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Abstract
The solidification of a liquid by nucleation is an important first order phase transition process. It is known that in order for elemental liquids to solidify homogeneously, it is necessary to supercool the liquid to a characteristic temperature below the thermodynamic melting point . Approximately 60 years ago Turnbull [J. Appl. Phys.21, 1022 (1950)] established the empirical rule that is approximately given by for several elemental metallic liquids. We show here that the magnitude of and for the metals can be accurately predicted from classical nucleation theory (CNT) provided the excess volume resulting from the density difference between liquid and solid be accounted for. Specifically, the density change accompanying the formation of a microscopic nucleus of the solid from the liquid results in a volume change in the surrounding liquid. When this is included in the free energy calculations within CNT, the resulting predictions for and for several metals with ranging from to 2900 K are in very good agreement with experimental measurements. This theory also shows that there is a universal character in the minimum nucleation barrier energy and the critical radius. The minimum barrier energy occurs at temperature for all the elemental liquids investigated, while the temperature dependencies of the barrier energy and the critical radius appear identical when expressed as a function of the scaled temperature .
R.K. acknowledges the support by the National Science Foundation through CAREER Grant No. DMI0449258.
I. INTRODUCTION
II. THEORY
A. Modified CNT
III. RESULTS
A. Characteristic undercooling in bulk liquids
B. Critical activation barrier energy and radius for solidification
IV. DISCUSSION AND CONCLUSION
Key Topics
 Liquid solid interfaces
 32.0
 Nucleation
 30.0
 Free energy
 23.0
 Carbon nanotubes
 18.0
 Fluid drops
 18.0
Figures
In homogeneous phase transformation under CNT, a spherical nucleus of the stable phase forms with radius when it can overcome the maximum energy barrier . In the existing CNT, the magnitude of is governed on the fundamental level by only two terms: a decrease due to the volume energy in going from the metastable to stable phase which varies as (solid curve) and an increase due to the new interface formed between the two phases which increases as (dotted curve). However, if the free energy calculation includes the excess volume due to the density difference between solid and liquid, then an additional internal energy term, which varies as (dashed curve), is present. For the elemental metallic liquids this contributes a positive quantity to the free energy change. The various curves shown here are for bulk gold metal undercooled to a temperature of 1004 K.
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In homogeneous phase transformation under CNT, a spherical nucleus of the stable phase forms with radius when it can overcome the maximum energy barrier . In the existing CNT, the magnitude of is governed on the fundamental level by only two terms: a decrease due to the volume energy in going from the metastable to stable phase which varies as (solid curve) and an increase due to the new interface formed between the two phases which increases as (dotted curve). However, if the free energy calculation includes the excess volume due to the density difference between solid and liquid, then an additional internal energy term, which varies as (dashed curve), is present. For the elemental metallic liquids this contributes a positive quantity to the free energy change. The various curves shown here are for bulk gold metal undercooled to a temperature of 1004 K.
Homogeneous nucleation of a solid nucleus within a liquid drop enclosed inside a closed thermodynamic system. (a) Initial drop of liquid volume and density . (b) A solid nucleus of volume and density forms from a volume of liquid at the center of the bulk liquid drop. If the solid has higher density than the liquid, i.e., , then and the surrounding liquid will rearrange to form a final drop whose outer radius will be smaller then the initial drop.
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Homogeneous nucleation of a solid nucleus within a liquid drop enclosed inside a closed thermodynamic system. (a) Initial drop of liquid volume and density . (b) A solid nucleus of volume and density forms from a volume of liquid at the center of the bulk liquid drop. If the solid has higher density than the liquid, i.e., , then and the surrounding liquid will rearrange to form a final drop whose outer radius will be smaller then the initial drop.
Calculation of the characteristic undercooling temperature . (a) Thermodynamic Gibbs volume energy curves and for Ag. The intersection of the curves is the thermodynamic melting temperature . (b) The characteristic undercooling temperature for each metal is obtained from the plot of as a function of temperature. The location of the crossover from negative to positive values gives . Representative plots for Zr, Ag, and Hg are shown.
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Calculation of the characteristic undercooling temperature . (a) Thermodynamic Gibbs volume energy curves and for Ag. The intersection of the curves is the thermodynamic melting temperature . (b) The characteristic undercooling temperature for each metal is obtained from the plot of as a function of temperature. The location of the crossover from negative to positive values gives . Representative plots for Zr, Ag, and Hg are shown.
(a) Plot of characteristic undercooling as a function of melting temperature for several liquids. The theoretical values estimated from our theory are shown as open triangles, while the experimentally determined values are shown as solid circles. Very good agreement between theory and experiment is evident. The trend line (dashed line) shows that most of the metals have consistent with Turnbull’s (Ref. 18) observations. (b) Plot of the characteristic undercooling temperature vs . The linear trend line (dashed line) has a slope of 0.84.
