^{1,a)}and Harald Behrens

^{2}

### Abstract

Structural relaxation in silicate glasses with different histories was experimentally examined by differential scanning calorimetry and measurements of molar volume under ambient pressure. Temperature and pressure-dependent rates of changes in molar volume and generation of excess enthalpy were determined for sodium trisilicate, soda lime silicate, and sodium borosilicate (NBS) compositions. From the derived data, Prigogine-Defay ratios are calculated and discussed. Changes of excess enthalpy are governed mainly by changes in short-range structure, as is shown for NBS where boron coordination is highly sensitive to pressure. For all three glasses, it is shown how the relaxation functions that underlie volume, enthalpy, and structural relaxation decouple for changes in cooling rates and pressure of freezing, respectively. The magnitude of the divergence between enthalpy and volume may be related to differences in structural sensitivity to changes in the space on different length scales. The findings suggest that the Prigogine-Defay ratio is related to the magnitude of the discussed decoupling effect.

The authors wish to thank Dr. S. Reinsch, Federal Institute of Materials Testing, BAM, Berlin, Germany, and Dr. J. Deubener, Clausthal University of Technology, Germany, for providing the employed base glasses and standard data.

I. INTRODUCTION

II. EXPERIMENTAL

A. Glass preparation

B. Compression

C. Quenching under ambient pressure

D. DSC measurements

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Glasses
- 72.0
- Enthalpy
- 30.0
- High pressure
- 17.0
- Reaction enthalphies
- 13.0
- Differential scanning calorimeters
- 12.0

## Figures

Schematic of the performed experiments: samples were initially (1) equilibrated under high pressure, above their glass transition temperature , subsequently cooled under pressure (2). This was followed by the first DSC scan, during which samples recovered to state (3), and a final reference DSC scan.

Schematic of the performed experiments: samples were initially (1) equilibrated under high pressure, above their glass transition temperature , subsequently cooled under pressure (2). This was followed by the first DSC scan, during which samples recovered to state (3), and a final reference DSC scan.

Observed differences in isobaric heat capacities between first and second DSC scans, , for NBS [(a) and (b)], NS3 [(c) and (d)], and NCS [(e) and (f)]. Analyses of noncompressed but differently cooled glasses are shown in (a), (c), and (d); compressed glasses cooled at one constant rate of were employed for (b), (d), and (e). DSC experiments were performed under ambient pressure, employing a heating rate of . Labels indicate original cooling rates and pressures of freezing, respectively.

Observed differences in isobaric heat capacities between first and second DSC scans, , for NBS [(a) and (b)], NS3 [(c) and (d)], and NCS [(e) and (f)]. Analyses of noncompressed but differently cooled glasses are shown in (a), (c), and (d); compressed glasses cooled at one constant rate of were employed for (b), (d), and (e). DSC experiments were performed under ambient pressure, employing a heating rate of . Labels indicate original cooling rates and pressures of freezing, respectively.

Molar volume of liquids at under ambient pressure. Values were calculated from molar volumes (measured under ambient conditions) as described in the text (NSC, squares; NBS, circles; and NS3, triangles). Lines are linear regressions; the slopes directly yield thermal expansivity of the respective liquid (Table II).

Molar volume of liquids at under ambient pressure. Values were calculated from molar volumes (measured under ambient conditions) as described in the text (NSC, squares; NBS, circles; and NS3, triangles). Lines are linear regressions; the slopes directly yield thermal expansivity of the respective liquid (Table II).

Molar volume of liquids as a function of pressure at specific reference temperatures (: for NS3, for NCS, and for NBS). A detailed description of the data treatment is given in the text. Symbols: NSC, squares; NBS, circles; and NS3, triangles. Lines are linear regressions; their slopes directly yield the differences in isothermal compressibility between liquid and glassy states (Table II).

Molar volume of liquids as a function of pressure at specific reference temperatures (: for NS3, for NCS, and for NBS). A detailed description of the data treatment is given in the text. Symbols: NSC, squares; NBS, circles; and NS3, triangles. Lines are linear regressions; their slopes directly yield the differences in isothermal compressibility between liquid and glassy states (Table II).

Changes in excess enthalpy, , from first to second scan as derived from integration of data shown in Fig. 2, in dependence on pressure of freezing for constant cooling rate (; NSC, squares; NBS, circles; and NS3, triangles). Slopes of best-fit lines yield pressure derivatives of enthalpy generation for constant cooling rate. Values are given in Table II.

