Inducing thermodynamically blocked atomic ordering via strongly driven nonequilibrium kinetics.

Ultrafast light-matter interactions enable inducing exotic material phases by promoting access to kinetic processes blocked in equilibrium. Despite potential opportunities, actively using nonequilibrium kinetics for material discovery is limited by the poor understanding on intermediate states of driven systems. Here, using single-pulse time-resolved imaging with x-ray free-electron lasers, we found intermediate states of photoexcited bismuth nanoparticles that showed kinetically reversed surface ordering during ultrafast melting. This entropy-lowering reaction was further investigated by molecular dynamics simulations to reveal that observed kinetics were thermodynamically buried in equilibrium, which emphasized the critical role of electron-mediated ultrafast free-energy modification in inducing exotic material phases. This study demonstrated that ultrafast photoexcitations of electrons provide an efficient strategy to induce hidden material phases by overcoming thermodynamic barriers via nonequilibrium reaction pathways.

INTRODUCTION

Solid structures reveal stable configurations of electronic and ionic energies geometrically, and abundant systems in nature evidence that equilibrium crystal shapes display minimal free-energy configuration via macroscopic surface morphology (–). Material formations and phase changes, as such, are dictated by the hills and valleys of the system’s free energy, which provides a guide for designing new functional materials (). However, the kinetic reactions for a desired structure and functionality involve overcoming thermodynamic barriers, which limit the viability and practicality of this approach in novel functional materials. Manipulating free energy with external stimuli by driving a system far from thermodynamic equilibrium can provide an efficient path for active controls of the material phases. The investigation in such a direction requires ab initio understanding of the intermediate states in nonequilibrium conditions (–), which has been hindered by indirect evidence with model-dependent interpretation. Accordingly, enormous effort in material design has been confined to thermodynamically favorable kinetic reactions in near-equilibrium conditions (, ). New developments of extreme light sources provide the potential to directly induce and probe nonequilibrium kinetics using ultrafast light-matter interactions (, –). In particular, electron photoexcitation enables the nonadiabatic modification of the electronic potential energy in the femtosecond (fs) time scale, with which ionic configurations in large scales can be manipulated efficiently, thereby overcoming the thermodynamic barriers.

With the pump-probe single-pulse x-ray imaging (), we investigate the ultrafast melting of bismuth (Bi) nanoparticles (NPs) illuminated by single femtosecond optical laser pulses. By photoexciting bonding electrons to modify the potential energy with a nonlinear response on the laser fluence, we observe ultrafast atomic ordering at the surface far above the melting temperature as a reverse kinetic process. Photoinduced entropy-lowering kinetics emerge during archetypical solid-liquid phase transformations, which manifest that the complex nature of electron-driven lattice dynamics in out of equilibrium provides new paths to induce new macroscopic states.

RESULTS

Experiments were performed using x-ray free-electron lasers (XFELs) at the SPring-8 Angstrom Compact Free Electron Laser and Pohang Accelerator Laboratory (, ). A femtosecond Ti-sapphire laser with an 800-nm wavelength in the near–infrared (IR) region was used as a pumping source. Then, the specimens excited by single fs-IR laser pulses were probed using single XFEL pulses (Fig. 1A and Materials and Methods). Specimens in spherical NPs of crystalline Bi with 170 (±6)–nm diameters were loaded using thin Si3N4 membranes (electron microscope image and diffraction pattern in Fig. 1A, fig. S1, and Materials and Methods). We collected single-pulse diffraction patterns by scanning the membrane relative to the micrometer-sized focused XFEL pulses (Materials and Methods). Every single specimen exposed to the IR and subsequent XFEL pulses were destroyed, and repeated observation of the same specimen was not allowed.

Fig. 1.

Faceting above the melting point in single Bi NPs.