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(a) Plot of characteristic undercooling as a function of melting temperature for several liquids. The theoretical values estimated from our theory are shown as open triangles, while the experimentally determined values are shown as solid circles. Very good agreement between theory and experiment is evident. The trend line (dashed line) shows that most of the metals have consistent with Turnbull’s (Ref. 18) observations. (b) Plot of the characteristic undercooling temperature vs . The linear trend line (dashed line) has a slope of 0.84.
(a) Variation of the critical activation barrier energy as a function of temperature for various elemental liquids. The large barrier observed at higher temperatures corresponds to the characteristic undercooling temperature . The activation barrier has a broad minimum at a temperature below for the metals. (b) The location of the minimum in the nucleation barrier is plotted vs . The best fit to the data (dashed line) gives a linear trend with a slope of 0.27 and intercept of 31 K.
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(a) Variation of the critical activation barrier energy as a function of temperature for various elemental liquids. The large barrier observed at higher temperatures corresponds to the characteristic undercooling temperature . The activation barrier has a broad minimum at a temperature below for the metals. (b) The location of the minimum in the nucleation barrier is plotted vs . The best fit to the data (dashed line) gives a linear trend with a slope of 0.27 and intercept of 31 K.
(a) Characteristic behavior of the barrier energy vs the scaled temperature for several metals. All the curves have similar shapes and location of the energy minima suggesting that the energetics of the first order solidification process has a universal character. (b) Characteristic behavior of the critical nucleus vs the scaled temperature for several metals. Once again, all the curves have similar shapes as well as magnitude of (which varies between and 100 nm) over a large temperature range below the characteristic undercooling temperature.
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(a) Characteristic behavior of the barrier energy vs the scaled temperature for several metals. All the curves have similar shapes and location of the energy minima suggesting that the energetics of the first order solidification process has a universal character. (b) Characteristic behavior of the critical nucleus vs the scaled temperature for several metals. Once again, all the curves have similar shapes as well as magnitude of (which varies between and 100 nm) over a large temperature range below the characteristic undercooling temperature.
Tables
The various metals, their solid phase, and their densities in the liquid and solid states used to estimate the characteristic undercooling . The density values are evaluated at the melting point and were obtained from Ref. 24 unless otherwise indicated. The crystal phase determined the values of used from Ref. 23. The values were obtained from Ref. 22, while was evaluated from the model presented here.
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The various metals, their solid phase, and their densities in the liquid and solid states used to estimate the characteristic undercooling . The density values are evaluated at the melting point and were obtained from Ref. 24 unless otherwise indicated. The crystal phase determined the values of used from Ref. 23. The values were obtained from Ref. 22, while was evaluated from the model presented here.
Table showing surface tension, characteristic undercooling temperature , and temperature of minimum barrier energy . The surface tension values were obtained from Ref. 29 and were used to estimate the barrier energy and critical radius. The values were calculated from Ref. 22, while the is the theoretical value for the characteristic undercooling estimated from our theory. is the temperature at which the nucleation barrier is a minimum.
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Table showing surface tension, characteristic undercooling temperature , and temperature of minimum barrier energy . The surface tension values were obtained from Ref. 29 and were used to estimate the barrier energy and critical radius. The values were calculated from Ref. 22, while the is the theoretical value for the characteristic undercooling estimated from our theory. is the temperature at which the nucleation barrier is a minimum.
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Abstract
The solidification of a liquid by nucleation is an important first order phase transition process. It is known that in order for elemental liquids to solidify homogeneously, it is necessary to supercool the liquid to a characteristic temperature below the thermodynamic melting point . Approximately 60 years ago Turnbull [J. Appl. Phys.21, 1022 (1950)] established the empirical rule that is approximately given by for several elemental metallic liquids. We show here that the magnitude of and for the metals can be accurately predicted from classical nucleation theory (CNT) provided the excess volume resulting from the density difference between liquid and solid be accounted for. Specifically, the density change accompanying the formation of a microscopic nucleus of the solid from the liquid results in a volume change in the surrounding liquid. When this is included in the free energy calculations within CNT, the resulting predictions for and for several metals with ranging from to 2900 K are in very good agreement with experimental measurements. This theory also shows that there is a universal character in the minimum nucleation barrier energy and the critical radius. The minimum barrier energy occurs at temperature for all the elemental liquids investigated, while the temperature dependencies of the barrier energy and the critical radius appear identical when expressed as a function of the scaled temperature .
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