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Journal of Thermal Analysis and Calorimetry

, Volume 133, Issue 1, pp 371–377 | Cite as

Structural relaxation of lead and barium-free crystal glasses

  • Mária Chromčíková
  • Eleonóra Gašpáreková
  • Andrea Černá
  • Branislav Hruška
  • Marek Liška
Article
  • 91 Downloads

Abstract

The structural relaxation of Na2O–K2O–CaO–ZrO2–SiO2 (NKCZ), Na2O–K2O–ZnO–ZrO2–SiO2 (NKzZ), Na2O–CaO–ZnO–ZrO2–SiO2 (NCzZ), K2O–CaO–ZnO–ZrO2–SiO2 (KCzZ), and Na2O–K2O–CaO–ZnO–ZrO2–SiO2 (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 SiO2 by ZrO2 (i.e., the increase of the ZrO2 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 K2O for Na2O. Substitution of ZrO2 for SiO2 decreases the dynamic viscosity limit η 0 that corresponds to an activation energy increase of temperature dependence of isostructural viscosity. Increased content of ZrO2 in the glass caused the increase of the value of the modulus K.

Keywords

Glass transition Structural relaxation Thermomechanical analysis 

Introduction

Silicate glasses containing zirconia play a significant role both in the igneous petrology [1, 2] and glass technology [3]. Due to the non-toxicity and extremely high chemical durability in alkaline conditions, these glasses are used for the production of alkali-resistant fibers for Portland cement composites [4]. Both the thermal expansion coefficient and the glass transition temperature are positively correlated with the ZrO2 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 lead-free crystal glass [6, 7]. In addition to ZrO2, other oxides of heavy elements as ZnO and TiO2 are used to substitute harmful lead- and barium-oxide.

In connection with the development of the lead- and barium-free crystal glass [1, 7], the silicate glasses containing ZrO2 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 Na2O–K2O–CaO–ZrO2–SiO2 (NKCZ), Na2O–K2O–ZnO–ZrO2–SiO2 (NKzZ), Na2O–CaO–ZnO–ZrO2–SiO2 (NCzZ), K2O–CaO–ZnO–ZrO2–SiO2 (KCzZ), and Na2O–K2O–CaO–ZnO–ZrO2–SiO2 (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 time-dependent 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].

Let us consider the sample length, l, as a function of thermodynamic temperature, T, and Tool’s fictive temperature T f:
$${\text{d}}\,l\, = \,\left( {\frac{\partial \,l}{\partial \,T}} \right)_{{{\text{T}}_{\text{f}} }} {\text{d}}T\, + \,\left( {\frac{\partial \,l}{{\partial \,T_{\text{f}} }}} \right)_{\text{T}} {\text{d}}T_{\text{f}} \, = \,l\,\left( {\alpha_{\text{g}} {\text{d}}T\, + \,\Delta \alpha_{{}} {\text{d}}T_{\text{f}} } \right)$$
(1)
where
$$\Delta \alpha = \,\alpha_{\text{m}} - \alpha_{\text{g}}$$
(2)
where α g and α m are the thermal expansion coefficient of glass and metastable equilibrium melt, respectively. The values of both thermal expansion coefficients are considered as temperature independent in the present work.
In the case when viscous flow takes place, an additional source of changes of the sample length has to be included into Eq. (1):
$$\frac{1}{l}\left( {\frac{\partial l}{\partial t}} \right)_{{{\text{T}},{\text{T}}_{\text{f}} }} = \frac{\sigma }{3\eta }$$
(3)
where σ is the axial stress, t is the time, and η stands for the viscosity. Let us suppose a change of the sample state from an initial state 1 to final state 2. The relative length change of the sample during the above transition, i.e., the sample strain, can be expressed as:
$$\varepsilon = \frac{{l_{2} - l_{1} }}{{l_{1} }} \approx \int\limits_{{T_{1} }}^{{T_{2} }} {\alpha_{\text{g }} \,{\text{d}}T} + \int\limits_{{T_{\text{f,1}} }}^{{T_{\text{f,2}} }} {\Delta \alpha_{{}} \,{\text{d}}T_{\text{f}} } - \left( {1 + \int\limits_{{T_{1} }}^{{T_{2} }} {\alpha_{\text{g }} \,{\text{d}}T} + \int\limits_{{T_{\text{f,1}} }}^{{T_{\text{f,2}} }} {\Delta \alpha_{{}} \,{\text{d}}T_{\text{f}} } } \right)\,\,\int\limits_{{t_{1} }}^{{t_{2} }} {\frac{\sigma }{{3\,\eta (T,T_{\text{f}} )}}} \,{\text{d}}t$$
(4)
The time course of fictive temperature T f is obtained within the frame of the Tool–Narayanaswamy–Mazurin model with the Kohlrausch–Williams–Watts (KWW) relaxation function [18, 19]:
$$M(\xi ) = \exp \left( { - \xi^{\text{b}} } \right)\quad 0 < b \le 1$$
(5)
where b is a constant determining the width of the spectrum of relaxation times (b = 1 corresponds to the single relaxation time) and ξ is the dimensionless relaxation time:
$$\xi (t) = \int\limits_{0}^{t} {\frac{{{\text{d}}t^{\prime}}}{{\tau (t^{\prime})}} = } \int\limits_{0}^{t} {\frac{K}{{\eta (t^{\prime})}}{\text{d}}t^{\prime}}$$
(6)
where τ is the relaxation time. The viscosity dependence on temperature and Tool’s fictive temperature can be expressed by the Mazurin’s approximation:
$$\log \eta \,(T,T_{\text{f}} ) = \left( {A + \frac{B}{{T - T_{0} }}} \right)\frac{{T_{\text{f}} }}{T} + \log \eta_{0} \left( {1 - \frac{{T_{\text{f}} }}{T}} \right)$$
(7)

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 Na2CO3 (AFT, p.a.), K2CO3 (Fluka, p.a.), CaCO3 (AFT, p.a.), ZnO (Fluka, p.a.), ZrSiO4 (Aldrich, p.a.), and SiO2 (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.

