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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 (T_UC) below the thermodynamic melting point (T_MP). Approximately 60 years ago Turnbull [J. Appl. Phys. 21, 1022 (1950)] established the empirical rule that T^*=|T_UC−T_MP| is approximately given by 0.18  T_MP for several elemental metallic liquids. We show here that the magnitude of T^* and T_UC 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 T^* and T_UC for several metals with T_MP ranging from ~200 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 T_N~0.27  T_MP 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 T_UC/T_MP. ©2008 American Institute of Physics

Contents

• INTRODUCTION
• THEORY
  • A.Modified CNT
• RESULTS
  • A.Characteristic undercooling in bulk liquids
  • B.Critical activation barrier energy and radius for solidification
• DISCUSSION AND CONCLUSION
• § 5
• REFERENCES
• FIGURES
• TABLES
• FOOTNOTES

Nucleation initiated phase transformation occurs in several natural and man made environments. It is involved in water turning to ice in biological systems,^1,2 solid phase regrowth of amorphous silicon,³ vapor phase growth of semiconductor nanowires,⁴ and templated growth of functional materials.^5,6,7 Therefore, understanding this process in quantitative detail is of broad fundamental and applied interest. Over the past century, classical nucleation theory (CNT) has been widely used to describe the energetics and kinetics of phase transformation.^8,9,10 This phenomenological theory has been successfully applied to a wide array of problems, including pattern formation in epitaxial systems,¹¹ nucleation in polymer material,¹² and formation of optically active oxide nanoparticles.^13,14 One critical feature of this theory is that an energy barrier must be overcome in order to form the stable microscopic nucleus of the new phase in the existing phase. In the case of solid to liquid phase transformation, the physical manifestation of the nucleation barrier is that the solid can be stable to temperatures well above its equilibrium solidification temperature, i.e., superheating.^15,16 Likewise, in liquid to solid transformations, the liquid can be supercooled well below the equilibrium solidification point. This phenomenon was observed three centuries ago in the case of solidification of water to ice by Fahrenheit.¹⁷ Despite the general understanding of phase transformation via CNT, a general shortcoming is its inability to predict certain quantitative features of phase transformation, such as the characteristic undercooling temperature at which the liquid must solidify. Thus, to date, only experimental measurements have been able to establish the magnitude of the characteristic undercooling temperatures, such as those done approximately 60 years ago by Turnbull and Cech^18,19 for a large number of elemental metallic liquids and more recently by Vinet et al.^20,21 These cumulative works as well as more recent measurements have been tabulated in various reviews and books.^10,22

In this work we show that thermodynamics in conjunction with CNT can give useful quantitative information about the energetics of homogeneous nucleation and solidification. The important step that enables this is to account for all energy contributions to the formation of a nucleus, including that arising from the density change during phase transformation. In the case of solidification, the internal Gibbs energy due to the excess volume of the liquid resulting from the density change contributes a substantial component to the total free energy, which, in the past, was typically evaluated only from the interface and volume energies of the two phases involved.^8,9 The resulting modifications in CNT permits prediction of the characteristic undercooling T^* and characteristic undercooling temperature T_UC for elemental metallic liquids, and our results are in very good agreement with the experimental measurements made over the past 60 years. The empirical observation of Turnbull,¹⁸ i.e., that most metals have T^*~0.18  T_MP, is also observed from our theory. This theory also shows that there is a universal character to the minimum nucleation barrier energy and the critical radius. The minimum barrier energy occurs at the temperature T_N~0.27  T_MP for all the elemental liquids investigated, while the temperature dependencies of the barrier energy and critical radius appear similar when expressed as a function of the scaled temperature T_UC/T_MP.

In the current standard form of CNT,^8,9 the free energy change due to nucleation is estimated from two fundamental contributions. The first is the decrease in the Gibbs volume energy g=g_S−g_L when atoms from the metastable liquid (g_L) go into the stable solid phase (g_S). The second is the increase due to the formation of the new interface between the solid and the liquid. CNT assumes that there is no contribution to the free energy change from the region surrounding the nanoscopic liquid volume which changes into solid. This implies that any volume change resulting from the liquid to solid transformation will be accommodated by the surrounding material without any additional energy. From a thermodynamic viewpoint, this situation is similar to an open system in which energy and matter are transferred freely into or out of the process without influencing nucleation. For a spherically shaped volume of the solid nucleating inside a bulk liquid, the magnitude of the negative volume free energy term will increase as the cube of the sphere radius, i.e., (4/3  r³)g, while the positive interfacial free energy term will increase as the square of the radius 4r²_SL, where _SL represents the solid-liquid interfacial free energy. In this picture the total free energy change due to the formation of a nucleus is therefore

