# Viscosity and configuration entropy of glasses for CHROMPIC vitrification

## Abstract

The composition and temperature dependence of viscosity and configuration entropy of glasses with chemical composition close to that used for CHROMPIC radioactive waste vitrification were studied. The composition of fifteen studied glasses was derived from the composition of currently used glass frit by increasing/decreasing the content of each particular oxide and retaining the same relative amounts of the other components. The high-temperature viscosity in the range (10^{1.8}–10^{3.0}) dPa s was measured by the falling ball method while the low-temperature viscosity in the range (10^{8}–10^{12}) dPa s was measured by the thermomechanical analysis. The experimental viscosity data were smoothed by the Vogel–Fulcher–Tammann viscosity equation. The temperature–composition dependence of viscosity was described by the Adam and Gibbs viscosity equation. The influence of increasing/decreasing content of particular oxides on the values of parameters of the Adam and Gibbs viscosity equation and on the value of configuration entropy was quantified and discussed.

## Keywords

Viscosity Adam and Gibbs viscosity equation Configuration entropy## Introduction

Vitrification is a high-temperature process of liquid radioactive waste fixation into a glass matrix [1, 2]. CHROMPIC is a high-level radioactive waste (HLW) containing potassium chromate and potassium dichromate, which has been produced by cooling of fuel elements after their removal from reactor A1 of nuclear power plant Jaslovské Bohunice. CHROMPIC is fixed into a boroaluminosilicate glass matrix at the vitrification line in Jaslovské Bohunice [1].

Viscosity is one of the key parameters controlling the process of high-level radioactive waste vitrification. The main aim of the present work resides in the investigation of the composition–viscosity relationships for model glasses. The composition of fifteen studied glasses was derived from the composition of currently used glass frit (mass%) 3.5%Li_{2}O–9%Na_{2}O–14.5%B_{2}O_{3}–5.5%Al_{2}O_{3}–5.5%TiO_{2}–4.5%Fe_{2}O_{3}–57.5%SiO_{2} by increasing and decreasing the content of each particular oxide and retaining the same relative amounts of the other components. To envisage the viscosity compositional dependence from the point of structure–property relationships the measured experimental data were analyzed by the Adam and Gibbs (AG) viscosity equation and the configuration entropy was evaluated. Although the Adam–Gibbs theory has had much success in explaining many diverse features of the viscosities of silicate melts, some details relating structure and configuration entropy remain unknown. To further our understanding of the relationships between melt structure, entropy, and viscosity, we have evaluated the parameters of AG viscosity equation for all studied glasses.

## Method

*A*,

*B*, and

*T*

_{0}are adjustable parameters. For particular composition the experimental values of viscosity measured at different temperatures by different methods can be smoothed and interpolated by the VFT equation with parameters determined by the nonlinear regression analysis of experimental data. These smoothed data can be further used for study of compositional dependence of viscosity points (so-called isokomes). Moreover, the glass transition temperature can be estimated as the temperature where the viscosity reaches the value of 10

^{13}dPa s:

*A*

_{AG}and

*B*

_{AG}are adjustable parameters and

*S*

_{conf}is the configuration entropy at thermodynamic temperature

*T*. The

*B*

_{AG}value is proportional to the height of activation barrier for the smallest cooperatively rearranging system (CRS) while the configuration entropy is inversely proportional to the size of the CRS. The configuration entropy can by expressed as:

*C*

_{P}is defined as the difference between the heat capacity of the melt,

*C*

_{P,m}, and that of the glass,

*C*

_{P,g}, at the glass transition temperature

*T*

_{g}:

_{P}is assumed in Eq. (4).

*η*

*T*

_{g}= 10

^{13}dPa s into Eq. (3) the value of configuration entropy at

*T*

_{g}can be expressed as:

*C*

_{P}values can be obtained experimentally by the DSC method combining Eq. (6) with Eqs. (3) and (4) leads to the viscosity equation containing only two unknown parameters:

*A*

_{AG}and

*B*

_{AG}can be found by regression analysis of smoothed viscosity data. Moreover, when a set of viscosity curves for various glass compositions is treated it is possible to use composition-independent

*A*

_{AG}value [13]. Thus if the sum of squares between measured–smoothed and calculated viscosity values is minimized for

*N*

_{g}glass compositions the number of unknown parameters is

*N*

_{g}+ 1 (i.e., for each glass composition its own

*B*

_{AG}value plus one common

*A*

_{AG}value).

