Heterogeneous to homogeneous melting transition visualized with ultrafast electron diffraction

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Science  29 Jun 2018:
Vol. 360, Issue 6396, pp. 1451-1455
DOI: 10.1126/science.aar2058

Golden ultrafast melting

Understanding fast melting of metals is important for applications such as welding and micromachining. However, fast melting leaves simulation as the only option for probing the process. Mo et al. performed ultrafast electron diffraction experiments on laser-pulsed gold films. This allowed detailed mapping of the melting process, which proceeds through two distinct regimes while the bonding behavior changes in unexpected ways. The results require adding new physical processes to high-energy melting models.

Science, this issue p. 1451


The ultrafast laser excitation of matters leads to nonequilibrium states with complex solid-liquid phase-transition dynamics. We used electron diffraction at mega–electron volt energies to visualize the ultrafast melting of gold on the atomic scale length. For energy densities approaching the irreversible melting regime, we first observed heterogeneous melting on time scales of 100 to 1000 picoseconds, transitioning to homogeneous melting that occurs catastrophically within 10 to 20 picoseconds at higher energy densities. We showed evidence for the heterogeneous coexistence of solid and liquid. We determined the ion and electron temperature evolution and found superheated conditions. Our results constrain the electron-ion coupling rate, determine the Debye temperature, and reveal the melting sensitivity to nucleation seeds.

Modern ultrafast laser techniques can bring materials into states far from thermal equilibrium. These ultrafast processes yield extreme material conditions with thermal energy comparable with the Fermi energy and the ion-ion coupling parameter exceeding unity, which is referred to as warm dense matter (1, 2). These conditions exist as a transient state in a variety of processes ranging from laser micromachining (3) to inertial confinement fusion experiments (4).

In the case of semiconductors, ultrafast optical irradiation can cause strong bond softening and nonthermal melting owing to the changes in the potential-energy surface of the lattice by the excited valence electrons (5, 6). By contrast, melting of metals is a purely thermal process governed by the energy coupling between the excited electrons and relatively cold lattices (7, 8). Two-temperature modeling coupled to molecular dynamics (TTM-MD) simulations predicted the existence of distinct melting regimes in ultrafast laser–excited gold (9). At low energy densities, the simulations predict that the slow ion heating rate will allow the solid-liquid phase transition to occur as heterogeneous melting initiated on liquid nucleation sites on surfaces, grain boundaries, or defects, resulting in a slow melting process limited by the subsonic melt-front propagation speed. However, higher energy densities can cause extremely high heating rates that exceed 1014 K/s, producing a superheated state in which homogeneous nucleation occurs catastrophically throughout the sample. Early electron-diffraction experiments observed long melt times in aluminum but did not observe the heterogeneous coexistence (10). Other experiments were performed in the homogeneous melting regime (7, 8, 11, 12), but determining melt times and testing theoretical predictions (9, 1315) have been elusive. In addition, whether the highly excited electron system can cause bond hardening (8, 16) or softening (17) in gold remains controversial, further complicating the understanding of ultrafast laser–induced solid-liquid phase transitions in metals.

Visualizing solid-liquid phase transitions and accurately measuring melt times in the heterogeneous and homogeneous melting regimes required the development of ultrafast electron diffraction (UED) with mega–electron volt energies (1820). Because of the reduced space charge effect, this device provides high peak currents (~100 mA), enabling measurements with extremely high signal-to-noise ratios. The electron beam is produced by means of ultrafast ultraviolet laser irradiation of a copper cathode and accelerated with a linac accelerator-type radio frequency gun; the same laser is split off to heat the sample, providing accurate cross timing between laser pump and the electron probe of <30 fs [root mean square (RMS)] (21, 22). Furthermore, mega–electron volt electrons form a nearly flat Ewald sphere on the reciprocal space, allowing simultaneous access to multiple orders of diffraction peaks (23). Last, multiple elastic scattering effects are less probable in nanometer-thin films at these energies because of their relatively large elastic mean-free-path (24).