Given that g is negative and _SL is positive for all T<T_MP, the behavior of G/kT as a function of radius r, where k is the Boltzmann's constant and T is the temperature, is readily evaluated from the sum of the volume energy (solid line in Fig. 1) and surface energy (dotted line in Fig. 1) contributions, respectively. The resulting sum (curve in open circles in Fig. 1) gives the total free energy change, and it shows a maximum value G^*/kT at the critical radius r^*. Mathematically, this critical r^* occurs at the location where dG/dr=0, giving

and therefore the height of the critical nucleation barrier is

At the outset this model suggests a simple way to estimate the critical nucleus and barrier height. However, a deeper investigation into the behavior of r^* and G^* brings out some of the quantitative failings of the existing CNT. The first is that at the thermodynamic melting temperature T_MP, which is defined as the temperature at which g=0, the critical radius and the energy barrier have singularity. Second, if one uses the practical form of g=h_fT/TMP,⁸ where h_f is the heat of formation of the solid from the liquid and T=T−T_MP is the degree of undercooling, one finds that the barrier energy will monotonically decrease with increasing undercooling, and there is no way to predict a characteristic undercooling temperature below which solidification must proceed. As we show next, these issues can be resolved if the free energy change associated with the liquid to solid transition is correctly evaluated.

Figure 1.

In contrast to the situation presented above, here we consider nucleation energetics in a closed thermodynamic system that can exchange energy, but not matter, with its surroundings. A system of this kind would be a liquid drop in equilibrium with its vapor and enclosed in a rigid container, such as that would be used in undercooling experiments involving levitated drops.¹⁰ The important difference from CNT case can be qualitatively understood from the following analysis for the formation of a nucleus in the interior of the drop. When a nanoscopic volume of the liquid changes to solid with a different density, this density difference must be accommodated by the surrounding liquid material (see Sec. IV). In the case of CNT it was implicitly assumed that the surrounding did not influence nucleation energetics. However, as we show below, for the final equilibrium state consisting of a solid nucleus in the liquid drop, the density change introduces a correction to the net free energy associated with nucleation and results in surprisingly accurate predictions of the characteristic undercooling temperature.

Consider homogeneous solidification inside a freely deformable spherical drop of initial volume V₀ made from a liquid of density _L and enclosed in a closed chamber, as shown in Fig. 2(a). We assume that there is no change in concentration in going from the liquid to solid phase. Let a spherically shaped solid nucleus of volume v_s form inside the bulk spherical liquid drop, as shown in Fig. 2(b). The solid nucleus v_S will occupy a region originally occupied by the liquid of volume v_L, but because of the density change in going from liquid to solid, the two volumes will be related as v_S=_L/_Sv_L, where _S and _L are the densities of the solid and liquid states, respectively. The resulting excess volume given by v_L−v_S will be eventually adjusted by rearrangement of liquid in the surrounding drop. As shown in Fig. 2(b) for the situation of liquids with lower density than the solid (i.e., _L<_S, as is the case for all the elemental metallic liquids), the final surrounding drop size will decrease from its initial size. More importantly, when the total free energy difference is estimated, as shown below, the difference v_L−v_S introduces an additional internal energy contribution of magnitude (_L−_S/_L)g_Lv_S. For the bulk spherical liquid drop with volume V₀=4R and surface area A₀=4R made from a liquid of density _L levitated in vacuum such that the outer surface has a liquid-vapor surface tension _LV, the total free energy of this initial drop will be the sum of the volume free energy and the surface energy, giving

where g_L=h_L−Ts_L is the thermodynamic Gibbs volume energy of the liquid state. Let the solid nucleus form at the center of the spherical drop. Because of mass conservation and density change, the solid nucleus will be related to the original liquid volume by _Sv_S=_Lv_L, where v_L will be the volume of the liquid transformed. Now the remaining bulk liquid volume will be V=V₀−v_L, and the final bulk spherical drop size will be given by V_final=V₀−v_L+v_S in order to account for the volume change in going from the liquid to solid nucleus and the resulting liquid flow. Based on mass conservation this final bulk drop volume can be expressed as V_final=V₀+(/_L)v_S, where =_L−_S. As a result, the final spherical bulk drop radius R_final will be related to the initial spherical radius R_bulk as R=R+(3/4)(/_L)v_S. This expression can be simplified by expanding and ignoring higher order terms, giving

Figure 2.