## Experimental

The composition of fifteen studied glasses was derived from the glass frit composition used for CHROMPIC vitrification (mol%): 7.4%Li_{2}O–9.2%Na_{2}O–13.2%B_{2}O_{3}–3.4%Al_{2}O_{3}–4.4%TiO_{2}–1.8%Fe_{2}O_{3}–60.6%SiO_{2} by increasing/decreasing the content of each particular oxide and retaining the same relative amounts of the other components. Samples are abbreviated by the element symbol followed by “+” or “−” sign (for decreased/increased content of particular oxide) or by “0” for zero content of particular oxide (e.g., Si+ means increased SiO_{2} content and Li0 lithium-free glass). Glasses were prepared from high-purity oxides and carbonates by slow heating of the reaction mixture up to 600 °C to remove water. Subsequently the mixture was heated up to 1250–1400 °C (according to the composition) in a Pt-20% Rh crucible in super-Kanthal furnace. Homogeneity was ensured by mixing of the glass melt by Pt-20% Rh blender. Melt was poured from the crucible onto a stainless steel plate. The samples were tempered in a muffle furnace for 1 h at 560 °C, after which the furnace was switched off and samples allowed remain there until completely cool.

_{4}by inductively coupled plasma optical emission spectroscopy (Varian-Vista MPX/ICP-OES). The chemical composition of studied glasses is summarized in Table 1. As the glasses were melted in oxidizing atmosphere the amount of two valent reduced iron was neglected.

Composition (mol %) and abbreviation of studied glasses

No. | Glass | Li | Na | B | Al | TiO | Fe | SiO |
---|---|---|---|---|---|---|---|---|

1 | Base | 7.21 | 9.16 | 13.11 | 2.63 | 4.16 | 1.71 | 62.02 |

2 | Li0 | 0.00 | 9.95 | 14.13 | 2.83 | 4.53 | 1.87 | 66.69 |

3 | Li− | 5.87 | 8.95 | 12.74 | 2.65 | 4.22 | 1.72 | 63.84 |

4 | Na− | 7.61 | 5.15 | 13.66 | 2.74 | 4.28 | 1.79 | 64.79 |

5 | Na+ | 6.85 | 13.09 | 12.57 | 2.49 | 3.99 | 1.64 | 59.38 |

6 | B− | 7.53 | 9.16 | 9.90 | 2.69 | 4.31 | 1.75 | 64.66 |

7 | B+ | 6.90 | 8.71 | 16.31 | 2.32 | 3.88 | 1.62 | 60.27 |

8 | Al0 | 7.54 | 9.58 | 13.61 | 0.00 | 4.23 | 1.79 | 63.25 |

9 | Al+ | 6.96 | 8.84 | 12.63 | 5.75 | 3.86 | 1.65 | 60.32 |

10 | Ti0 | 7.63 | 9.51 | 13.68 | 2.83 | 0.00 | 1.82 | 64.52 |

11 | Ti+ | 6.95 | 8.72 | 12.53 | 2.51 | 8.48 | 1.63 | 59.19 |

12 | Fe0 | 7.50 | 9.45 | 13.34 | 2.74 | 3.40 | 0.00 | 63.56 |

13 | Fe+ | 6.94 | 8.93 | 12.85 | 2.69 | 4.13 | 3.49 | 60.97 |

14 | Si− | 8.32 | 10.56 | 14.91 | 3.03 | 4.60 | 1.93 | 56.66 |

15 | Si+ | 6.16 | 7.87 | 11.34 | 2.36 | 3.57 | 1.49 | 67.21 |

Glass transition temperature, *T* _{g}, was measured by thermomechanical analysis (Netzsch, TMA 402) on prismatic samples by cooling from sufficiently high temperature by the cooling rate of 5 °C min^{−1}.

