Structural relaxation of lead and bariumfree crystal glasses
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Abstract
The structural relaxation of Na_{2}O–K_{2}O–CaO–ZrO_{2}–SiO_{2} (NKCZ), Na_{2}O–K_{2}O–ZnO–ZrO_{2}–SiO_{2} (NKzZ), Na_{2}O–CaO–ZnO–ZrO_{2}–SiO_{2} (NCzZ), K_{2}O–CaO–ZnO–ZrO_{2}–SiO_{2} (KCzZ), and Na_{2}O–K_{2}O–CaO–ZnO–ZrO_{2}–SiO_{2} (NKCzZ) glasses were studied by thermomechanical analysis. The structural relaxation was described by the Tool–Narayanaswamy–Mazurin model (TNMa). The relaxation function of Kohlrausch, Williams, and Watts (KWW) was used. The parameters of relaxation model were calculated by nonlinear regression analysis of thermodilatometric curves measured under cyclic time–temperature regime by thermomechanical analyzer under the constant load. The values of the exponent b of the KWW equation, modulus K, limit dynamic viscosity η _{0} of the Mazurin’s expression for relaxation time, and constant B of the Vogel–Fulcher–Tammann viscosity equation were optimized. It was found that TNMa relaxation model very well describes the experimental data. A more detailed analysis of the obtained results showed that the equimolar substitution of SiO_{2} by ZrO_{2} (i.e., the increase of the ZrO_{2} content in the glass) decreases the parameter b, therefore the continuous distribution of the relaxation times spectrum is widening. A wider spectrum of relaxation times was obtained even in the case of substitution of ZnO for CaO and K_{2}O for Na_{2}O. Substitution of ZrO_{2} for SiO_{2} decreases the dynamic viscosity limit η _{0} that corresponds to an activation energy increase of temperature dependence of isostructural viscosity. Increased content of ZrO_{2} in the glass caused the increase of the value of the modulus K.
Keywords
Glass transition Structural relaxation Thermomechanical analysisIntroduction
Silicate glasses containing zirconia play a significant role both in the igneous petrology [1, 2] and glass technology [3]. Due to the nontoxicity and extremely high chemical durability in alkaline conditions, these glasses are used for the production of alkaliresistant fibers for Portland cement composites [4]. Both the thermal expansion coefficient and the glass transition temperature are positively correlated with the ZrO_{2} content in silicate glass [1, 5]. In addition to the chemical durability, the high density and high value of refractive index and dispersion predetermined these glasses for production of ecologically friendly barium and leadfree crystal glass [6, 7]. In addition to ZrO_{2}, other oxides of heavy elements as ZnO and TiO_{2} are used to substitute harmful lead and bariumoxide.
In connection with the development of the lead and bariumfree crystal glass [1, 7], the silicate glasses containing ZrO_{2} and ZnO were studied [1]. The compositional dependence of the structural relaxation is one of the parameters that are used for optimization of the glass forming and annealing time–temperature schedule. On the other hand, this dependence can be interpreted in terms of structural changes caused by the changing glass composition. This work is therefore devoted to the structural relaxation of Na_{2}O–K_{2}O–CaO–ZrO_{2}–SiO_{2} (NKCZ), Na_{2}O–K_{2}O–ZnO–ZrO_{2}–SiO_{2} (NKzZ), Na_{2}O–CaO–ZnO–ZrO_{2}–SiO_{2} (NCzZ), K_{2}O–CaO–ZnO–ZrO_{2}–SiO_{2} (KCzZ), and Na_{2}O–K_{2}O–CaO–ZnO–ZrO_{2}–SiO_{2} (NKCzZ) glasses. The aim of the present work resides in evaluation of the compositional dependence of parameters of Tool–Narayanaswamy–Mazurin’s (TNMa) relaxation model [3, 11].
Theoretical part
Generally, the structural relaxation is understood as the timedependent response of the glass structure, characterized by the Tool’s fictive (structural) temperature, during the temperature change in the temperature range close to the glass transition temperature (T _{g}) [8, 9, 10]. The kinetics of structural relaxation is commonly described by Tool–Narayanaswamy–Moynihan’s (TNMo) or Tool–Narayanaswamy–Mazurin’s (TNMa) models [9, 11, 12]. Both models are based on the concept of Tool’s fictive temperature [13]. The method of thermomechanical analysis in which simultaneously with the relaxation of the glass structure the viscous flow under the applied constant axial load takes place [14, 15] is used in the present work.
The structural (or volume) relaxation is typically studied by dilatometry. In the present paper, we use the method of thermomechanical analysis, where during the zigzag time–temperature regime, the length of the prismatic sample exposed to constant axial stress undergoes simultaneously the changes caused by viscous flow and by structural relaxation [16]. The method itself, as well as the optimization of the experimental schedule, is discussed in our previous paper [17].
The modulus K relating the viscosity and relaxation time in Eq. (6) is considered as characteristic material constant dependent on the glass composition [8]. The Vogel Fulcher Tamman viscosity equation (VFT) is used for the viscosity temperature dependence of metastable equilibrium melt in Eq. (7).
In principle, an arbitrary subset of the above model parameters, i.e., a subset chosen from {K, b, A, B, T _{0}, η _{0}, α _{g}, and α _{m}}, can be estimated by means of the nonlinear regression analysis. Obviously, the lower is the number of estimated parameters the more statistically robust are the results of the regression analysis. Therefore, those parameters that can be estimated with sufficient accuracy by an independent experiment are not optimized in the regression analysis.
The values of the exponent b of the KWW equation, modulus K, limit dynamic viscosity η _{0} of the Mazurin’s expression for relaxation time, and constant B of the Vogel–Fulcher–Tammann viscosity equation were optimized in the present work.
Experimental
The glass batches were prepared by mixing of powdered carbonates and oxides of p.a. purity Na_{2}CO_{3} (AFT, p.a.), K_{2}CO_{3} (Fluka, p.a.), CaCO_{3} (AFT, p.a.), ZnO (Fluka, p.a.), ZrSiO_{4} (Aldrich, p.a.), and SiO_{2} (AFT, min. 96.5%). Sodium sulfate (AFT, p.a.) and potassium sulfate (Lachema, p.a.) were used as fining agents. Glasses were melted in Pt–10%Rh crucible in superkanthal furnace at the temperature of 1600 °C for 2–3 h in ambient atmosphere. The homogeneity was ensured by repeated hand mixing of the melt. The glass melt was then poured onto a stainless steel plate. The samples were tempered in a muffle furnace for 1 h at 650 °C, after which the furnace was switched off and samples allowed remain there until completely cool.