Using these quantities the final free energy of the bulk drop containing the solid nucleus can be expressed as

where the first two terms are identical to those in Eq. (4) but modified to account for the new liquid volume and outer surface area of the bulk drop, the third term is the increase in energy due to the new solid-liquid interface with area a_S=4r, and the final term is the contribution from the thermodynamic Gibbs volume energy of the solid nucleus of volume v_S=4/3  r. The total free energy change is therefore

and using A_final−A₀=4(R−R)~(2/_L)(v_S/R_bulk) and V−V₀=−_S/_Lv_S, one gets

The critical radius r is obtained from dG/dr=0 as

and substituting for r^* in Eq. (7), one gets the critical barrier energy G as

where g_true=g+/_Lg_L+(2/_L)(_LV/R_bulk).

The important result to note is that r and G differ from their CNT values only in the form of the denominator, i.e., g_true instead of g. The consequences of this form for the case of bulk-sized droplets are discussed next.

In the above calculation we have assumed that the change in size of the drop due to condensation/evaporation as a result of the change in equilibrium vapor pressure at the top liquid-vapor surface is negligible. The equilibrium vapor pressure can change due to two reasons. The first is due to a change in the liquid temperature dT following a nucleation event, and the second is due to a change in the Laplace pressure of the liquid drop due to the change in radius, as discussed in Sec. II A. However, as we show next, both these effects are negligible for drops of microscopic size (i.e., ~1  µm or larger). The temperature change dT in the surrounding liquid will be due to the heat of fusion released by the formation of the nanoscale solid nucleus. An estimate of dT can be made from energy balance considerations, which gives dT=(h_fusv_SS)/(_LC_p,LV_finalL). For metals, the order of magnitude of the various quantities involved is heat of fusion h_f~10⁶  J/kg, _L~_S~10⁴  kg/m³, and heat capacity of the liquid C_p,L~10²  J/kg K. Consequently, the rise in temperature will be dT~(v_S/V_final), and if one assumes a solid nucleus of radius ~1  nm and liquid drop of radius ~1  µm, one gets a temperature rise of dT~O(10⁻⁹)  K, which is clearly a negligible change. Similarly, one can estimate the magnitude of the change in Laplace pressure dP due to change in radius of the drop as dP=2_LV(3/4_L)(v_S/R) that gives a change of approximately dp~10⁻¹⁸  N/m². Given that the equilibrium vapor pressure for the flat surface of metals at their T_MP is typically 10⁻⁶  N/m², the contribution from the Laplace pressure change can also be neglected in an estimate of the condensation/evaporation process. What is more important to note is that in the limit of bulk size, the Laplace term in g_true can also be neglected. The Gibbs volume energy g_L for elemental metallic liquids is typically of order ~10⁴  J/mole, while the Laplace pressure is of the order _L−V/R_bulk~1  J/mole for a 1  µm radius particle, and therefore the Laplace term can be easily ignored for drops of bulk size leaving only the g and g_L terms in the denominators of Eqs. (8),(9). Therefore, the net consequence of performing the bulk free energy calculation that includes the density difference between the liquid and solid is that an additional contribution of (/_L)g_L must be added to the volume free energy contribution.

In this limit of bulk size, the behaviors of Eqs. (8),(9) are now intricately linked to the magnitude of the Gibbs volume energies g_S and g_L and to the density change. Here we specifically analyze the behavior for elemental metallic liquids in which the density change in going from liquid to solid is negative, i.e., <0. Of the several implications of this modified description of homogeneous solidification within CNT, we specifically discuss those that either have supporting experimental evidence or which can be tested by future experiments. The immediate consequence of course is that there is no singularity in the critical radius or the barrier height at T=T_MP because g_true=g+(/_L)g_L0 when g=0. In fact, a singularity occurs at the temperature when g+(/_L)g_L=0. From thermodynamics we know that for temperatures below the melting point, g_L will in general be less negative than g_S, as shown in Fig. 3(a) for the case of Ag metal. Therefore g_S−g_L will be negative for all T<T_MP. However, the (/_L)g_L term contributes a positive value thus shifting the denominator of Eq. (9) [or Eq. (8)] to less negative values. In fact, more importantly, at some characteristic temperature T_UC below the T_MP, the sign of the denominator changes from positive to negative values, as shown in Fig. 3(b). When this happens, the sign of Eq. (8) is now positive, as would be required to have a real critical radius. Therefore, this temperature signifies a characteristic point, which is defined as the characteristic undercooling temperature T_UC. To determine accurately this characteristic undercooling temperature, one must use the first principles definition of g(T)=g_S(T)−g_L(T) rather than the approximate form involving the enthalpy of formation. We have made use of the empirical tabulations from existing thermodynamic databases that represent the Gibbs energy g as a polynomial function of T for the liquid and solid phases.²³ From these polynomials, the undercooling can be obtained in two steps. First the accuracy of the polynomial is verified by ensuring that the T_MP is accurately predicted. This is done by determining the temperature at which g_L=g_S, which is the thermodynamic definition of the T_MP. A typical calculation is shown in Fig. 3(a) for the case of Ag. The crossover occurs at a temperature of 1233 K, which is in good agreement with the known T_MP of Ag. Next, the function g+(/_L)g_L is evaluated as a function of T, and the characteristic undercooling temperature T_UC is determined by the location at which the curve crosses over from positive to negative values. This is shown in Fig. 3(b) for several different metals. All the calculations are done using the bulk liquid and solid densities at the T_MP taken from Ref. 24.