Changes in excess enthalpy, , from first to second scan as derived from integration of data shown in Fig. 2, in dependence on pressure of freezing for constant cooling rate (; NSC, squares; NBS, circles; and NS3, triangles). Slopes of best-fit lines yield pressure derivatives of enthalpy generation for constant cooling rate. Values are given in Table II.

Change in excess enthalpy, , from first to second scan as derived from integration of data shown in Fig. 2, in dependence on real fictive temperature, , for noncompressed but differently cooled glasses (NSC, squares; NBS, circles; and NS3, triangles). Slopes of best-fit lines yield derivatives of enthalpy generation with for constant pressure. Values are given in Table II.

Change in excess enthalpy, , from first to second scan as derived from integration of data shown in Fig. 2, in dependence on real fictive temperature, , for noncompressed but differently cooled glasses (NSC, squares; NBS, circles; and NS3, triangles). Slopes of best-fit lines yield derivatives of enthalpy generation with for constant pressure. Values are given in Table II.

Plots of vs according to Eq. (11) for NBS glasses. Parameters for Eq. (11) are for the equivalence, for the equivalence, and for the equivalence. Markers do not represent specific data points but are only used to label the different curves.

Plots of vs according to Eq. (11) for NBS glasses. Parameters for Eq. (11) are for the equivalence, for the equivalence, and for the equivalence. Markers do not represent specific data points but are only used to label the different curves.

Slopes according to Eq. (11) for NBS, NCS, and NS3 glasses and different underlying relaxation functions. Squares, relaxation of molar volume; triangles, enthalpy relaxation, expressed as shifts in fictive temperature; and circle, boron coordination. Error bars are the result error propagation calculation for linear fits of the data shown in Figs. 3–6 and 9.

Slopes according to Eq. (11) for NBS, NCS, and NS3 glasses and different underlying relaxation functions. Squares, relaxation of molar volume; triangles, enthalpy relaxation, expressed as shifts in fictive temperature; and circle, boron coordination. Error bars are the result error propagation calculation for linear fits of the data shown in Figs. 3–6 and 9.

Change in excess enthalpy, , from first to second scan as derived from integration of data shown in Fig. 2, in dependence on real fictive temperature, , for compressed glasses (NSC, squares; NBS, circles; and NS3, triangles). Values of for constant cooling rate but changing pressures (Table I) were calculated from Eq. (2). Slopes of best-fit lines yield derivatives of enthalpy generation with for constant cooling rate.

Change in excess enthalpy, , from first to second scan as derived from integration of data shown in Fig. 2, in dependence on real fictive temperature, , for compressed glasses (NSC, squares; NBS, circles; and NS3, triangles). Values of for constant cooling rate but changing pressures (Table I) were calculated from Eq. (2). Slopes of best-fit lines yield derivatives of enthalpy generation with for constant cooling rate.

## Tables

Freezing conditions and , experimentally determined densities and molar volumes, respectively, apparent and real fictive temperatures [Eqs. (1) and (2)], and change in excess enthalpy from first to second DSC scan (Fig. 2). Error ranges are given in the text.

Freezing conditions and , experimentally determined densities and molar volumes, respectively, apparent and real fictive temperatures [Eqs. (1) and (2)], and change in excess enthalpy from first to second DSC scan (Fig. 2). Error ranges are given in the text.

Compilation of experimental relationships between molar volume, excess enthalpy, fictive temperature, and pressure. Data for NBS glasses, including the dependence on the equilibrium constant of the boron speciation reaction, were adopted from Ref. 5. See text for details.

Compilation of experimental relationships between molar volume, excess enthalpy, fictive temperature, and pressure. Data for NBS glasses, including the dependence on the equilibrium constant of the boron speciation reaction, were adopted from Ref. 5. See text for details.

Experimentally determined jumps in isobaric heat capacity , molar compressibility , and thermal expansivities and at the glass transition in NBS, NCS, and NS3 glasses. Together with molar volume , extrapolated to fictive temperature for a cooling rate of at ambient pressure, the Prigogine-Defay ratio was calculated from these data.

Experimentally determined jumps in isobaric heat capacity , molar compressibility , and thermal expansivities and at the glass transition in NBS, NCS, and NS3 glasses. Together with molar volume , extrapolated to fictive temperature for a cooling rate of at ambient pressure, the Prigogine-Defay ratio was calculated from these data.

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