(A) Schematics of time-resolved single-pulse XFEL imaging experiments. Electron microscope images show the spherical morphology and crystalline quality of the Bi specimen confirmed by the diffraction pattern. (B) Time-resolved single-pulse diffraction patterns of single Bi NP exposed by single fs-IR laser pulses. Diffraction patterns are shown in a logarithmic scale with the color map scale bar to the right. Four- or sixfold streaks (e.g., white arrow at 20 ps) were developed for delays of ˃20 ps. (C) Reconstructed images of melting developing with facet formation, beginning at ~20 ps. The color map for projected density is scaled to visualize images with the same total density. With the progress of internal melting, density significantly varies while forming a central void after ~80 ps. Scale bar, 100 nm. (D) The polyhedrons composed of small areas of (100), (110), and (111) facets reproduce the experimental diffraction pattern and projected density (experiments: top; model simulations: bottom). The polyhedron evolved with more-developed facet area along with radial expansion, and internal density reduction gives a good reproduction of experimental results observed later in melting transition.

We collected time-resolved coherent diffraction patterns that revealed the ultrafast evolution of single Bi NPs exposed to single fs-IR pulses at a 140–mJ cm−2 laser fluence (Fig. 1B). We selected Bi as a model covalent bonding crystal displaying intriguing phenomena of subpicosecond melting under intense fs-IR laser illumination, which is attributed to the abrupt change in electronic potential energy surface (PES) caused by photodepletion of boding electrons above a stability limit (). Spherical Bi NPs were prepared via solution-based growth processes (Materials and Methods). These solution-based particle synthesis protocols usually produce spherical morphology in the resulting nanocrystals especially for elemental metals with low melting temperatures including Bi (melting temperature of 575 K). The temperatures to thermally decompose particle growth precursors are usually close to melting temperatures, which leads to spherical morphology of the resulting crystals (). As the elapsed time after the IR laser illumination increased to tens of picosecond, the Airy pattern from the intact spherical particle started to deform. Measured diffraction patterns showed new developments of strong streaks (white arrow in Fig. 1B). These streaks implied a morphological change in the specimens to accompany sharp edges at the surface, which was a well-known feature of knife-edge diffraction in classical wave optics. The streak started developing at 20 ps and strengthened over time to show clear four- or sixfold symmetry.

Projected charge densities of the specimens were obtained from numerical phase retrievals of measured single-pulse diffraction patterns (Fig. 1C and Materials and Methods) (, ). A few tens of images were collected at each delay time (table S1), and the one that represented the average behavior of images at each given delay was chosen to exhibit ultrafast kinetics of NP melting after its interaction with a single fs-IR laser pulse. For these, we have selected high-resolution images that faithfully deliver the average degree of characteristic features in NP radius, void sizes and locations, internal density, and developments of streaks and small speckles among the collected single-pulse data (Fig. 1C and movies S1 and S2). As inferred from the streaks in the diffraction patterns, reconstructed images showed the formation of edges. The Bi NP melting proceeded with the formation of facets from ~20 ps, which continued to develop by transforming initially spherical NPs into polyhedrons. The faceting consistently appeared as a characteristic feature of this intense laser-induced melting reaction (figs. S2, S15, and S16). During facet formation, the internal density of the NP accompanied local variation in the projected density. More apparently from ~80 ps, the NP displayed the internal void(s) formation, which was also previously observed in femtosecond laser–induced ultrafast melting of Au NP as well as numerical simulations (, , ). Focused plasmonic heating led to local accumulation of the ionic pressure, and the voids were formed to release the pressure. Overall morphology of the NP changed as the melting proceeded by accompanying the volume expansion and shape distortion. While faceting continued, the NP radially expanded from ~20 ps; the change continued monotonically to reach ~123% at 120 ps. From the temporal evolution of the radius (fig. S3), we have estimated the expansion speed of the Bi NP as ~200 m s−1, slower than the elastic wave propagation speed of Bi with its sound velocity of 1790 m s−1 ().

Starting from a low-temperature spherical shape, the evolution of the nanocrystal proceeded with a high-temperature faceted structure on ultrafast melting. This observation of crystal melting accompanying the development of new facets is in stark contrast to the well-established knowledge on crystal shape transformations. According to the crystal shape theory, small crystals are usually formed in polyhedron shapes to lower the binding energy and natural anisotropy in the surface energy as explained by the Wulff construction (, ). On increasing temperature, these small crystals undergo melting transition by smoothing the sharp corners and edges of the polyhedron and eventually transform into a spherical shape near the melting point, as observed almost universally (–). Hence, the observed facet formation during ultrafast nonequilibrium melting of the Bi NPs at elevated temperature is very unusual and clearly follows reversed kinetics of the long-standing crystal shape theory. Note that this structural change with faceting far above the bulk melting point is fundamentally distinct from the surface melting–induced faceting reported before (–). In the present case, the faceting occurred along with internal melting even at 1000 K, which is far above Bi’s bulk melting temperature of 545 K (fig. S4). The observed high-temperature faceting evidenced incipience of entropy-lowered ordered phase at the surface (), which was realized by driving the system far from equilibrium states through ultrafast photoexcitation of electrons in Bi bonding orbitals.