NKCZx stands for 7.5Na_{2}O·7.5K_{2}O·10CaO·xZrO_{2}·(75 − x)SiO_{2} where x = 1, 3, 5, 7;

NKzZx stands for 7.5Na_{2}O·7.5K_{2}O·10ZnO·xZrO_{2}·(75 − x)SiO_{2} where x = 1, 3, 5, 7;

NCzZx stands for 15Na_{2}O·5CaO·5ZnO·xZrO_{2}·(75 − x)SiO_{2} where x = 1, 3, 5, 7;

KCzZx stands for 15K_{2}O·5CaO·5ZnO·xZrO_{2}·(75 − x)SiO_{2} where x = 1, 3, 5, 7;

NKCzZx stands for 7.5Na_{2}O·7.5K_{2}O·5CaO·5ZnO·xZrO_{2}·(75 − x)SiO_{2} where x = 1, 3, 5, 7;
Thus, the comparison of results obtained for different series can be used as a measure of the influence of isomolar substitutions of Na_{2}O by K_{2}O (NCzZx vs. KCzZx) and CaO by ZnO (NKCZx vs. NKzZx).
The chemical composition of studied glasses was determined after their decomposition by the mixture of HF and HClO_{4} by inductively coupled plasma optical emission spectroscopy (VARIAN—Vista MPX/ICPOES). The content of SiO_{2} has not been analyzed.
Prismatic glass samples with approximate dimensions of (5 × 5 × 20) mm^{3} were prepared by cutting and grinding (MTH KOMPAKT 1031). The thermodilatometric curves were recorded by the thermomechanical analyzer (NETZSCH TMA 402 F1 Hyperion) under the constant axial load of 5 g. The heating/cooling rate of 5 °C min^{−1} was applied.
The thermal expansion coefficients of the glass, α_{g}, and of the metastable melt, α _{m}, were calculated from the first temperature derivative of the cooling dilatometric curve. The α _{g} value was obtained from the lowtemperature plateau of the derivative curve, while the α _{m} value was obtained from the maximum value reached in the hightemperature part due to the decay of the viscous flow that was the prevailing effect at the highest temperatures.
The thermomechanical experimental study of structural relaxation was performed by the vertical thermomechanical analyzer (Netzsch TMA 402 F1 Hyperion) on prismatic samples under a constant load of 50 g. Two cycles consisting from cooling followed by heating and cooling were performed—the first one with the heating/cooling rate of ±20 °C min^{−1} and the second with the heating/cooling rate of ±10 °C min^{−1}. The same values of upper and lower temperature limits were used for each heating and cooling ramp. For all samples, the lower temperature limit of 350 °C was used. The values of upper temperature limit were determined individually for each series (710 °C for NCzZx, 730 °C for NKCZx, NKzZx, and NKCzZx, 810 °C for KCzZx). The nonlinear regression analysis of experimental data was performed by the own FORTRAN computer code based on the minimization (by the simplex method followed by the pit mapping) of the sum of squares of deviations between measured and calculated sample relative deformation ε = Δl/l _{0}.
Results and discussion
Composition (mol%) and abbreviation of studied glasses
Glass  Na_{2}O  K_{2}O  CaO  ZnO  ZrO_{2}  SiO_{2} 