Figure 3.

In Table I we have tabulated the theoretically estimated and experimentally measured magnitudes of T^* for several metals with melting point ranging from ~200 to 2900 K, along with their _L and _S at T_MP. The experimental measurements were taken from Ref. 22 in which updated values from various works have been tabulated. In Fig. 4, the theoretical (open triangles) and experimental values (solid circles) are plotted as a function of melting temperature. Clearly, the theoretical predictions of characteristic undercooling are in very good agreement for the elemental metals, and the model also accurately captures the experimentally observed trend that T^* is generally 0.18  T_MP (dashed line), as was established by Turnbull.¹⁸ While Turnbull¹⁸ attributed this empirical behavior to the strong correlation to the ratio of the surface tension to the enthalpy of fusion, the calculations presented here suggest that the trend can also be explained by the physical picture based on the density change and the resulting excess volume contribution to the free energy change during solidification. In Table II, the theoretical and experimental values of T_UC are tabulated for several metals, and these are shown in Fig. 4(b). As expected, very good agreement between experiment (closed circles) and theory (open triangles) was observed over a large melting temperature range, and T_UC increased linearly with T_MP with a slope of 0.84 [shown by the dashed trend line in Fig. 4(b)].

Figure 4.

The behavior of the critical activation barrier energy G/kT as a function of temperature can be readily obtained from Eq. (9). The surface tension values _SL used to estimate the barrier energy are given in Table II. In Fig. 5(a), log(G/kT) is plotted as a function of temperature for several elemental liquids. Various important features can be noted. The first is that the energy barrier is very large at the characteristic undercooling temperature, as seen by the rapid increase in the barrier energy at higher temperatures for the various liquids. The barrier energy drops rapidly with increasing undercooling, a feature that is consistent with the experimental observations of the rapid increase in nucleation rate below T_UC.¹⁰ Second, there is a broad minima in the barrier energy at lower temperatures, and the temperature at which the minimum value of the activation barrier occurs (T_N) can be readily determined and is shown plotted in Fig. 5(b) as a function of T_MP for several liquids. These values are also tabulated in Table II. The behavior for all the liquids presented here falls on a linear trend with a slope of 0.27 implying that the minimum barrier occurs at temperature T_N~0.27  T_MP [shown by the dashed trend line in Fig. 5(b)]. In fact, one of the significant findings from this modified CNT model is the universal character of nucleation energetics for the elemental liquids. In Fig. 6(a) the critical barrier energy is plotted as a function of the scaled temperature T_UC/T_MP for several liquids. The similarity of the shapes as well as the similar locations of the energy barrier minima implies that the energetics of nucleation in the elemental liquids possesses a universal behavior. The size of the critical nucleus r with temperature can also be readily evaluated. In Fig. 6(b) r is plotted as a function of the scaled temperature T_UC/T_MP for several liquids. Again, a universal behavior is seen for the liquids. The r decreases monotonically with temperature, and its typical magnitude ranges from ~0.2  nm at the lowest temperatures up to ~100  nm near the characteristic undercooling temperature for all the liquids presented.

Figure 5.

Figure 6.

