APPLIED PHYSICS LETTERS VOLUME 82, NUMBER 12 24 MARCH 2003 Superheating systematics of crystalline solids Sheng-Nian Luoa) and Thomas J. Ahrens Lindhurst Laboratory of Experimental Geophysics, Seismological Laboratory, California Institute of Technology, Pasadena, California 91125 ͑Received 1 October 2002; accepted 30 January 2003͒ ␪ϭ Ϫ Systematics of superheating ( T/Tm 1) of crystalline solids as a function of heating rate (Q) are established as ␤ϭA(Q)(␪ϩ1)␪2, where the normalized energy barrier for homogeneous ␤ϭ ␲␥3 ⌬ 2 nucleation is 16 sl/(3kTm Hm), T is temperature, Tm melting temperature, A a Q-dependent ␥ ⌬ parameter, sl interfacial energy, Hm heat of fusion, and k Boltzmann’s constant. For all elements and compounds investigated, ␤ varies between 0.2 and 8.2. At 1 and 1012 K/s, Aϭ60 and 31, ␪ ϭ0.05– 0.35 and 0.06–0.45, respectively. Significant superheating is achievable via ultrafast heating. We demonstrate that the degree of superheating achieved in shock-wave loading and intense laser irradiation as well as in molecular dynamics simulations (Qϳ1012 K/s) agrees with the ␪ – ␤ – Q systematics. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1563046͔ ␪ϭ Ϫ ϭ ␤ ␪ Superheating ( T/Tm 1) of a crystalline solid occurs then I can be written as I I0 f ( , ), with when the long-range order of the crystalline structure is ␤ maintained up to certain temperature T above the equilib- ͑␤ ␪͒ϭ ͫϪ ͬ ͑ ͒ f , exp ͑␪ϩ ͒␪2 . 3 rium melting temperature Tm . Previously, the details of crys- 1 tal melting and the temperature range over which solids may Nucleation rate I is controlled by f (␤,␪), essentially by ␤ at be superheated have been investigated.1–7 In contrast to un- a given temperature. Equations ͑1͒–͑3͒ are also applicable to dercooling of liquid prior to crystallization, experimental su- the undercooling case. perheating of crystals is difficult as grain boundaries and free To estimate the magnitude of ␤, we note that ␥ surfaces lower the energy barriers for melt nucleation.8,9 Spe- sl ϳ0.1 J/m2, T ϳ103 K and ⌬H ϳ109 J/m3, yields ␤ cial experimental designs5 and rapid heating are required to m m ϳ1.2. Based on previous data,14 ␤ for elements is calculated superheat crystalline solids. Catastrophic melting1,2 and ho- ͑Fig. 1͒. For Group IVB-IIB elements, ␤ is 0.9–3.1, except mogeneous nucleation3,4 theories have been utilized to define for Hg ͑6.3͒. For most transition metals, ␤ϳ1.8. Due to the the limits of superheating, and a wide range of superheating unproportionally lower ⌬H and T , Group IIIA-VIA ele- (␪ϳ0.1– 2.0) is predicted. Here, we will investigate the sys- m m ments have larger ␤ values ͑2.5–8.2͒ except Al ͑1.5͒ and Se tematics of nucleation energy barrier for elements and com- ͑0.2͒. Figure 1 demonstrates the periodic nature of ␤ for pounds, and the corresponding superheating as a function of elements due to their periodic variations in electronic struc- heating rate. We also compare theory to superheating ture, with peaks occurring mostly at Group IIIA-VIA ele- achieved in experiments and simulations. ments and Hg. ␤ for compounds such as some alkali halides Homogeneous nucleation of melt may be described via ␥ 9–14 and silicates is similar. In general, sl increases with Tm and classical theories. Let I be the rate per unit volume of ⌬ ␥ ⌬ 14,15 Hm , because sl , Tm , and Hm are fundamentally related steady-state homogeneous nucleation of melt in solid: ␤ ␥ to binding energy. Thus, although is sensitive to sl , varia- ⌬ Gc IϭI expͩ Ϫ ͪ , ͑1͒ 0 kT ⌬ where Gc is the critical Gibbs free energy for nucleation, and k is Boltzmann’s constant. The prefactor I0 can be ob- tained experimentally or theoretically. Consider a spherical liquid nucleus of critical radius within a crystal lattice, the 14,15 ⌬ ϭ ␲␥3 ⌬ 2 critical nucleation energy Gc 16 sl/(3 Gsl) where ␥ ⌬ sl is the solid–liquid interfacial energy, and Gsl is the Gibbs free energy difference per unit volume between solid ⌬ Ϫ and liquid state, approximated as Hm(T Tm)/Tm where ⌬ ͑ Hm is the heat of fusion assuming heat capacities of liquid and solid are approximately equal16͒. If we define the nor- malized energy barrier for nucleation ␤ as 16␲␥3 ␤͑␥ ⌬ ͒ϭ sl ͑ ͒ sl , Hm ,Tm ⌬ 2 , 2 3 HmkTm FIG. 1. Normalized energy barrier ␤ as a function of atomic number Z. a͒Electronic mail: [email protected] Values for V, Mo, and W are estimated. 