NKCZ1  7.64  7.36  8.61  –  0.95  75.44 
NKCZ3  7.54  7.44  8.84  –  2.86  73.32 
NKCZ5  7.74  6.89  9.97  –  5.12  70.28 
NKCZ7  7.66  7.01  9.98  –  7.18  68.17 
NKzZ1  7.45  7.41  –  10.08  0.98  74.08 
NKzZ3  7.30  7.06  –  9.63  2.61  73.40 
NKzZ5  7.66  7.13  –  10.92  5.22  69.07 
NKzZ7  7.83  7.00  –  11.03  7.54  66.60 
NCzZ1  13.51  –  4.80  4.58  0.89  76.22 
NCzZ3  13.72  –  4.72  5.01  2.71  73.84 
NCzZ5  15.78  –  5.13  5.58  5.42  68.09 
NCzZ7  15.78  –  5.11  5.43  7.40  66.29 
KCzZ1  –  15.81  5.03  5.27  1.01  72.88 
KCzZ3  –  13.95  4.47  4.95  2.66  73.97 
KCzZ5  –  13.64  4.97  5.48  5.20  70.72 
KCzZ7  –  13.88  4.98  5.43  7.25  68.46 
NKCzZ1  8.22  8.01  4.40  5.40  1.05  72.92 
NKCzZ3  6.61  6.96  4.41  4.81  2.58  74.63 
NKCzZ5  7.82  7.18  5.18  5.63  5.35  68.85 
NKCzZ7  7.86  6.99  5.00  5.53  7.40  67.22 
Parameters of VFT viscosity equation obtained by the regression analysis of viscosity experimental data, standard deviation of approximation of log(η/dPa s), s _{apr}, and experimental values of thermal expansion of glass, α _{ g }, and metastable melt, α _{m}. The standard deviations s(10^{7} α _{g}) are less than 1 K^{−1}, and the standard deviations s(10^{7} α _{m}) are less than 2 K^{−1}
Glass  A  B/K  T _{0}/K  s _{apr}  10^{7} α _{g}/K^{−1}  10^{7} α _{m}/K^{−1} 

NKCZ1  −2.002 ± 0.086  5478 ± 133  451 ± 8  0.034  91  318 
NKCZ3  −2.271 ± 0.090  5796 ± 142  467 ± 8  0.031  88  295 
NKCZ5  −2.154 ± 0.161  5076 ± 243  554 ± 15  0.058  73  285 
NKCZ7  −2.373 ± 0.090  5056 ± 131  596 ± 8  0.034  81  275 
NKzZ1  −2.232 ± 0.105  6117 ± 174  416 ± 10  0.036  78  245 
NKzZ3  −2.049 ± 0.077  5523 ± 118  504 ± 7  0.028  85  243 
NKzZ5  −3.090 ± 0.062  6706 ± 105  482 ± 6  0.018  85  243 
NKzZ7  −2.679 ± 0.155  5771 ± 239  564 ± 13  0.043  83  242 
NCzZ1  −1.133 ± 0.124  4127 ± 172  525 ± 11  0.056  74  271 
NCzZ3  −1.967 ± 0.095  5070 ± 138  504 ± 8  0.038  79  303 
NCzZ5  −2.713 ± 0.157  5576 ± 241  510 ± 14  0.056  79  253 
NCzZ7  −3.134 ± 0.086  6085 ± 136  521 ± 7  0.026  77  258 
KCzZ1  −1.523 ± 0.093  4415 ± 130  533 ± 8  0.043  88  287 
KCzZ3  −1.578 ± 0.103  4473 ± 140  563 ± 9  0.049  87  275 
KCzZ5  −2.951 ± 0.638  6308 ± 102  563 ± 53  0.106  78  273 
KCzZ7  −1.167 ± 0.747  4036 ± 103  718 ± 63  0.111  71  280 
NKCzZ1  −2.239 ± 0.085  5662 ± 131  520 ± 8  0.026  105  301 
NKCzZ3  −1.776 ± 0.070  5008 ± 100  593 ± 6  0.022  83  294 
NKCzZ5  −2.711 ± 0.106  5957 ± 170  496 ± 10  0.036  78  271 
NKCzZ7  −3.383 ± 0.217  6794 ± 355  494 ± 18  0.053  77  252 
The values of thermal expansion coefficients were obtained with relatively good accuracy—the standard deviations s(10^{7} α _{g}) are less than 1 K^{−1}, and the standard deviations s(10^{7} α _{m}) are less than 2 K^{−1}. Thus, the temperature independent values of these quantities can be used in the relaxation model.
Values of optimized parameters of TNMa relaxation model obtained by nonlinear regression analysis of the thermomechanical experimental data, standard deviations of approximation, s _{apr}, of Δl/l _{0} experimental values, and Fisher’s Fstatistics
Glass  B/K  log{K/dPa}  b  log{η _{0}/dPa s}  10^{5} s _{apr}  F 