The high-temperature viscosity in the range 10^{1.8}–10^{3.0} dPa s was measured by the falling ball method while the low-temperature viscosity in the range 10^{8}–10^{12} dPa s was measured by the thermomechanical analysis (Netzsch, TMA 402) on prismatic samples under a constant load at isothermal conditions. The experimental viscosity data were smoothed by the VFT viscosity equation.

DSC measurements were taken on TA Q2000 (up to 600 °C, in N_{2} atmosphere). The heating rate of 30 °C min^{−1} and the cooling rate of 10 °C min^{−1} were used. Each sample was measured three times. From the heating curve the difference between the thermal capacities of the glass and the metastable equilibrium melt was determined at the *T* _{g} temperature.

## Results and discussion

*s*

_{apr}) and Fisher

*F*-statistics it can be concluded that VFT equation very well describes the measured viscosity temperature dependences and thus can be used for smoothing of experimental data. For the purpose of HLW vitrification technology the compositional dependence of constant viscosity points can be simply obtained from these data by

Parameters of VFT equation (Eq. (1)), Fisher *F*-statistics, standard deviation of approximation, glass transition temperature determined from viscosity (log[*η*(*T* _{g} ^{VFT} ) dPa s] = 13) and measured by thermodilatometry (*T* _{g} ^{DIL} ), configuration heat capacity ∆*C* _{P}

Glass | | | | | | | | Δ |
---|---|---|---|---|---|---|---|---|

Base | −0.98 | 2.901 | 557 | 18.209 | 0.03 | 763 ± 1 | 760 ± 2 | 30.15 ± 0.27 |

Li0 | −1.20 | 3.967 | 533 | 19.813 | 0.02 | 811 ± 1 | 798 ± 4 | 26.94 ± 0.59 |

Li− | −1.13 | 2.980 | 557 | 4.303 | 0.05 | 767 ± 2 | 762 ± 2 | 30.54 ± 0.37 |

Na− | −1.07 | 3.215 | 540 | 43.236 | 0.02 | 767 ± 1 | 759 ± 3 | 25.30 ± 0.18 |

Na+ | −1.55 | 3.053 | 552 | 9.151 | 0.04 | 761 ± 1 | 756 ± 2 | 34.41 ± 0.03 |

B− | −1.17 | 2.963 | 555 | 4.946 | 0.05 | 763 ± 2 | 758 ± 2 | 29.63 ± 0.36 |

B+ | −1.30 | 2.898 | 560 | 9.423 | 0.04 | 762 ± 1 | 756 ± 1 | 33.00 ± 0.30 |

Al0 | −1.07 | 2.622 | 584 | 9.118 | 0.04 | 770 ± 1 | 766 ± 2 | 30.79 ± 0.33 |

Al+ | −0.72 | 2.665 | 572 | 2.883 | 0.06 | 766 ± 2 | 757 ± 2 | 30.12 ± 0.22 |

Ti0 | −1.23 | 3.100 | 543 | 25.314 | 0.02 | 759 ± 1 | 755 ± 3 | 28.76 ± 0.05 |

Ti+ | −1.55 | 3.186 | 551 | 9.977 | 0.04 | 769 ± 1 | 766 ± 2 | 26.20 ± 0.59 |

Fe0 | −1.41 | 3.131 | 552 | 1.475 | 0.03 | 768 ± 1 | 768 ± 2 | 29.15 ± 0.19 |

Fe+ | −1.11 | 2.762 | 558 | 5.311 | 0.05 | 753 ± 1 | 751 ± 3 | 32.88 ± 0.21 |

Si− | −1.28 | 2.621 | 572 | 4.977 | 0.05 | 754 ± 1 | 754 ± 3 | 34.00 ± 0.25 |

Si+ | −0.71 | 2.832 | 569 | 3.720 | 0.05 | 775 ± 2 | 767 ± 3 | 26.16 ± 0.32 |

*T*

_{g}

^{DIL}values obtained by thermodilatometry. It can be seen that all

*T*

_{g}

^{VFT}values are little bit higher than

*T*

_{g}

^{DIL}values, with the mean difference close to 5 K. This difference is not very important when the standard deviations of individual

*T*

_{g}

^{DIL}values are taken into account. Reported standard deviations of the

*T*

_{g}

^{VFT}values were calculated from the standard deviations of the VFT fit (

*s*

_{apr}). For analysis of viscosity data by the AG viscosity equation (Eq. (7)) the

*T*

_{g}

^{VFT}values were used.