The calculation presented in this work differs from the previous standard calculations of nucleation energetics within CNT (Refs. 8,9) in one significant respect. The essential difference is that here we estimate the free energy change by including the contribution to the energy from the density change, which is an excess energy term of magnitude (/_L)g_L. The reason for including this excess is simply the fact that the liquid surrounding the solid nucleus cannot be ignored in the free energy calculation. In addition, since nucleation is often assumed to have a time-dependent behavior, it is also important to analyze the results in the framework of the actual microscopic processes involved. If one assumes that the precursor to a nucleation event is a density fluctuation that provides the right concentration of atoms for the solid phase, then two key events must occur in order to form the solid nucleus. The density fluctuation introduces a stress in the surrounding medium due to the imbalance in the density of liquid atoms. Since the final state of nucleus and liquid will be stress free, this stress must eventually be relaxed. A second event is the rearrangement of atoms within the density fluctuation to form the crystal (nucleus) of the solid phase. The free energy calculation presented in this work assumed that the nucleation barrier is a result only of the energy difference in the solid nucleus and the liquid phase, thus implying that the rearrangement of atoms was the more relevant of the two processes. This assumption can be justified by a time scale analysis of the two events. The atomic rearrangement will be characterized by the time scale _D=(r^*)²/D_at, where D_at is the characteristic atomic self-diffusion coefficient for the solid phase and r^* is the characteristic nucleus size. On the other hand, stress relaxation will occur by interatomic vibrations that will propagate through the surrounding material and will have a characteristic time of _S~r^*/v, where v is the velocity of sound. If stress relaxation takes longer than atomic rearrangement within the nucleus, then nucleation energetics must include effects due to stress buildup/decay in the liquid.²⁵ Meanwhile, if atomic rearrangement is the slower process, then the approach presented in this work would more accurately describe the nucleation energetics. The lower limit for the characteristic time scale for atomic rearrangement is _D=(r^*)²/D_at~10⁻¹¹  s, where D_at is typically of the order of ~10⁻⁹  m²/s for the elemental metals at their melting temperature²⁶ and r^* was taken to be the size of a single unit cell of the crystal, i.e., 10⁻¹⁰  m. On the other hand, _S can be estimated by choosing the typical sound velocity in liquid metals, which is of the order 10³  m/s (Ref. 27) giving _S~10⁻¹³  s, which is much smaller than _D. This time scale is also consistent with experimental measurements of stress relaxation following density fluctuations in elemental metallic liquids.²⁸ This analysis, along with the very good agreement between our model and the experimental measurements of characteristic undercooling (T^* and T_UC), can be said to support the model presented in this work post facto.

In conclusion, for nearly a century, a clear understanding of phase transition via nucleation has generally been complicated by the often inseparable role of energetics and kinetics. In order to advance our understanding of this important phenomenon, it is therefore necessary to gain quantitative insight into the independent role of energetics or kinetics in processes such as solidification. In this work we have shown that thermodynamics in conjunction with CNT can give useful quantitative information about the energetics of homogeneous nucleation and solidification. The important step that enables this advance is to account for the free energy due to the excess volume of the liquid resulting from the density change, in addition to the interface and volume energies of the two phases involved. The resulting modification to CNT resolves some important issues with respect to the first order solidification phase transition. This theory accurately captures the characteristic undercooling temperature T^* and T_UC for elemental metallic liquids, as exemplified by the excellent agreement with Turnbull's¹⁸ empirical observation. This theory also shows that there is a universal character in the critical nucleation barrier energy and the critical radius. The minimum barrier energy occurs at temperature T_N~0.27  T_MP for all the elemental liquids investigated, while the temperature dependencies of the barrier energy and critical radius appear identical when expressed as a function of the scaled temperature T_UC/T_MP. This theory could be used to further our understanding of novel materials and processes, including their use in evaluating and estimating the characteristic undercooling in unexplored liquids, to predict the dependence of nucleation rate on undercooling temperature and to estimate the influence of system size on nucleation. Based on the quantitative success of these results, advances in the synthesis, processing, and modeling of nano- and bulk materials controlled by nucleation phenomenon could be anticipated.

R.K. acknowledges the support by the National Science Foundation through CAREER Grant No. DMI-0449258.