The chemical composition of the studied glasses was designed as five compositional series with stepwise isomolar substitution of SiO2 by ZrO2. The molar amount of ZrO2 reached the values of {1.0, 3.0, 5.0, 7.0} mol%. Thus, the abbreviation:
  • NKCZx stands for 7.5Na2O·7.5K2O·10CaO·xZrO2·(75 − x)SiO2 where x = 1, 3, 5, 7;

  • NKzZx stands for 7.5Na2O·7.5K2O·10ZnO·xZrO2·(75 − x)SiO2 where x = 1, 3, 5, 7;

  • NCzZx stands for 15Na2O·5CaO·5ZnO·xZrO2·(75 − x)SiO2 where x = 1, 3, 5, 7;

  • KCzZx stands for 15K2O·5CaO·5ZnO·xZrO2·(75 − x)SiO2 where x = 1, 3, 5, 7;

  • NKCzZx stands for 7.5Na2O·7.5K2O·5CaO·5ZnO·xZrO2·(75 − x)SiO2 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 Na2O by K2O (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 HClO4 by inductively coupled plasma optical emission spectroscopy (VARIAN—Vista MPX/ICP-OES). The content of SiO2 has not been analyzed.

Prismatic glass samples with approximate dimensions of (5 × 5 × 20) mm3 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, αg, 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 low-temperature plateau of the derivative curve, while the α m value was obtained from the maximum value reached in the high-temperature part due to the decay of the viscous flow that was the prevailing effect at the highest temperatures.

The low-temperature viscosity (108–1012) dPa s was measured by the vertical thermomechanical analyzer (Netzsch TMA 402) on prismatic samples under a constant load of 5 g. The isothermal rate of axial deformation of the sample was used for the determination of viscosity. The high -temperature viscosity (101.5–104.0) dPa s was measured by the falling ball method. The obtained experimental viscosity data were described by the VFT equation:
$$\log \left( \eta /{{\text{dPa}}\,{\text{s}}} \right) = {A + B} /\left( {T - T_{0} } \right)$$
(8)

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

Chemical composition of studied glasses measured by inductively coupled plasma optical emission spectroscopy is summarized in Table 1. The differences between theoretical and real compositions reaching typically the level of 1–2 mol% are acceptable. The main differences are probably caused by volatilization of alkaline oxides.
Table 1

Composition (mol%) and abbreviation of studied glasses

Glass

Na2O

K2O

CaO

ZnO

ZrO2

SiO2

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

The VFT parameters A, B, and T 0 found by the regression analysis of viscosity experimental data are, together with the thermal expansion coefficients, summarized in Table 2. The values of standard deviation of approximation of log(η/dPa s) are on the level of experimental error of viscosity measurement for all studied glasses. For KCzZ5 and KCzZ7, relatively high values of T 0 standard deviations are observed. For these samples, only limited number of low-temperature (i.e., high viscosity) data was obtained due to crystallization of the samples during the TMA experiment.
Table 2

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(107 α g) are less than 1 K−1, and the standard deviations s(107 α m) are less than 2 K−1

Glass

A

B/K

T 0/K

s apr

107 α g/K−1

107 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(107 α g) are less than 1 K−1, and the standard deviations s(107 α m) are less than 2 K−1. Thus, the temperature independent values of these quantities can be used in the relaxation model.

The parameters of TNMa relaxation model were obtained by nonlinear regression analysis by minimization of the sum of squares of differences between the calculated and measured values of relative deformation. The values of the VFT parameter B (Eq. 8), the modulus K (Eq. 6), KWW exponent b (Eq. 5), and the log η 0 (Eq. 7) were optimized. Obtained results are summarized in Table 3. The reported statistical characteristics (standard deviation of approximation and the Fishers F-statistics) confirmed high quality of the fit of experimental data by the TNMa relaxation model. This situation is illustrated in Figs. 15 where the measured and calculated thermomechanical curves are compared to glasses with lowest and highest content of ZrO2. Further, it can be seen that the optimized values of the B parameter (Tab 3.) coincide with the (starting) values obtained by the VFT fit of viscosity data when the standard deviations reported in Table 2 are taken into account. This can be considered as an additional validation of the Mazurin’s part of the TNMa model.
Table 3

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 F-statistics

Glass

B/K

log{K/dPa}

b

log{η 0/dPa s}

105 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

Fig. 1

Comparison of experimental (points) and calculated (line) strain values for the selected NKCZx glasses

Fig. 2

Comparison of experimental (points) and calculated (line) strain values for the selected NKzZx glasses

Fig. 3

Comparison of experimental (points) and calculated (line) strain values for the selected NCzZx glasses

Fig. 4

Comparison of experimental (points) and calculated (line) strain values for the selected KCzZx glasses

Fig. 5

Comparison of experimental (points) and calculated (line) strain values for the selected NKCzZx glasses

Conclusions

The TNMa model describes the structural relaxation of all studied glasses with relatively high accuracy. The substitution of SiO2 by ZrO2 (i.e., the increase of the ZrO2 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 K2O for Na2O. Substitution of ZrO2 for SiO2 decreases the dynamic viscosity limit η0 that corresponds to an activation energy increase of temperature dependence of isostructural viscosity. Increased content of ZrO2 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: 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

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