To guide the understanding on the facet formation with atomic-scale visualization, we provided the polyhedron models to be compared with the experimental results (Fig. 1D). The crystallographic notation of the cubic symmetry was used here. We have constructed polyhedron models by adjusting the size of the facets for different orientations to agree with both the measured diffraction patterns and projection images (Materials and Methods). Our manual adjustment of the facet sizes was to demonstrate the influence of facet formation on producing streaks on otherwise Airy-type diffraction patterns; quantitative analysis of the facet formation was performed, instead, by inspecting the anisotropy developed in the diffraction patterns with the streak formations (fig. S2A). The diffraction patterns taken at 40 ps (i.e., early stages of melting) were successfully reproduced using a polyhedron with small areas of facets developed along the major crystallographic orientations (Fig. 1D, left). To simulate experimental results in later stages of melting (100 ps), fully developed facets were incorporated into the polyhedron model with reduced internal density (Fig. 1D, right). The polyhedrons, being projected along the [111] direction, produced diffraction patterns with sixfold symmetric streaks resulting from [110] edges, which agreed with the experimental result. A projection along the [100] or [110] direction generated a fourfold symmetry in the diffraction pattern, which also agreed with the experimental result (fig. S5).

Furthermore, to understand the underlying mechanisms of the reverse kinetics, we performed two-temperature molecular dynamics (TTMD) simulations (Fig. 2). These MD simulations implemented the two-temperature model description of the energy transfer from the photoexcited electrons to ions to treat the electrons to gain energy from the fs-IR laser pulses, which was then transferred to the Bi ions via electron-phonon coupling (Materials and Methods). The TTMD simulations performed at a 140–mJ cm−2 laser fluence reproduced the experimental results well (Fig. 2A, fig. S6, and movies S4 and S5). The simulation confirmed that the behavior of radial expansion and internal density decrease resulted from releasing the ionic pressure accumulated through the electron-to-lattice energy transfer ().

Fig. 2.

TTMD simulations of the Bi NP on melting.

(A) The projected density of the Bi NP is obtained from TTMD simulations for a 140–mJ cm−2 laser fluence reproducing the experimental results in facet developments and internal density variations in melting (left). Corresponding facet formation is verified by calculating the surface curvature of the NP structures from the TTMD (right). (B) Obtained surface curvatures were compared directly with a model polyhedron with good consistency, supporting the polyhedron model interpretation of the experimental results. (C) Atomic ordering for the surface and internal Bi atoms is monitored, which confirms that the Bi atoms at the surface (green) order well amid the disorder in internal (violet) atoms on melting.

The high-temperature surface ordering behavior was further analyzed by calculating the surface curvature of the TTMD structures (Fig. 2, A and B, and Materials and Methods), which clearly displayed the gradual development of facets, and the curvatures estimated for the highly faceted NP at 120 ps agreed with the abovementioned polyhedron-guided interpretation of the experimental results (Fig. 2B). The atomic ordering at the surface was further verified by calculating local order parameters using the atomic positions acquired from the TTMD structures (Fig. 2C and fig. S7). This local order parameter was defined to measure the deviation of each atom from the perfect hexagonal crystalline order, normalized by the average value at 0 ps. This analysis clearly manifested the ordering of the surface atoms (green), despite the internal melting with increased disorder among internal atoms (violet). This order parameter analysis confirmed the recovery of crystalline stacking at the surface in the high-fluence ultrafast melting.