0003-6951/2003/82(12)/1836/3/$20.001836 © 2003 American Institute of Physics Downloaded 06 Mar 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp Appl. Phys. Lett., Vol. 82, No. 12, 24 March 2003 S. N. Luo and T. J. Ahrens 1837 ␤ ␥ tion in induced by that in sl could be offset by those in Tm ⌬ ␥ ϰ⌬ 2 2 and Hm . We also note that sl Hm /a , where a is the effective atomic surface area.12 Since ␤ lies between 0.2 and 8.2 for materials with a wide range of binding energy and other physical properties, we expect that ␤ remains at the same order under high pressures, ͑e.g., shock compression͒. Given the ␤ systematics, next we will develop a Q-dependent scheme to probe the systematics of superheat- ing characteristic of elements and compounds. The under- cooling experiments on various elements and compounds were documented,14 and serve as the basis for our calculating the superheating at various heating rates. The parameters for ␥ ⌬ undercooling such as sl , Hm , Tm , and molar volume V can be regarded as equal to those for superheating. In the case of superheating, the probability14 x of v moles parent phase containing no new phase ͑liquid͒ under heating rate Q is FIG. 2. Superheating vs normalized energy barrier for elements. Solid curve ϩ is the analytical fitting of ␪–␤ pairs at Qϭ1K/s ͑circles͒ to Eq. ͑5͒ with vT I V (1ϩ␪ ) m 0 Aϭ60. Dashed curve corresponds to Qϭ1012 K/s with Aϭ31. Superheat- xϭexpͫ Ϫ ϩ ͵ f ͑␪,␤͒d␪ͬ, ͑4͒ Q 1 ing of Ni is possibly overestimated ͑see Ref. 14͒. where ϩ denotes superheating, and the expression is similar ͑ Ϫ tematics ͓Eq. ͑5͒ and Fig. 2͔ based on undercooling for undercooling case denoted with ). If we assume I0 , v, and x for superheating are similar to those for undercooling, experiments and homogeneous nucleation theory. superheating ␪ϩ can be estimated from undercooling ␪Ϫ During shock-wave loading via planar impact or intense with Eq. ͑4͒. Given undercooling results14 under typical laser irradiation, a solid is subjected to ultrafast heating. It is cooling rate QϪϭ1 K/s, superheating (␪ϩ) under certain well known that pronounced superheating may occur in mo- ϩ ͑ ͒ heating rate Q can be obtained by solving lecular dynamics MD simulations of perfect crystals with ϩ 17 Ϫ 1 ϩ (1ϩ␪ ) three-dimensional periodic boundaries. Melt nucleation (1/Q ) ͐ Ϫ f (␪,␤)d␪ϭ (1/Q ) ͐ f (␪,␤)d␪. For (1Ϫ␪ ) 1 during such processes can be regarded as homogeneous. In superheating, we will consider two extreme cases: Qϩ ϩ ϩ contrast to low heating rate experiments at 1 K/s order, in ϭ1 K/s ͑corresponding to ␪ ) and 1012 K/s (␪ ). These s ns shock-wave loading ͑e.g., planar impact͒, the shock-front heating rates should be regarded as typical but not exact, rise time is on the order of 1 ns, and temperature increase because a factor of 102 difference in Q would yield a negli- upon shock compression is on the order of 103 –104 K; that gible difference in ␪ given a reasonable value of f (␤,␪). is, Qϳ1012 K/s. Q is similar in intense laser irradiation, de- Superheating at these heating rates is calculated in Fig. 2 for pending on energy deposited, irradiation time, and material elements. Group IIIA-VIA elements ͑except Se͒ can be su- properties. Similar values of Q apply for MD simulations. perheated more than transition elements ͑except Hg͒. The Melting under ultrafast heating has been investigated experi- numerical relationship between ␤ and ␪ can be fitted with a mentally employing both planar impact and intense laser ir- simple analytical form radiation. Sound speed and temperature measurement at ␤ϭA͑Q͒͑1ϩ␪͒␪2, ͑5͒ shock state unequivocally demonstrated superheating of met- als, alkali halides, and silicates.6,18–25 Time-resolved electron where A is a Q-dependent parameter. Aϭ60 and 31 at Q diffraction and mass spectroscopy demonstrated the occur- ϭ1 and 1012 K/s respectively, and decreases with increasing rence of superheating during intense laser irradiation.7,26,27 In Q. As a preliminary result, A(Q) is fitted as A(Q) Table I, we list the superheating results from planar impact 1 ϭ60– 2.4 log Q. While its exact physical meaning is not and laser irradiation experiments. For planar impact, if TH is 2 clear, A could be regarded as defining a relative characteris- the lowest temperature on liquid Hugoniot and TH the high- tic time scale for nucleation at various heating rates. For est temperature of superheated solid at corresponding shock ␪ϩ 2 silicates and alkali halides, the degree of superheating agrees pressure PH , then superheating on Hugoniot H is (TH ␪ Ϫ 1 1 ␪ϩ ␪ϩ ␪ϩ with the fits for elements. Superheating is inherently lim- TH)/TH . The comparison of H with ns shows that H ␤ ͑ ␪ϩ ited by material property , and increases monotonically except for Mg2SiO4) is close to ns , the prediction of su- with ␤ and Q ͑Fig. 2͒. Values of ␤ϭ0.2– 8.2 correspond to perheating systematics, given the modest variation of energy superheating of 0.05–0.35 at 1 K/s and 0.06–0.45 at barrier ␤ with pressure under shock loading and uncertainties 1012 K/s, respectively.