig. 5

MCR—Scores

Fig. 6

Comparison of selected experimental spectra with the spectra calculated by MCR and Malfait method

The thermodynamic model of Shakhmatkin and Vedishcheva (SVTDM) was evaluated for each temperature. The molar Gibbs energies of considered components were taken from the FACT database [12]. Ten following system components (found as stable crystalline phases of the Na2O–B2O3 binary phase diagram [25]) were considered: Na2O, B2O3, 3Na2O·B2O3 (N3B), 2Na2O·B2O3 (N2B), Na2O·B2O3 (NB), Na2O·2B2O3 (NB2), Na2O·3B2O3 (NB3), Na2O·4B2O3 (NB4), Na2O·5B2O3 (NB5), and Na2O·9B2O3 (NB9). The thermodynamic model was evaluated for the glass composition of 0.25 mol Na2O + 0.75 mol B2O3. Only four components—NB, NB2, NB3, NB4—were found with not negligible abundance in the studied temperature range (Fig. 7). Strong positive correlation (Table 1) was found between the equilibrium molar amounts of NB4 and NB (both increasing with temperature). The NB3 equilibrium molar amount decreases with temperature, while the NB2 molar amount reaches maximum of 0.064 mol at 670 °C. This maximal amount of NB2 is significantly lower than the maximal amounts of NB3 and NB4. Thus, the two independent components identified by PCA can be identified as NB3 and NB4 (accompanied by the NB linearly dependent on NB4). The temperature dependence of MCR Score 2 corresponds to those of NB4 and NB, while the temperature dependence of MCR Score 1 corresponds to those of NB3. Therefore, in the next step the Malfait’s spectral decomposition was performed considering the equilibrium molar amounts of NB3 and NB4. Our own FORTRAN program JaneDove was used for this purpose. This way two partial Raman spectra (PRS) were obtained—first one corresponding to NB3 with small admixture of NB2 and the second one corresponding to NB4 with significant admixture of NB and small admixture of NB2. The calculated spectra reproduced the experimental data with very high accuracy comparable with those of the MCR results (Fig. 6). It was found that MCR loadings are practically identical with the partial Raman spectra obtained from the results of thermodynamic model by the method of Malfait (Fig. 4). The obtained results validate the correctness of the SVTDM and confirmed the structural information acquired from the thermodynamic model. The statistical significance of the obtained results can be seen from the comparison between experimental and calculated Raman spectra.
Fig. 7

Temperature dependence of equilibrium molar amounts of main system components

Table 1

Correlation coefficients between equilibrium molar amounts of system components with not negligible abundance

Variable

n(NB)

n(NB2)

n(NB3)

n(NB4)

n(NB)

1.00

−0.63

−0.92

0.96

n(NB2)

−0.63

1.00

0.28

−0.38

n(NB3)

−0.92

0.28

1.00

−0.99

n(NB4)

0.96

−0.38

−0.99

1.00

Conclusions

The partial Raman spectra calculated from the thermodynamic model of Shakhmatkin and Vedishcheva by the method of Malfait coincide with the loadings calculated by MCR method. Both treatments, i.e. MCR and Malfait based on the results of SVTDM, reproduced the experimental spectra with high accuracy. The obtained results confirmed the thermodynamic model.

Notes

Acknowledgements

This work was supported by the Slovak Grant Agency for Science under the Grant VEGA 2/0088/16 and by the Slovak Research and Development Agency Project ID: APVV-0487-11.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Vitrum Laugaricio – Joint Glass Center of IIC SAS, TnU AD, and FChPT STUTrenčínSlovakia
  2. 2.Institute of MineralogyUral Branch RASMiassRussian Federation

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