The observed reverse kinetic reaction showed distinct nonlinear dependence on the laser fluence, thereby signifying the nonequilibrium nature. The same experiment was repeated using a 32–mJ cm−2 laser fluence (23% of the previous laser fluence), which exhibited markedly different melting transitions. Bi NPs melted under this lower laser fluence preserved the spherical morphology without faceting (Fig. 3A, fig. S17, and movie S3). No streak signal was found, whereas the development of small speckles on top of the main circular fringe occurred from ~80 ps because of inhomogeneous density variation in melting (Fig. 3A, top). Reconstructed images showed that the specimens melted and remained spherical (Fig. 3A, middle). The internal density variation became notable at ~80 ps and evolved into a large central void at 180 ps. This low-fluence reaction without the discernible signature of faceting was also reproduced by the TTMD simulations performed at a lower laser fluence of 17 mJ cm−2 (Fig. 3A, bottom, and movies S6 and S7). This investigation demonstrated that the laser fluence was the main determinant of the faceting during ultrafast melting of Bi NPs.

Fig. 3.

Laser fluence dependent two different melting kinetics.

(A) Time-resolved single-pulse diffraction patterns of Bi NP at a 32–mJ cm−2 laser fluence (top). Reconstructed images show that the melting proceeds by forming internal voids without faceting (middle). Scale bar, 100 nm. TTMD simulations confirm the melting without faceting for low laser fluence (bottom). (B) Gibbs free energies calculated to display the free-energy difference relative to the equilibrium melting structures obtained for laser fluences of 140 and 32 mJ cm−2, respectively. This shows that free-energy reduction compared with equilibrium melting is more significant for high fluence (by 150 meV per atom; orange) than for low fluence (by 40 meV per atom; blue) melting. The height of the squares represents the SDs for 10 independent simulations. (C) Free-energy landscape with a reaction barrier, EA ~120 meV (Supplementary Materials), illustrates the presence of two types of reaction pathways for the ultrafast melting transition. Free energy is estimated to show that Bi in polyhedrons has lower free energy of 54 meV per atom than the spherical shape. In the free energy picture of the nonequilibrium melting, the excited state is drawn in solid line, distinguished from the ground state (broken line).

To understand this uncommon melting reaction with nonlinear dependence on the laser fluence (Fig. 1, B and C; Fig. 3A; and fig. S8), we analyzed the thermodynamic free energy for the two structures. The free-energy difference between the two configurations of spherical and polyhedral structures at zero temperature was estimated to show that, as anticipated from the equilibrium crystal theory, the polyhedral structure had lower free energy of 54 meV per atom (fig. S9). This indicated that the spherical Bi NP was initially in a metastable state. It also provides important message that synthesizing NPs in the ground state configuration can be more challenging because of the thermodynamic conditions required for NP growth, which become thermodynamic barriers to the formation of ground state configuration, as detailed in the report on polyhedral Bi NP synthesis (). To directly compare the kinetic reaction of ultrafast nonequilibrium melting with that of equilibrium melting, we generated structures by slowly increasing the lattice temperature of the spherical Bi NP up to the target temperatures starting from 300 K (fig. S10). It showed that the Gibbs free energies of these equilibrium melting structures (Geq) were higher than those of TTMD-generated, nonequilibrium melting structures (Gneq): ΔG = Gneq − Geq < 0 (Fig. 3B). Moreover, the high-fluence TTMD structures with faceting gave lower ΔG compared to the low-fluence structures without faceting by ~0.1 eV.

These calculations, corroborating the experimental results, established a simplified free-energy picture that captured the essence of this laser fluence–dependent melting kinetics (Fig. 3C). Being also supported by a previous near-equilibrium kinetics study (), the low-fluence melting without facet formation evidenced that there was a reaction energy barrier from the metastable spherical shape to a polyhedron of the lower-energy state. As the shape change accompanied atomic-scale rearrangements at the surface, the energy to activate the atomic diffusion at the surface should be supplied to form facets, which became the reaction energy barrier (). In this fs-IR laser–induced melting, atoms at a spherical surface should gain enough kinetic energy from the photoexcited electrons, by overcoming the reaction energy barrier, to be released from the crystal bonding and move around the surface diffusively to become a polyhedron with the facets.

