^{1,a)}

### Abstract

I employ the van der Waals theory of Baus and co-workers to analyze the fast, adiabatic decay of a supercooled liquid in a closed vessel with which the solidification process usually starts. By imposing a further constraint on either the system volume or pressure, I use the maximum-entropy method to quantify the fraction of liquid that is transformed into solid as a function of undercooling and of the amount of a foreign gas that could possibly be also present in the test tube. Upon looking at the implications of thermal and mechanical insulation for the energy cost of forming a solid droplet within the liquid, I identify one situation where the onset of solidification inevitably occurs near the wall in contact with the bath.

I gratefully acknowledge many enlightening discussions with Paolo V. Giaquinta. I also express my thanks to Franco Aliotta, Rosina C. Ponterio, Franz Saija, and Cirino Vasi (CNR-IPCF, Messina) for introducing me to the fascinating world of adiabatic freezing. I am also grateful to an anonymous referee who helped me to improve the paper considerably by pointing out the limits of a purely thermodynamic approach to adiabatic freezing.

I. INTRODUCTION

II. MODEL AND METHOD

III. RESULTS

A. Constant volume

B. Constant volume with a foreign gas in the vessel

C. Constant pressure

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Entropy
- 23.0
- Liquid solid interfaces
- 15.0
- Solidification
- 10.0
- Crystalline solids
- 8.0
- Crystallization
- 8.0

##### B01J19/06

## Figures

Theoretical phase diagram for a system of particles interacting through the potential (2.2) with ϕ(*x*) = *x* ^{−6} and *z* _{1} = 12. *T* _{ c } is the critical temperature and β_{ c } = (*k* _{ B } *T* _{ c })^{−1}. The critical-point coordinates, ρ_{ c } and *T* _{ c }, follow from requiring that the first- and second-order density derivatives of the fluid pressure be simultaneously zero. One thus finds ρ_{ c } = ρ_{0}/3 and *k* _{ B } *T* _{ c } = (8/27)*a*ρ_{0}, with *a* = (2π/3)εσ^{3}. Top: phase diagram on the density-temperature plane, showing the extent of the coexistence regions; the triple temperature is between 0.6 and 0.65 of *T* _{ c }. Bottom: phase diagram on the temperature-pressure plane, reporting as blue crosses also the (*T*, *P*) points characterizing the solid-liquid coexistence states borne out of the decay of the metastable-liquid states at various *T* _{in} values, for *x* _{ g } = 0.001 (see Sec. III B ).

Theoretical phase diagram for a system of particles interacting through the potential (2.2) with ϕ(*x*) = *x* ^{−6} and *z* _{1} = 12. *T* _{ c } is the critical temperature and β_{ c } = (*k* _{ B } *T* _{ c })^{−1}. The critical-point coordinates, ρ_{ c } and *T* _{ c }, follow from requiring that the first- and second-order density derivatives of the fluid pressure be simultaneously zero. One thus finds ρ_{ c } = ρ_{0}/3 and *k* _{ B } *T* _{ c } = (8/27)*a*ρ_{0}, with *a* = (2π/3)εσ^{3}. Top: phase diagram on the density-temperature plane, showing the extent of the coexistence regions; the triple temperature is between 0.6 and 0.65 of *T* _{ c }. Bottom: phase diagram on the temperature-pressure plane, reporting as blue crosses also the (*T*, *P*) points characterizing the solid-liquid coexistence states borne out of the decay of the metastable-liquid states at various *T* _{in} values, for *x* _{ g } = 0.001 (see Sec. III B ).

Final equilibrium state after the adiabatic decay of the metastable liquid under constant-volume conditions, for *T* _{ m } = 0.8 *T* _{ c }. Top: temperature; bottom: pressure.

Final equilibrium state after the adiabatic decay of the metastable liquid under constant-volume conditions, for *T* _{ m } = 0.8 *T* _{ c }. Top: temperature; bottom: pressure.