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Fig. 1. (Color online) In homogeneous phase transformation under CNT, a spherical nucleus of the stable phase forms with radius rr^* when it can overcome the maximum energy barrier G^*(r)/kT. In the existing CNT, the magnitude of G^*/kT 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 r³ (solid curve) and an increase due to the new interface formed between the two phases which increases as r² (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 r³ (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. First citation in article

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Fig. 2. (Color online) Homogeneous nucleation of a solid nucleus within a liquid drop enclosed inside a closed thermodynamic system. (a) Initial drop of liquid volume V₀ and density _L. (b) A solid nucleus of volume v_S and density _S forms from a volume of liquid v_L at the center of the bulk liquid drop. If the solid has higher density than the liquid, i.e., _S>_L, then v_S<v_L and the surrounding liquid will rearrange to form a final drop whose outer radius will be smaller then the initial drop. First citation in article

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Fig. 3. (Color online) Calculation of the characteristic undercooling temperature T_UC. (a) Thermodynamic Gibbs volume energy curves g_S and g_L for Ag. The intersection of the curves is the thermodynamic melting temperature T_MP. (b) The characteristic undercooling temperature for each metal is obtained from the plot of (g_S−g_L)+(/_L)g_L as a function of temperature. The location of the crossover from negative to positive values gives T_UC. Representative plots for Zr, Ag, and Hg are shown. First citation in article

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Fig. 4. (a) Plot of characteristic undercooling T^*=|T_UC−T_MP| as a function of melting temperature T_MP 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 T^*=0.18  T_MP consistent with Turnbull's (Ref. 18) observations. (b) Plot of the characteristic undercooling temperature T_UC vs T_MP. The linear trend line (dashed line) has a slope of 0.84. First citation in article

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Fig. 5. (Color online) (a) Variation of the critical activation barrier energy log(G/kT) as a function of temperature for various elemental liquids. The large barrier observed at higher temperatures corresponds to the characteristic undercooling temperature T_UC. The activation barrier has a broad minimum at a temperature below T_UC for the metals. (b) The location of the minimum in the nucleation barrier T_N is plotted vs T_MP. The best fit to the data (dashed line) gives a linear trend with a slope of 0.27 and intercept of 31 K. First citation in article

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Fig. 6. (Color online) (a) Characteristic behavior of the barrier energy log(G/kT) vs the scaled temperature T_UC/T_MP 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 r vs the scaled temperature T_UC/T_MP for several metals. Once again, all the curves have similar shapes as well as magnitude of r (which varies between ~0.2 and 100 nm) over a large temperature range below the characteristic undercooling temperature. First citation in article

Table I. The various metals, their solid phase, and their densities in the liquid and solid states used to estimate the characteristic undercooling T*. The density values are evaluated at the melting point TMP and were obtained from Ref. 24 unless otherwise indicated. The crystal phase determined the values of gS used from Ref. 23. The T values were obtained from Ref. 22, while T was evaluated from the model presented here.
Metal	Phase	TMP (K)	L (gm/cm3)	S (gm/cm3)	T (K)	T (K)
Mo	BCC	2896	9.34	10.08 a	520	449
Nb	BCC	2750	7.95	8.26 b	525	516
Zr	HCP	2128	6.24	6.34	347	337
Fe	FCC	1809	7.03	7.28	420	346
Co	FCC	1766	7.76	8.03	333	295
Ni	FCC	1727	7.91	8.26	321	325
Cu	FCC	1356	8	8.84	236	227
Au	FCC	1336	17.36	18.25	230	319
Ag	FCC	1234	9.3	9.7	227	227
Al	FCC	933	2.39	2.54	160	152
Pb	FCC	600	10.68	11.05	152	133
Hg	HCP	2001	13.69	14.2	54	48
aReference 30.
bReference 31.

First citation in article

Table II. Table showing surface tension, characteristic undercooling temperature TUC, and temperature of minimum barrier energy TN. The surface tension values were obtained from Ref. 29 and were used to estimate the barrier energy and critical radius. The T values were calculated from Ref. 22, while the T is the theoretical value for the characteristic undercooling estimated from our theory. TN is the temperature at which the nucleation barrier is a minimum.
Metal	TMP (K)	SL (J/m2)	T (K)	T (K)	T (K)
Zr	2128	0.193	1781	1791	610
Fe	1809	0.204	1508	1463	510
Co	1766	0.234	1433	1471	521
Ni	1727	0.255	1406	1403	490
Mn	1514	0.206	1185	1373	490
Cu	1356	0.177	1120	1129	390
Au	1336	0.132	1106	1017	350
Ag	1234	0.126	1007	1006	370
Al	933	0.121	772	781	270
Pb	600	0.033	448	467	160
Hg	2001	0.024	146	153	120

First citation in article

^*Author to whom correspondence should be addressed. Electronic mail: ramki@utk.edu. Present address: Department of Materials Science and Engineering & Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, TN 37996.

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