We used 35-nm-thick 100-oriented single-crystalline (SC) or 30-nm-thick polycrystalline (PC) gold foils for the electron diffraction measurements. We uniformly excited these freestanding foils with 130-fs [full width at half maximum (FWHM)] 400-nm laser pulses at 4° incidence angle with flat-top–like intensity profiles of ~420 μm diameters. The RMS intensity variation of the optical pump within the probed area is better than 5%, ensuring uniform excitations in the transverse direction. We expect uniform heating in longitudinal direction because of the ballistic energy transport from nonthermal electrons excited by the laser pulses (12, 25, 26). We performed time-resolved electron diffraction measurements in normal incidence transmission geometry with 3.2 MeV electrons. We focused these relativistic electron bunches onto the target with diameters of ~120 μm (FWHM), bunch charges of ~20 fC, and pulse durations of ~350 fs (FWHM) (22).

We show three distinct melting regimes of the laser-excited SC gold, with raw diffraction patterns measured at various delay times for three selective absorbed energy densities ε (Fig. 1). At the highest energy density of 1.17 MJ/kg (Fig. 1, A to D), we first observed the decrease of Laue diffraction peaks (LDPs) intensity owing to the Debye-Waller effect immediately after laser excitation. At 2 ps delay, the heights of diffraction peaks relative to the adjacent backgrounds show obvious drops compared with the reference data taken before the arrival of the laser pulse (–2 ps) (Fig. 1D). At 7 ps delay, the data shows a weak liquid diffraction ring, which is a signature of the formation of a disorderd state. At 17 ps, the complete disappearance of the LDPs and the appearance of the two liquid Debye-Scherrer rings demonstrate that the sample is completely molten. Such a fast melting process is indicative of homogeneous melting according to MD simulations (9, 27).

Fig. 1 Mega–electron volt electron diffraction studies of the ultrafast solid-liquid phase transition in single-crystalline gold.

(A to C) Snapshots of the raw diffraction patterns at selective pump-probe delay times for homogeneous melting at ε = 1.17 MJ/kg. (E to G) Heterogeneous melting at ε = 0.36 MJ/kg. (I to K) Incomplete melting at ε = 0.18 MJ/kg. The radially averaged lineouts of the displayed diffraction patterns together with the reference lineouts taken at negative delay are shown in (D), (H), and (L) for these different energy densities, respectively. The color bars represent the scattering intensity in arbitrary units.

At intermediate energy density of 0.36 MJ/kg (Fig. 1, E to H), the low-order LDPs from regions of solid gold and the primary diffraction ring from liquid gold are simultaneously visible at the delay time of 20 ps. Such heterogeneous coexistence persists over long time scales until a 800 ps delay, long after electron-ion equilibration time of ~50 ps, demonstrating the solid-liquid coexistence at heterogeneous melting conditions.

At an even lower energy density of 0.18 MJ/kg (Fig. 1, I to L), the data show strong LDPs over a longer duration even at 100 ps, when ion temperature Ti should have reached its apex, but the melt front is propagating at a very slow rate. At 1000 ps, the sample is still in a solid-liquid coexistence regime and does not show complete disappearance of solid diffraction peaks, even at delay times as large as 3000 ps. We categorized this case as incomplete melting regime because the energy density deposited in the sample is below the requirement of complete melting expected at ~0.22 MJ/kg (28).

Our experiment provided high-quality liquid diffraction data spanning over a large reciprocal space that allowed us to determine its corresponding ion temperature. We realized this by comparing with the theoretical liquid scattering signal based on density functional theory (DFT)–MD simulations (23). We performed this analysis for ε = 1.17 MJ/kg, which yields a best fit Ti = 3500 K ± 500 K at the delay time of 17 ps, indicating a superheated state. The error bar here represents one standard deviation (SD) uncertainty.

Fig. 2 Energy density dependence of the lattice heating and disordering process.

(A to C) experimental data of (220) decay (red solid squares) at different excitation energy densities, compared with three different models to calculate Debye-Waller factor (23): ΘD(Ti) derived from (220) decay with constant gei (blue solid and dashed lines), ΘD(Te) from (16) with constant gei (magenta line), and ΘD(Te) used in (8) with gei from DFT calculations (gray line) (9). The (220) intensities were normalized with respect to those values from the laser-off diffraction pattern of the same sample. The error bars represent 1 SD uncertainties. (D to F) Temporal evolution of Te and Ti simulated by means of TTM with different gei at energy densities corresponding to data of (A) to (C). (G to I) Temporal evolution of ΘD (±SD) derived from the experimental (220) decay (red squares) up to Embedded Image, and the linear fit through individual data points as a function of Ti [blue solid line with the gray area representing error bar (±SD)], which are compared with the x-ray measurements at equilibrium conditions from (31), shown by the green dots; the DFT calculations from (16), shown by the magenta line; and results adopted from (8), shown by the gray line. In (A) to (C), the solid blue lines represent the data determined below the nominal Embedded Image of 1340 K, and the dashed lines represent Debye-Waller factor based on linearly extrapolated ΘD as a function of Ti.