Final equilibrium state after the adiabatic decay of the metastable liquid under constant-volume conditions, for *T* _{ m } = 0.8 *T* _{ c } and for two different amounts of foreign gas in the vessel (crosses, *x* _{ g } = 0.001; squares, *x* _{ g } = 0.1). Top: temperature; bottom: pressure.

Final equilibrium state after the adiabatic decay of the metastable liquid under constant-volume conditions, for *T* _{ m } = 0.8 *T* _{ c } and for two different amounts of foreign gas in the vessel (crosses, *x* _{ g } = 0.001; squares, *x* _{ g } = 0.1). Top: temperature; bottom: pressure.

Top: Solid fraction in the equilibrium state resulting from the adiabatic decay of the metastable liquid under constant-volume conditions, for *T* _{ m } = 0.8 *T* _{ c } and for two different amounts of foreign gas in the vessel (crosses, *x* _{ g } = 0.001; squares, *x* _{ g } = 0.1). Bottom: Entropy of the solid-liquid mixture at *T* _{fin} (solid lines) vs. entropy of the supercooled liquid at *T* _{in} (dotted lines).

Top: Solid fraction in the equilibrium state resulting from the adiabatic decay of the metastable liquid under constant-volume conditions, for *T* _{ m } = 0.8 *T* _{ c } and for two different amounts of foreign gas in the vessel (crosses, *x* _{ g } = 0.001; squares, *x* _{ g } = 0.1). Bottom: Entropy of the solid-liquid mixture at *T* _{fin} (solid lines) vs. entropy of the supercooled liquid at *T* _{in} (dotted lines).

Final equilibrium state after the adiabatic decay of the metastable liquid under constant-volume conditions, for *T* _{ m } = 0.8 *T* _{ c } and for two different amounts of foreign gas in the vessel (top panel, *x* _{ g } = 0.001; bottom panel, *x* _{ g } = 0.1). Volume of the solid-liquid mixture (solid lines) vs. volume of the supercooled liquid at *T* _{in} (dotted lines).

Final equilibrium state after the adiabatic decay of the metastable liquid under constant-volume conditions, for *T* _{ m } = 0.8 *T* _{ c } and for two different amounts of foreign gas in the vessel (top panel, *x* _{ g } = 0.001; bottom panel, *x* _{ g } = 0.1). Volume of the solid-liquid mixture (solid lines) vs. volume of the supercooled liquid at *T* _{in} (dotted lines).

Final equilibrium state after the adiabatic decay of the metastable liquid at constant pressure, for *T* _{ m } = 0.8 *T* _{ c }. Top: volume of the solid-liquid mixture at *T* _{ m } (solid line) vs. volume of the liquid at *T* _{in} (dotted line); bottom: solid fraction in the mixture.

Final equilibrium state after the adiabatic decay of the metastable liquid at constant pressure, for *T* _{ m } = 0.8 *T* _{ c }. Top: volume of the solid-liquid mixture at *T* _{ m } (solid line) vs. volume of the liquid at *T* _{in} (dotted line); bottom: solid fraction in the mixture.

Difference in specific entropy between the droplet-liquid mixture at *T* _{fin} and the original metastable liquid at *T* _{in}, as a function of the droplet “radius,” . Two values of *N* are considered, 1000 (red curves, left) and 10 000 (blue curves, right), for *T* _{ m } = 0.8 *T* _{ c }. For each *N*, various *T* _{in}/*T* _{ c } values were considered: from top to bottom, 0.57, 0.60, 0.63 for *N* = 10^{3}; and 0.60, 0.65, 0.70 for *N* = 10^{4}.

Difference in specific entropy between the droplet-liquid mixture at *T* _{fin} and the original metastable liquid at *T* _{in}, as a function of the droplet “radius,” . Two values of *N* are considered, 1000 (red curves, left) and 10 000 (blue curves, right), for *T* _{ m } = 0.8 *T* _{ c }. For each *N*, various *T* _{in}/*T* _{ c } values were considered: from top to bottom, 0.57, 0.60, 0.63 for *N* = 10^{3}; and 0.60, 0.65, 0.70 for *N* = 10^{4}.

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