MELTINGATTHELIMITOFSUPERHEATING Sheng-Nian Luo,1 Thomas J. Ahrens,2 and Damian C. Swift1
1P-24 Plasma Physics, Los Alamos National Laboratory, Los Alamos, NM 87545 2Seismological Laboratory, California Institute of Technology, Pasadena, CA 91125 Abstract. Theories on superheating-melting mostly involve vibrational and mechanical instabilities, catastrophes of entropy, volume and rigidity, and nucleation-based kinetic models. The maximum achievable superheating is dictated by nucleation process of melt in crystals, which in turn depends on material properties and heating rates. We have established the systematics for maximum super- heating by incorporating a dimensionless nucleation barrier parameter and heating rate, with which systematic molecular dynamics simulations and dynamic experiments are consistent. Detailed mi- croscopic investigation with large-scale molecular dynamics simulations of the superheating-melting process, and structure-resolved ultrafast dynamic experiments are necessary to establish the con- nection between the kinetic limit of superheating and vibrational and mechanical instabilities, and catastrophe theories.
INTRODUCTION cluding Lindemann and Born’s criteria, order- Melting and freezing as first-order phase disorder transition, catastrophes of entropy, vol- changes and their related kinetics, are of ubiqui- ume and rigidity, and nucleation-based kinetic tous theoretical and experimental interest in con- models are briefly reviewed. We present certain densed matter physics, materials science and en- details on the recently developed systematics for gineering, geophysics and planetary sciences.[1] the maximum superheating and undercooling.[1] Metastable superheating and undercooling are Experimental, theoretical and simulation direc- inherent in melting and freezing processes. De- tions for future investigations of superheating- termining the degree to which a solid can be melting process are presented. superheated and a liquid undercooled, is a fun- damental and challenging issue. Experimen- SUPERHEATING-MELTING tal investigation of the maximum superheating THEORIES is particularly difficult due to the existence of Solids differ distinctly from liquids in both heterogeneous nucleation sites (e.g. free sur- their long-range order and ability to resist shear- faces and defects), and the difficulty in achieving ing. The definitions and criteria for melting high heating rates while making sensible mea- mostly involve vibrational and mechanical insta- surements. Theoretical efforts in understand- bilities and order-disorder transitions.[1] Linde- ing superheating-melting have been seriously un- mann’s vibrational criterion[1] states that melt- dermined by the paucity in superheating data. ing occurs at the onset of an instability when Molecular dynamics (MD) simulations have been the atomic displacements (e.g. the root-mean- utilized to probe melting and freezing processes squared displacements) during thermal vibra- at atomic level, and serve an important comple- tions exceed a certain threshold. Born’s mechan- mentary approach to theoretical and experimen- ical criterion[1] states that the stability against tal techniques. shearing stress vanishes (e.g. for cubic lattice Previous superheating-melting theories[1] in- c44 =0wherec44 is the elastic constant in
Downloaded 06 Mar 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp Voigt’s notation) and such shearing instability molar volumes of liquid (glass) and crystal are v is essentially melting. Melting is also interpreted equal. Beyond Tm, liquid would become denser v as structure transition from order to disorder as for normal materials, thus melt preempts at Tm. proposed by Lennard-Jones and Devonshire.[1] Isochoric melting defines a smaller superheating v Such a order-disorder transition is arguably at- Tm at the volume catastrophe. Tallon further tributed by Cahn[1] to the spontaneous produc- argued that rigidity instability occurs where the tion of intrinsic lattice defects. density of the superheated crystal equal to that of the liquid at the freezing point. This defines r C' a superheating state Tm at the catastrophe of L' rigidity. The hierarchy of catastrophes of a suc- s Tm G' cession of stability limits for crystalline state is elastic rigidity (T r ), volume (T v )andentropy v m m Tm s (Tm).[1]
s r
Entropy Tg Tm 10 undercooling superheating v Tg Q = 1 K/s C L 8 Q = 1 K/s Ga G Tm