NKCZ1  5375 ± 0.33  9.61 ± 0.18  0.72 ± 0.12  0.01 ± 1.67  73  1023 
NKCZ3  5651 ± 0.26  10.05 ± 0.09  0.67 ± 0.05  −5.70 ± 1.89  27  2167 
NKCZ5  4910 ± 0.27  10.65 ± 0.18  0.78 ± 0.15  0.08 ± 2.42  44  1318 
NKCZ7  4891 ± 0.19  11.30 ± 0.10  0.51 ± 0.04  −11.94 ± 3.46  14  3758 
NKzZ1  5944 ± 0.31  9.90 ± 0.21  0.61 ± 0.10  −3.06 ± 2.78  37  1508 
NKzZ3  5253 ± 0.36  10.83 ± 0.07  0.41 ± 0.02  −14.44 ± 2.33  84  2651 
NKzZ5  6404 ± 0.40  10.80 ± 0.10  0.43 ± 0.04  −18.30 ± 4.23  13  1768 
NKzZ7  5520 ± 0.24  11.30 ± 0.07  0.40 ± 0.02  −15.67 ± 2.66  8  3981 
NCzZ1  3945 ± 0.25  10.47 ± 0.18  0.75 ± 0.09  2.78 ± 1.75  41  1230 
NCzZ3  4855 ± 0.36  9.74 ± 0.15  0.53 ± 0.05  −0.11 ± 1.67  30  1615 
NCzZ5  5388 ± 0.26  10.88 ± 0.24  0.61 ± 0.22  −4.51 ± 2.25  47  1525 
NCzZ7  5838 ± 0.17  11.24 ± 0.11  0.51 ± 0.06  −20.72 ± 5.03  18  3937 
KCzZ1  4928 ± 0.36  10.30 ± 0.06  0.61 ± 0.02  −4.00 ± 1.25  11  2425 
KCzZ3  5046 ± 0.24  10.58 ± 0.06  0.60 ± 0.03  −11.79 ± 1.94  14  2834 
KCzZ5  6082 ± 0.35  11.43 ± 0.08  0.54 ± 0.10  −28.58 ± 9.50  25  1550 
KCzZ7  3938 ± 0.21  11.32 ± 0.16  0.37 ± 0.10  −16.21 ± 11.27  16  2490 
NKCzZ1  5399 ± 0.34  9.78 ± 0.32  0.79 ± 0.25  1.66 ± 1.96  99  934 
NKCzZ3  4717 ± 0.29  10.17 ± 0.26  0.55 ± 0.07  −5.42 ± 4.41  33  1639 
NKCzZ5  5489 ± 0.31  10.38 ± 0.13  0.58 ± 0.05  −6.96 ± 3.05  27  1602 
NKCzZ7  6119 ± 0.28  10.47 ± 0.11  0.55 ± 0.07  −6.20 ± 2.60  29  1796 
Conclusions
The TNMa model describes the structural relaxation of all studied glasses with relatively high accuracy. The substitution of SiO_{2} by ZrO_{2} (i.e., the increase of the ZrO_{2} content in the glass) decreases the parameter b, therefore the continuous distribution of the relaxation times spectrum is widening. A wider spectrum of relaxation times was obtained even in the case of substitution of ZnO for CaO and K_{2}O for Na_{2}O. Substitution of ZrO_{2} for SiO_{2} decreases the dynamic viscosity limit η_{0} that corresponds to an activation energy increase of temperature dependence of isostructural viscosity. Increased content of ZrO_{2} in the glass caused the increase of the modulus K.
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: APVV048711.
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