We characterized the initial and final temperatures of the dynamic melting process, and thus the electron-ion coupling rate gei is constrained with a pair of coupled equations of the commonly used TTM to describe the temperature evolution of both electron and ion subsystems in ultrafast laser–excited materials (23). We used the temperature-dependent electron- and ion-specific heat Ce(Te) and Ci(Ti) of gold from (29) and (30), respectively. To first order, we assumed a temporally constant gei and determined its value by solving for the ion temperature at complete melt, taking into account the energy consumed by latent heat. The TTM yields gei = (4.9 ± 1) × 1016 W/m3/K at 1.17 MJ/kg. We compared the temporal evolution of Te and Ti using this value for gei with those based on simulated Te-dependent values for gei from (29) at 1.17 MJ/kg (Fig. 2D). We found that Te-dependent gei overestimates the ion temperature at complete melting by more than 60%.

We estimated the temporal evolution of the Debye temperature ΘD (23), a manifestation of interatomic potential (16), directly from the measured LDP decay using Ti determined from the TTM. We observed rapid decay of ΘD (Fig. 2, G to I). This differs dramatically from both the bond-hardening model based on the Te-dependent phonon spectrum in nonequilibrium conditions (16) and those values used in (8). Neither of these models agreed with our measured (220) LDP decay (Fig. 2, A to C). Below the nominal melting temperature (Embedded Image K), ΘD showed striking agreement with the x-ray measurements of gold under thermal equilibrium (31), suggesting that Ti is still the dominant factor for ΘD in nonequilibrium gold at much higher Te. Our finding is thus different from the previously reported bond-softening model of gold (17), which ascribed the effect to the highly elevated Te.

We arrived at the following picture for ultrafast melting of gold. We fit our entire dataset over three melting regimes with only one single assumption, that gei is weakly dependent on energy density, modestly increasing from 2.2 × 1016 W/m3/K at the lowest energy density to 4.9 × 1016 W/m3/K for the highest energy density. For example, using gei = 2.2 × 1016 W/m3/K for the lowest energy density from (26) leads to Debye temperature decay that is consistent with x-ray measurements at equilibrium conditions (Fig. 2I). For the heterogeneous melting regime, we linearly interpolated gei as a function of energy density between 0.18 and 1.17 MJ/kg, resulting in gei = 2.7 × 1016 W/m3/K at 0.36 MJ/kg. This value for gei yields a ΘD that is also consistent with data from (31) below Embedded Image (Fig. 2H).

An important observable to quantify the lattice dynamics in laser-induced melting processes is the complete melting time, τmelt, corresponding to the duration over which the long-range order is completely lost after laser arrival. We identified τmelt through the complete disappearance of (200) diffraction peaks, whose intensity is most resistant to thermal vibrations and disordering effects, together with the appearance of the two broad peaks of the liquid structure factor. For comparison, we also measured the complete melting time for the 30-nm-thick PC gold thin films. Both SC and PC samples show similar trends for τmelt (Fig. 3). As energy density decreased, τmelt first exhibited a gentle increase but then rose dramatically by orders of magnitude as energy density dropped below ~0.4 MJ/kg. The complete melting threshold was found at Embedded Image MJ/kg, which is similar to the expected value of ~0.22 MJ/kg (28). We attributed the observed different characteristic time scales of τmelt to homogeneous melting and heterogeneous melting, the two mechanisms of ultrafast melting. TTM-MD simulations (9, 27, 32) showed that the maximum velocity of melt front propagation is below 15% of the sound speed (~500 m/s for gold), above which homogeneous liquid nucleation dominates the melting process. Using this estimate suggests a minimum expected time for completion of the heterogeneous melting of 45 ps, including the time to reach Embedded Image (~10 ps). This estimation agrees with our observation of the transition between the two melting mechanisms.

Fig. 3 Energy density dependence of ultrafast laser–induced melting mechanisms in gold.

The measured melting time of SC gold and PC gold are represented by red squares and blue circles, respectively, as compared with TTM-MD simulation by Lin et al. (9) and Mazevet et al. (13). The vertical error bars are given by the time step intervals around the observed melting times, whereas the horizontal error bars represent 1 SD uncertainty of the measured absorbed energy density. Three melting regimes—homogeneous, heterogeneous, and incomplete melting—are identified from the measurements and indicated by the various background colors. The data located inside the gray shaded area are beyond the instrument limit of 3 ns for our experiments, and the two data points on the left are from measurements of below damage threshold.

Quantitatively, in the heterogeneous melting regime, our PC results are consistent with electron diffraction measurements of (33, 34) and indicate a shorter melting time than that of SC samples. We can explain this by the increased liquid nucleation seeds at grain boundaries of nanocrystalline structures and the additional crystal defects in PC samples (28). The nucleation seed density in PC samples can be estimated with the measured τmelt and calculated melt front velocities (10). For example, in the case of 0.28 MJ/kg, it takes ~20 ps to reach Embedded Image, but complete melting occurs at 130 ps. Combining this observation with melt front velocities ranging from 150 to 300 m/s (35) results in an average distance between nucleation seeds ranging from 35 to 70 nm, corresponding to a nucleation seed density ranging from 1 × 104 to 7 × 104 μm–3.

Our data from SC samples showed functional agreement with the TTM-MD simulation results of SC gold by Lin et al. (9) and Mazevet et al. (13). However, the threshold of the transition between heterogeneous melting and homogeneous melting was found to be higher than predicted (9). We speculate that this could be in large part due to the embedded-atom method potential used in the simulations: The resultant melting temperature is 963 K, and the threshold for complete melting is 0.13 MJ/kg, both of which are much lower than experimental observations. Meanwhile, for homogeneous melting, the slightly lower τmelt calculated from Mazevet’s simulations could be due to the thin-film geometry not being considered. Moreover, in the TTM part of both simulations, (i) a simple free-electron gas model–based electron heat capacity was used, which was found to overestimate Te (36), and (ii) the electron-ion coupling rate was set to the value consistent with low-temperature incomplete melting conditions.

Previous MD simulations correctly predicted the existence of the transition between the heterogeneous and homogeneous melting regimes, as shown with our experiments. However, our data reveals missing physical phenomena that will need to be included in the modeling of ultrafast melting dynamics. The observation of heterogeneous coexistence reveals a new method for addressing important questions related to the determination of nucleation seeds for melting. This will provide critical information to test and improve the kinetic theories of melting and advance the material processing related to solid-liquid phase transition to atomic-level precision.

Supplementary Materials

Materials and Methods

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References (3750)

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References and Notes

  1. Materials and methods are available as supplementary materials.
Acknowledgments: We thank the SLAC management for the strong support. The technical support by the SLAC Accelerator Directorate, Technology Innovation Directorate, LCLS Laser Science Technology Division, and Test Facilities Department is gratefully acknowledged. We also thank the technical support on sample manufacturing from the Center for Integrated Nanotechnologies, a U.S. Department of Energy (DOE) nanoscience user facility jointly operated by Los Alamos and Sandia National Laboratories. Funding: This work was supported by DOE contract DE-AC02-76SF00515 and the DOE Fusion Energy Sciences under FWP 100182 and partially supported by DOE BES Accelerator and Detector program, the SLAC UED/UEM Initiative Program Development Fund. The support from Natural Sciences and Engineering Research Council of Canada is also acknowledged. K.S.-T. acknowledges financial support from the German Research Council through project C01 “Structural Dynamics in Impulsively Excited Nanostructures” of the Collaborative Research Center SFB 1242 “Non-Equilibrium Dynamics of Condensed Matter in the Time Domain.” B.B.L.W. and R.R. acknowledge support from the DFG via the Research Unit FOR 2440. Author contributions: S.H.G., M.Z.M., and Z.C. designed the study. M.Z.M., Z.C., R.K.L., M.D., L.B.F., J.B.K., A.H.R., X.Z.S., K.S.-T., Q.Z., X.J.W., and S.H.G. performed the experiments. Z.C. and M.Z.M. designed the samples. J.K.B., P.S., M.S., Y.Y.T., and Y.Q.W. fabricated and characterized the samples. B.B.L.W. and R.R. performed the DFT-MD simulations. M.Z.M., Z.C., K.S.-T., A.N., and S.H.G. performed the data analysis.  M.Z.M., Z.C., and S.H.G. wrote the manuscript, with input from all authors. Competing interests: The authors declare that they have no competing interests. Data availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the supplementary materials. Additional data related to this paper are available from M.Z.M. upon reasonable request. 
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