However, the faceting observed in the high-fluence melting added more mechanistic insights than the near-equilibrium picture provided. As reported in previous studies on Bi with the bond softening, we expected a nonadiabatic modification of the electronic potential energy under intense laser fluence to weaken the interatomic bonding (, , ). This, in effect, lowered the reaction energy barrier, as the breaking of the atomic bonding to induce surface diffusion became easier with weakened interatomic bonding. This potential energy modification by the photoexcitation of bonding electrons and the nonequilibrium thermal activation of the Bi atoms facilitated the reverse kinetic reaction of high-temperature faceting. The resulting reaction of faceted melting was thermodynamically inaccessible in equilibrium condition; delivering higher energy to the NP under near-equilibrium condition only led to simple melting with complete loss of surface morphology instead of facet formation, as verified by the TTMD simulations under near-equilibrium conditions.

To verify the abovementioned free-energy description and, more specifically, the impact of electron-mediated interatomic potential energy change in inducing high-temperature faceting, we conducted an experiment using a much higher laser fluence of 1.4 J cm−2. From the experiment at this high-fluence laser illumination, we observed markedly accelerated faceting, featured by the streaks, developed as early as ~200 fs (Fig. 4A), which indicated that macroscopic atomic rearrangements occurred faster than the time scale for any thermal motion of atoms to begin. To extract ultrafast dynamics of the facet formation, we estimated the anisotropy in the single-pulse diffraction, and mean values of data at the same delay were plotted to understand the temporal evolution of the faceting (Fig. 4B). The anisotropy is defined as (Cmax − Cmin)/(Cmax + Cmin), where Cmax (min) is the maximum (minimum) value of the angular cross-correlation of the diffraction pattern, which quantitatively assesses the degree of streaked signals departed from the isotropic Airy pattern (Supplementary Materials). Obtained anisotropy manifested the gradual development of the streaking that started from ~200 fs to become most intense to ~1.5 ps. As the internal melting proceeded, the streak signals were smeared out, leading to reduced anisotropy values. Reconstructed projection images displayed morphological distortion and internal density variations in melting with the faceting (Fig. 4C).

Fig. 4.

Nonthermal faceting via nonequilibrium reaction kinetics.

(A) Diffraction patterns with streak developments in femtoseconds for experiments at a 1.4–J cm−2 laser fluence. (B) Development of the faceting quantified by calculating the anisotropy in the diffraction pattern. Calculated angular cross-correlation from diffraction patterns (green) were converted to anisotropy (violet) to show the development of streaks in patterns. Gradual facet development from ~200 fs is clearly observed with maximum development at 1.5 ps. With the internal melting, the anisotropy decreases after ~1.5 ps. (C) Projected density image of the Bi specimens on melting. The development of the faceting is more remarkable at 1.5 ps (dashed lines), with the progress of the internal melting at 2 ps.

Observed subpicosecond faceting under the intense laser fluence at 1.4 J cm−2 explicitly verified that this unusual faceting resulted from the electron-mediated ultrafast reaction, distinguished from laser-induced elastic pressure-driven reaction (figs. S4B and S11). Notably, this laser fluence expects to excite ~21% of valence electrons, which is much higher than the critical excitation level (~3%) for inducing nonthermal melting of Bi (, ). Hence, this ultrafast imaging result provided direct experimental evidence of nonthermal structural modification. Furthermore, it demonstrated that kinetic reactions buried in thermodynamic equilibrium could be accommodated at the time scale of electronic dynamics by nonadiabatically modifying the electronic potential energy via strongly driven nonequilibrium states.

DISCUSSION

In summary, we revealed the critical role of free energy in governing the reaction pathways during the solid-liquid phase transition of Bi excited by an fs-IR laser. The phase transition accompanied unusual kinetic reactions with nonlinear dependence on the laser fluence. Images of Bi NPs undergoing irreversible phase transition were obtained from single-pulse measurements at a sub–10-nm spatial and 200-fs temporal resolution to find a hidden reaction pathway. While providing the first direct visualization of thermodynamically buried melting kinetics in a strongly driven nonequilibrium condition, this research provides mechanistic insights into the active control of the material phases using the free energy. By allowing access to nonequilibrium states, an intense fs-IR laser enabled us to modify the interatomic potential energy to find a hidden, femtosecond rapid, reaction path to reach a low-energy configuration despite its blockage in the equilibrium process. As demonstrated, the nonequilibrium process can be actively used to search the exotic material phases buried in equilibrium conditions. Thus, the photoinduced modification of thermodynamic free energy can provide a breakthrough in the research on active material search via strongly driven nonequilibrium kinetics, which would outperform empirical material synthesis or computational exploration-based approaches.