Temperature Hg 6 Bi,Pb FIGURE 1. Schematic of entropy vs. temperature: A Q = 106 K/s Sn β Ge hierarchy of catastrophes as a succession of stability limits Te for the crystalline state. After Tallon.[1] 4 In,Fe Q = 1012 K/s Ni 6 Sb Q = 10 K/s These instabilities in thermal vibration and 2 Ru resistance to shearing, and breakdown of long- Se Q = 1012 K/s 0 range order are definitions of melting which in- 0 0.5 1.0 1.5 adequately describe the mechanism of melting, θ c = Tc / Tm i.e. the kinetics of melting. Similarly, equilib- rium thermodynamics simply states that liquid FIGURE 2. The systematics[1] of maximum superheat- ing and undercooling for elements: β =(A0−b lg Q)θc(1− has lower Gibbs free energy than solid above 2 θc) . Circles are experimental value of undercooling at melting temperature, Tm. Without considering cooling rate Q ∼ 1 K/s, and diamonds are calculated su- nucleation process, catastrophes of certain phys- perheating at Q ∼ 1 K/s. Solid and dotted curves are 6 12 ical quantities (e.g. molar entropy s, molar vol- plots with Q =1,10 and 10 K/s, respectively. Dotted θ − / ume v and rigidity r) are employed by Tallon[1] curves denote the undercooling portions for c =0 1 3. The maximum of β for undercooling occurs at θc =1/3 to define a hierarchy of the limit of superheat- for each Q. The elements within the double-headed arrow ing (Fig. 1). The entropy of solid (along CC ) areTi,Al,Au,Cu,Hf,Cd,Pd,Ag,Co,Pt,Ta,Rh,Zr, s s Mn, Si, Sb, Ni, In and Fe in β-increasing order. intersects that of liquid (LL )atTg and Tm s s (Tg Downloaded 06 Mar 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp (∆Hm)andTm, and heating rate Q.Anap- constant prefactor.[1] We define the energy bar- propriate estimation of superheating should be rier for nucleation, β, as a dimensionless quan- nucleation-based. Given γsl from undercooling tity, experiments and assuming nucleation rate I =1 −1 −3 3 s cm , Lu and Li[1] predicted less amount 16πγsl β(γsl, ∆Hm,Tm)= 2 (2) of superheating than Tallon’s hierarchy.[1] Reth- 3∆HmkTm felder et al.[1] estimated superheating assuming a critical volume of nucleation is formed during and the reduced temperature as θ = T/Tm,and a given time scale. Although both studies are β nucleation-based, their studies did not reveal the f(β,θ)=exp{− }. (3) θ(θ − 1)2 systematic nature of melting, and heating rate was not included. Thus, the nucleation process is essentially depen- We recently developed a framework[1] for the dent on β and θ. We denote the maximum super- systematics of maximum superheating and un- heating and undercooling as θc = Tc/Tm.Pre- dercooling which are based on undercooling ex- vious undercooling experiments yielded values of − − periments and classical nucleation theory, and in- θc ( denotes undercooling), γsl,∆Hm and Tm corporate heating rates. Systematic MD simula- (thus β) for elements and compounds.[1] θc obvi- tions and dynamic melting experiments demon- ously depends on heating (or cooling) rate (Q). − strate significant consistency with the systemat- Normally the reported experimental values of θc ics. Next we discuss the systematics for maxi- are for Q0 ∼ 1K/s. mum superheating. SYSTEMATICS FOR MAXIMUM 2.8 Al Heating SUPERHEATING Cooling ) The technical challenges of achieving homo- 3 2.6 geneous nucleation in melting experiments limit T1,m the amount of data of superheating which in turn limits development of a practical superheating- 2.4 T1,c melting theory. But a significant number of Density (g/cm freezing experiments have been conducted where T2,m appreciable undercooling has been observed with 2.2 homogeneous nucleation of crystals in liquids. As 250 750 1250 T (K) γsl,∆Hm and Tm are common to both melting and freezing, undercooling experiments would al- low us to make predictions on superheating. FIGURE 3. Typical single- and two-phase MD simu- Based on classical nucleation theories, the nu- lations of the melting and refreezing behavior: density T cleation rate[1] for both melting and freezing can vs. . A complete hysteresis of density forms dur- ing stepped heating-cooling process for Al. T1,m and be expressed and approximated as T1,c are the single-phase melting and freezing temper- ature at the superheated and undercooled states, respec- ∆Gc tively. T2,m is the equilibrium melting temperature from I = M(m, T )exp{− g(φ)}≈I0f(β,θ)(1) + the two-phase simulations. Thus, θc = T1,m/T2,m