Introduction

When compared with silicate or borate glasses, phosphate glasses possess valuable properties, such as high refractive index, low melting temperature, and low absorption in the ultraviolet spectral range [1, 2]. On the other hand, phosphate glasses are characteristic by low chemical durability caused by the hygroscopicity of phosphorus pentoxide. The chemical stability and supplementary properties can be improved by adding different oxides of different valence to the parent phosphate network [3, 4]. This large variability in composition makes phosphate glasses excellent candidates for various applications, namely in the field of metals sealing, photonics, medical and health care, vitrification of radioactive waste, etc. [5,6,7,8,9,10,11].

TDM-SV can be used to interpret the composition–structure–property relationships for a variety of multi-component glass-forming systems [12,13,14,15,16,17,18,19]. On the other hand, the model has to be checked by the comparison of its results with the structural data for various glasses and melts [17,18,19,20,21]. In our previous work [20, 22,23,24] we showed that the statistical analysis of the Raman spectra validated the TDM-SV for the Na2O-B2O3, CaO-P2O5, and ZnO-P2O5 binary glass systems. The main aim of this work is to confirm the validity of the model by the analysis of Raman spectra of compositional series of Mg-P2O5 binary glasses.

Method

TDM-SV

In previous time, TDM-SV was successfully applied to the study of silicate glasses [12,13,14,15,16,17,18,19,20]. In this model oxide glasses and melts are considered as the ideal equilibrium solutions consisting of products of chemical reactions between the oxides and from the un-reacted oxides. The stoichiometry of these products is given by the crystalline compounds found in the particular equilibrium phase diagram. Only the molar Gibbs energies of pure crystalline compounds, analytical composition of the system and temperature are used as input parameters. The equilibrium system composition is found by the minimization of the system’s Gibbs energy with respect to the molar amount of each of the system’s species [25]. The current thermodynamic databases containing molar Gibbs energy (such as FACT database [26, 27]) allow routine construction TDM-SV for many multi-component melts and glasses.

Malfait’s decomposition of Raman spectra and MCR

Malfait et al. [28,29,30] proposed the method of numerical analysis of Raman spectra based on the knowledge of the system composition. The basic assumption is that the Raman spectra are the sum of partial Raman spectra of individual species multiplied by their molar amount. Raman spectra of a series of glasses form linear vector space with dimensionality given by the number of independent species (i.e. species with different PRS that independently change their concentration). Arbitrary scaling is supposed for each experimental spectrum, i.e. it is known with the exception of a multiplication factor. The number of independent components can be determined by the PCA [31, 32]. It is worth noting that in the field of thermal analysis the application of the PCA may be very advantageous [33]. The set of experimental Raman spectra is decomposed by the MCR method [34, 35] on the so-called loadings (i.e. spectra of quasi pure components) and so-called scores (i.e. relative abundances of these components). MCR method is frequently used in the field of spectroscopy, e.g. in connection with the FT IR spectra analysis [36]. It is important to emphasize that the knowledge of the system composition is not needed for the MCR analysis. The TDM-SV has to be validated by comparison of MCR results with the results of Malfait’s spectra decomposition based on the results of TDM-SV.

Experimental

The glass batches were prepared from analytical grade ammonium dihydrogen phosphate (NH4H2PO4) and barium carbonate (MgCO3). Stoichiometric quantities of MgCO3 and NH4H2PO4 were placed into an alumina crucible after being mixed in an agate mortar. In order to remove the water, ammonia, and carbon dioxide, the sample was slowly heated to 700 °C in an electrical furnace. Subsequently, the calcination products were melted at the temperature range of 1100–1200 °C, depending on their chemical composition. The resulting melt was poured onto a preheated (300 °C) brass mold and annealed for 2 h at the temperature that was approximately 5 °C under the glass transition temperature (Tg). The glass composition was checked by the RTG fluorescence spectroscopy (Bruker, Tiger S8). Small amounts of Al2O3 on the level of 1 mol % were found. This is a typical situation for melting of phosphate glasses in alumina crucibles [37].

The glass transition temperature was measured by thermo-mechanical analyzer TMA NETZSCH 402 with the vertical orientation of the measured sample. Prismatic glass samples with dimensions of approximately (5 × 5 × 20) mm were used. The cooling dilatometric curve was recorded at a cooling rate of 5 °C min−1 in the temperature range below the glass softening temperature.

Raman spectra were obtained using Raman spectrometer (RENISHAW).

The 532 nm laser with the power of 28.5 mW was used with the spot of about 1 mm diameter. Baseline subtracted spectra were corrected by the Böse–Einstein population factor [38]:

$$I_{\text{cor}} = I_{\exp } vv_{0}^{3} \frac{1 - \exp [ - hcv/kT]}{{(v_{0} - v)^{4} }}$$
(1)

where Iexp is the observed Raman intensity, Icor is the corrected Raman intensity, ν0 and ν are the wavenumber of the excitation laser and the Raman shift, T—the thermodynamic temperature, h—Planck constant, k—Boltzmann constant, and c—the speed of light.

Results and discussion

The measured

The measured values of the glass transition temperature, Tg, are summarized in Table 1. It can be seen that the glass transition temperature increases with the MgO content. Small irregularities can be probably attributed to the Al2O3 contamination.

Table 1 Glass transition temperature, Tg, results of TDM-SV, and corresponding Q-distribution
Full size table

Six normalized baseline subtracted and thermally corrected Raman spectra of xMgO·(1−x) P2O5 (x = 0.30, 0.35, 0.40, 0.45, 050, 0.55) glasses were analyzed (Fig. 1). The spectra were digitized with a wavenumber step of 2 cm−1 between (150–1500) cm−1. The baseline subtraction was performed by the Renishaw software Wire ver. 4.2. It is worth noting that the baseline subtraction is not unique procedure and the result depends on various software user decisions.

Fig. 1
figure 1

Normalized and thermally corrected Raman spectra

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The TDM-SV was evaluated well above Tg at the temperature of 1000 K (Fig. 2). Six system components, defined as the stable crystalline phases of the MgO–P2O5 phase diagram, were considered: MgO (M), P2O5 (P), MgO·2P2O5 (MP2), MgO·P2O5 (MP), 2MgO·P2O5 (M2P), 3MgO·P2O5 (M3P). The FACT database [26] was used to gather the Gibbs energies of the compounds listed above. The TDM behaves quasi-binary (i.e. maximum two system components can be found with nonzero abundance for each glass composition) even at temperature well above Tg. Therefore, the mass conservation law can be used to evaluate the equilibrium molar amounts of system components for all studied glass compositions at Tg. For the xMgO·(1−x) P2O5 glass compositions we obtain the following relations:

Fig. 2
figure 2

Thermodynamic model of Shakhmatkin and Vedishcheva—equilibrium molar amounts of system components at temperature T = 1000 K

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  • for 0 ≤ x ≤ 1/3 only P and MP2 are present with a non-negligible equilibrium molar content, ni.

    $$n({\text{MP2}}) = x$$
    (2)
    $$n({\text{P}}) = 1 - 3x$$
    (3)

  • for 1/3 ≤ x ≤ 0.5 only MP2 and MP are present with non-negligible equilibrium molar content:

    $$n({\text{MP2}}) = 1 - 2x$$
    (4)
    $$n({\text{MP}}) = 3x - 1$$
    (5)

  • for 0.5 ≤ x ≤ 2/3 only MP and M2P are present with not negligible equilibrium molar content ni.

    $$n({\text{M2P}}) = 2x - 1$$
    (6)
    $$n({\text{MP}}) = 2 - 3x$$
    (7)

The resulting equilibrium molar amounts of system components and the corresponding Q-distributions are summarized in Table 1. Only three system components (i.e. M2P, MP, and MP2) and three Q-units (i.e. Q3, Q2, Q1) are present in nonzero equilibrium molar amounts for all studied glasses, with the exception of the xg(MgO) = 0.30 glass, where a small amount of P2O5 was found.

MATLAB software [32] was used for the principle component analysis. The indicator function [31, 32] reached a minimum for three components and the Malinowski significance level falls below 5% for three components. The real error found for three components (0.009) was at the level of spectral noise. Thus PCA resulted in three independent components. Obtained result is in agreement with the result of TDM-SV.

The MCR [34, 35] performed for three components resulted in the relative abundances and Raman spectra of each component (Fig. 3). 99.9% of the spectral data variance was reproduced by the MCR method. Based on the high positive correlation between equilibrium molar amounts and scores, the loadings were assigned to the particular system components. This way, Loading 1 was attributed to M2P, Loading 2, to MP and Loading 3, to MP2.

Fig. 3
figure 3

Comparison of normalized partial Raman spectra MP2, MP, and M2P with corresponding normalized MCR loadings

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The Malfait’s spectral decomposition was performed with our own FORTRAN program JaneDove, using the equilibrium molar amounts of M2P, MP, and MP2. Such way partial Raman spectra (PRS) of M2P, MP, and MP2 were obtained. The experimental data were reproduced by the calculated spectra with a very good accuracy on the level of experimental noise (Fig. 4).

Fig. 4
figure 4

Comparison of experimental and Malfait’s method calculated Raman spectra

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The normalized partial Raman spectra of M2P, MP, and MP2 are plotted in Fig. 3 where they are compared with the corresponding normalized MCR loadings. It can be seen that normalized MCR loadings are practically the same as the normalized PRS. The only exception is the non-physical negative part of PRS M2P ranging from 1215 to 1370 cm−1 caused probably by some singularity problems and small errors in baseline subtraction.

The MCR results are not unique. All experimental spectra and loadings (i.e. the spectra of “pure” component) can be scaled by multiplying or dividing them by an arbitrary positive constant. Such scaling of the experimental spectra and loadings is reflected in the corresponding change of the scores. This way, scores can be adjusted to reproduce the equilibrium molar amounts of system components. Adjustment of the scores is done by minimizing the sum of the squares of the differences between equilibrium molar amounts and corresponding scores. A good agreement between the adjusted scores and equilibrium molar amounts of system components is illustrated in Fig. 5.

Fig. 5
figure 5

Comparison of adjusted scores and equilibrium molar amounts of MP2, MP and M2P

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Conclusions

The PRS obtained from the TDM-SV by the Malfait’s method are in good agreement with the loadings obtained by the MCR method. MCR and Malfait’s decomposition based on the results of TDM-SV reproduced the experimental spectra with good accuracy. The MCR loadings coincide with the Malfait’s PRS and the adjusted scores agree well with the normalized molar amounts of system components. The obtained results confirm the TDM.