The values of Δ*C* _{P} obtained by DSC are summarized in Table 2. For each glass composition three samples were measured three times. After removing the maximal and minimal values the mean value and its standard deviation were evaluated. It can be concluded that the relative standard deviation of the mean Δ*C* _{P} value is on the level of 1–2%.

For each glass composition a set of smoothed viscosity data {*T* _{i}, log*η* _{i}}, *i* = 1, 2,… 21 was constructed using Eq. (8). The same equidistant set of log(*η*/dPa s) values was used for all glasses, namely log(*η*/dPa s) = {2.0, 2.5, 3.0, 3.5,…12.0}.

In the first step (denoted as A-single B-single) the nonlinear regression analysis of the above viscosity data was performed for each glass composition separately. In such a way the values of *A* _{AG}, *B* _{AG}, and *S* _{conf}(*T* _{g}) were obtained for each glass composition by minimizing the sum of squares between the 21 smoothed–experimental \( \log \eta \) values and \( \log \eta \) values calculated by Eq. (7). The obtained fits are characterized by the standard deviation of log(*η*/dPa s) approximation of the full set of viscosity data *s* _{apr} = 0.021.

In the next step (denoted as A-common B-single) the whole set of viscosity data for all studied glasses (i.e., 15 × 21 = 315 points) was treated by the nonlinear regression supposing the composition-independent *A* _{AG} value. In such a way the values of *B* _{AG} and *S* _{conf}(*T* _{g}) were obtained for each glass composition together with the common value of *A* _{AG} = −0.309. The obtained fit is characterized by the standard deviation of log(*η*/dPa s) approximation *s* _{apr} = 0.046.

*B*

_{AG}was expressed by the multilinear function of mol fractions of individual oxides:

*x*

_{i}is the mol fraction of the

*i*th oxide in the particular glass and

*b*

_{i}are unknown parameters determined by the regression analysis. In such a way the

*A*

_{AG}= −0.305 value was obtained together with the following values of

*b*

_{i}coefficients (values in kJ mol

^{−1}):

*b*(Li

_{2}O) = −364;

*b*(Na

_{2}O) = 220;

*b*(B

_{2}O

_{3}) = 181;

*b*(Al

_{2}O

_{3}) = 299;

*b*(TiO

_{2}) = −85;

*b*(Fe

_{2}O

_{3}) = 257;

*b*(SiO

_{2}) = 127. The obtained fit is characterized by the standard deviation of log(

*η*/dPa s) approximation

*s*

_{apr}= 0.093.

*A*

_{AG}and

*B*

_{AG}; common

*A*

_{AG}and single

*B*

_{AG}; common

*A*

_{AG}and

*B*

_{AG}expressed as multilinear function of glass composition) the experimental data are reproduced with high accuracy. Results obtained by these three approaches are compared in Figs. 1 and 2. With the exception of Na+ and Si+ glasses the difference between the

*B*

_{AG}and

*S*

_{conf}(

*T*

_{g}) values obtained by different methods is not significant and no trends are observed between the values obtained by different methods. Moreover, we can take the differences between

*B*

_{AG}and

*S*

_{conf}(

*T*

_{g}) values determined by three different methods as some rough estimate of their standard deviations.

^{+}and Na

^{+}in case of studied glasses) and partial molar entropies of structural units forming the glass network (e.g., SiO

_{4}, BO

_{4}, and AlO

_{4}tetrahedral Q-units). On the other hand the MD simulation of Gedeon [17] related the network part of configuration entropy to the ring distribution. The studied seven-component oxide system is probably too complex for straightforward expressing the configuration entropy in relation to the glass structure. On the other hand among the studied glasses we can identify only one glass composition with different character. It is the Li0 glass (Table 1) where the total molar amount of modifier oxides (only Na

_{2}O in this case) is significantly lower than molar amount the B

_{2}O

_{3}oxide. Thus in this case the amount of modifier oxides is in principle insufficient for transforming B

_{2}O

_{3}and Al

_{2}O

_{3}to tetrahedral coordination by the reaction:

*B*

_{AG}and

*S*

_{conf}(

*T*

_{g}) values significantly higher when compared with the other studied glasses. In fact, with the exception of Li0 glass, there are not significant differences between these quantities determined for studied glasses when we take the differences between values determined by three different methods as estimate of their standard deviations. On the other hand it can be seen that the changes in

*S*

_{conf}(

*T*

_{g}) and

*B*

_{AG}caused by changing glass composition are positively correlated. That is, when

*S*

_{conf}(

*T*

_{g}) increases/decreases then the

*B*

_{AG}changes in the same direction, i.e., increases/decreases. This can be understood as some type of compensation. When, e.g.,

*S*

_{conf}(

*T*

_{g}) increases then the viscosity decrease is partially compensated by the increase in

*B*

_{AG}and vice versa.

Parameters of AG viscosity equation obtained by three different methods

Glass | Method | ||||||
---|---|---|---|---|---|---|---|

A-single B-single | A-common B-single | A-common B-function | |||||

| | | | | | | |

Base | −0.16 | 107 | 10.64 | 112 | 11.01 | 105 | 10.38 |

Li0 | −0.30 | 143 | 13.24 | 143 | 13.28 | 142 | 13.15 |

Li− | −0.29 | 111 | 10.91 | 112 | 10.97 | 111 | 10.92 |

Na− | −0.22 | 102 | 10.10 | 105 | 10.33 | 100 | 9.80 |

Na+ | −0.65 | 128 | 12.28 | 115 | 11.39 | 111 | 10.92 |

B− | −0.32 | 107 | 10.53 | 107 | 10.50 | 102 | 10.04 |

B+ | −0.45 | 115 | 11.26 | 111 | 10.92 | 108 | 10.67 |

Al0 | −0.27 | 95 | 9.31 | 96 | 9.39 | 100 | 9.75 |

Al+ | 0.05 | 96 | 9.72 | 108 | 10.56 | 112 | 10.99 |

Ti0 | −0.36 | 110 | 10.89 | 109 | 10.75 | 113 | 11.22 |

Ti+ | −0.64 | 102 | 9.77 | 93 | 9.07 | 96 | 9.43 |

Fe0 | −0.52 | 112 | 10.79 | 105 | 10.29 | 104 | 10.17 |

Fe+ | −0.29 | 109 | 10.89 | 110 | 10.94 | 109 | 10.86 |

Si− | −0.46 | 105 | 10.32 | 100 | 9.98 | 102 | 10.19 |

Si+ | 0.07 | 90 | 9.02 | 102 | 9.86 | 109 | 10.56 |

## Conclusions

The viscosity temperature dependence of fifteen glasses with the composition close to that used for CHROMPIC vitrification was described by the Vogel–Fulcher–Tammann viscosity equation with high accuracy (comparable with the experimental error). These results enable the effective control of the vitrification process. The temperature–composition of viscosity was described by the Adam and Gibbs viscosity equation. The experimentally (DSC) determined configuration heat capacities (Δ*C* _{P}) together with the VFT smoothed viscosity experimental data enabled the determination of configuration entropy at *T* _{g} together with *A* _{AG} and *B* _{AG} parameters. The correctness of obtained parameters was confirmed by comparing of three different methods of regression analysis. The composition independence of *A* _{AG} parameter was confirmed by this way. The positive correlation between the changes in *S* _{conf}(*T* _{g}) and *B* _{AG} induced by the glass composition changes was found.

## 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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