and kT − θc = T1,c/T2,m. where M is a function of material properties (m) and temperature (T ). ∆Gc is the critical Gibbs As β is common to both melting and freezing, − free energy for nucleation, k Boltzmann’s con- values of β and θc (Q0) allow us to predict the − stant, and g(φ) a geometrical factor depending maximum undercooling θc (Q) and superheating + on the wetting angle φ of a heterogeneous nucle- θc (Q) at certain cooling and heating rate Q.For ant. For homogeneous nucleation, g(φ)=1,the steady-state homogeneous nucleation of crystals case assumed in the following discussions. I0 is a from liquid (or melt in solid), Kelton[1] proposed Downloaded 06 Mar 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp that the probability x for a given amount of par- tions of superheating and undercooling. Super- ent phase of volume v containing no new phase heating was observed previously in a few stud- under certain cooling (or heating) rate Q is ies.[1] We conducted systematic MD simulations with single- and two-phase techniques on fcc 1 vT I0 metals(Al,Ni,Cu,Rh,Pd,Ag,Ir,Pt,Auand x =exp{± m f(β,θ)dθ} (4) Pb) and Be.[1] A typical example is shown in Q θc Fig. 3 for Al where superheating and undercool- where + refers to superheating and − to under- ing can be determined. Current and previous cooling. The parameters for undercooling experi- simulations yielded values consistent with the ∼ 12 ments at Q 1K/s,suchasγsl,∆Hm, Tm (thus β − θc − Q systematics at Q ∼ 10 K/s. Thus β), and v can be regarded as equal to those for the empirical systematics are validated at atomic superheating and undercooling at different heat- level from MD simulations.[1] ing and cooling rates. By assuming x and I0 Superheating has been observed in planar is approximately equal for the undercooling and impact experiments with light-gas gun loading superheating cases, the maximum superheating and intense laser irradiation on silicates, alkali and undercooling under any Q can be calculated halides and metals.[1] A representative example − 12 from experimental value of θc (Q0) (Fig. 2). The of shock-induced superheating (Q ∼ 10 K/s) is numerical relationship[1] between β, θc and Q is shown in Fig. 4 for CsBr. Experimental super- fitted as heating values compare favorably to the predic- 2 tion of the systematics.[1] β =(A0 − b lg Q)θc(θc − 1) (5) DISCUSSION where A0 =59.4, b =2.33, and Q is normalized We have established the β − θc − Q system- by Q0 = 1 K/s. Eq. (5) is referred to as the atics for the maximum superheating and under- β − θc − Q systematics for the maximum super- cooling consistent with MD simulations and dy- heating and undercooling. namic experiments. Future experimental efforts will employ in-situ structure-resolved melting ex- periments with exploding wire and shockwave 6000 e techniques.[2] MD simulations and theoretical ef- forts are needed to establish a universal relation- c’ 5000 ship between kinetic limit of superheating and Lindemann MC d various definitions of melting, and catastrophe (K) c 4000 b T theories. The effects of heterogeneous nucleation sites at high heating rates, low dimensions and 3000 a Boness & Brown, 1993 anisotropy, are also of interest. 2000 0 20 40 60 80 100 ACKNOWLEDGMENTS P (GPa) This work has been supported by U.S. NSF Grant EAR-0207934. S.-N. Luo is sponsored by a Director’s Post-doctoral Fellowship at Los FIGURE 4. Shock-melting experiments on CsBr[1] Alamos National Laboratory (P-24 and EES-11). demonstrate simultaneous drop in shock temperature and sound-speed (not shown), signaling melting of shocked REFERENCES crystal at higher shock pressures than Pc (the long dashed curve). Solid curves indicate the Hugoniot states. The 1. Luo, S.-N., and Ahrens, T.J., Appl. Phys. Lett. 82, dashed curve is the Lindemann melting curve (MC).[1] 1836 (2003); Luo, S.-N. and Ahrens, T.J., Phys. Earth bc segment denotes superheated states. Planet. Int. (in press) (2003); Luo, S.-N., Ahrens, T.J., C¸a˘gın, T., Strachan, A., Goddard III, W.A., 68 − − and Swift, D.C., Phys. Rev. B , 134206 (2003), The β θc Q systematics for maximum su- and references therein. perheating and undercooling are empirical in na- 2. Luo, S.-N., Swift, D.C., Tierney, T., Xia, K., Tschau- ture. An independent verification is MD simula- ner, O., and Asimow, P.D., this conference. Downloaded 06 Mar 2006 to 131.215.240.9. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp