Thermal Expansion of Metal Oxides in Silicate Melts: A Simple Theoretical Analysis
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Bibliographic record
Abstract
The isobaric, isochemical coefficient of thermal expansion ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math>) is the proportionality constant relating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>∂</mml:mi></mml:math>V/V and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>∂</mml:mi></mml:math>T. Noting that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>∂</mml:mi></mml:math>V/V equals <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>∂</mml:mi></mml:math>lnV at constant pressure and composition (i.e., P and X): <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>V</mml:mi><mml:mi>∂</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mrow><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:mo>ln</mml:mo><mml:mi>V</mml:mi></mml:mrow><mml:mrow><mml:mi>∂</mml:mi><mml:mi>T</mml:mi></mml:mrow></mml:mfrac><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mrow><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> A polynomial expression for the temperature dependence of lnV yields direct and accurate calculation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> of melts and glasses. Using the approach, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> G of Na-silicate glasses is shown to be constant between 300 and ∼650 K (P and X constant), but it increases linearly with Na 2 O content, indicating that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> is controlled mostly by Na behaviour. Values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> for a melt or glass are necessarily (thermodynamically) related to the coefficients of thermal expansion of its ‘N’ melt components ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>α</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:math>) by: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>θ</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>θ</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi>θ</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mi>⋯</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>α</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:msub><mml:mi>θ</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:mrow></mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>θ</mml:mi></mml:math> represents volume fraction of a component. Calculated <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>α</mml:mi><mml:mrow><mml:msub><mml:mtext mathvariant="normal">Na</mml:mtext><mml:mn>2</mml:mn></mml:msub><mml:mtext mathvariant="normal">O</mml:mtext></mml:mrow></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>α</mml:mi><mml:msub><mml:mtext mathvariant="normal">SiO</mml:mtext><mml:mn>2</mml:mn></mml:msub></mml:msub></mml:math> values compare satisfactorily with reported values for melts containing ∼40–60 mole% Na 2 O but are less satisfactory for highly siliceous melts, implying need for refinement of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> values. The Coulombic force of attraction between cation and anion is the primary determinant of the magnitude of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> values of alkali and alkaline earths oxides. Their values represent a lower bound on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>α</mml:mi></mml:math> of 3d and 4f modifier oxides. Values larger than this lower bound likely result from a change in coordination number (CN) from higher to lower CN values of modifier cations. Experimental and molecular dynamics studies indicate that partial charges on NBO (O of Si-O-M moieties) and BO (O of Si-O-Si moieties) are about −0.96 and −0.64 respectively, thus NBOs should be incorporated into the first coordination spheres of modifier cations in preference to BOs. A simple mechanism is proposed whereby NBO-BO exchange occurs and CN of the modifier cation decreases with minimal topological adjustment or energetic cost to the melt.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.007 |
| Science and technology studies | 0.000 | 0.011 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it