MATERIALS AND METHODS

Synthesis of spherical Bi NPs

The spherical morphology of Bi NPs that is in a metastable state was synthesized by controlling the growth rate of Bi crystals. The Bi NPs were synthesized by solvothermal method. Bi(NO3)3·5H2O (0.5 mmol) was dissolved in 10 M−1 nitric acid. The mixed solution was stirred at room temperature for 2 hours. Then, 50 ml of ethylene glycol was added slowly to the solution with vigorous stirring for several hours. The final mixed solution was transferred into a stainless steel autoclave with Teflon liner. During the following preheating process, the solution was heated at 200°C and maintained for 2 hours in the sealed autoclave. Subsequently, it was cooled down to room temperature rapidly. Then, we repeated the preheating process and maintained for 28 hours. After spontaneous cooling to room temperature, the obtained black solution was centrifuged to collect the powder that was then washed with plenty of pure alcohol for 5 to 10 times. The final product was dried at 50°C for 2 days under vacuum condition. Details on the structure characterization of the specimens are described in the Supplementary Materials.

The morphology and size of the Bi NPs depend on the reaction time and temperature. During the synthesis process, ethylene glycol was added as the solvent. According to the previous studies, ethylene glycol can be adsorbed on the surface of Bi crystals through O─Bi bonds, which can significantly decrease growth rate and change crystal morphology (). In the first step, Bi tiny crystals were aggregated to large particles covered with ethylene glycol shell. Then, during the second step, the isotropic growth of Bi crystals was realized, resulting in the formation of sphere Bi NPs ().

Single-pulse time-resolved x-ray diffraction imaging

The experiments were performed using the multiple-application x-ray imaging chamber (MAXIC) installed at the BL3 of spring-8 angstrom compact free electron laser (SACLA) and a similarly designed chamber at the Nanocrystallography and Coherent Imaging (NCI) station of Pohang Accelerator Laboratory (PAL)-XFEL. The femtosecond Ti-sapphire optical laser (λ = 800 nm and pulse duration of 50 fs) was used as a pumping source, and the laser pulses were focused to 100-μm size at the position where the samples were located. Femtosecond x-ray laser pulses were micrometer focused using a pair of K-B mirrors, with a focal size of 1.5 μm by 1.5 μm at SACLA with 1-m-long focal length and 5 μm by 6 μm at PAL-XFEL with 5-m-long focal length. The spatial overlap of specimens, 100-μm focused IR laser and micrometer-focused XFEL, was confirmed visually using an in-line microscope by inspecting the laser burning scars left on the membrane. The time zero of the all-three aligned interaction spot was confirmed by monitoring the absorption profile of a thin GaN crystal mounted on the interaction spot (). The temporal resolution better than 0.5 ps was achieved at PAL-XFEL without an extra timing tool. Possible occurrence of the long-term drift in the timing of the delay line between XFELs and IR laser was monitored intermittently and corrected accordingly. The jittering of SACLA and PAL-XFELs is within several hundred femtoseconds and does not influence the present temporal resolution.

Single-particle sample loading and single-pulse data acquisition

Spherical Bi NPs of 170 (±6)–nm diameter were mounted using custom-designed Si3N4 membranes (Silson Ltd.) for the single-particle x-ray diffraction experiments. NPs were dispersed in purified deionized water, and the solution with the NPs was spread on the membrane using a spin-coating method. The concentration of NPs was adjusted to maximize the probability of single-particle hit by micrometer-focused x-ray pulses. Si3N4 membranes were plasma cleaned beforehand to enhance the wettability.

Single-pulse data acquisitions were performed by moving the sample stages with respect to the overlapped position of focused XFEL beam and the IR laser spot. The single IR laser pulse destroys the membrane completely, which prohibits repeated exposures of the same window. Visual inspection of the IR laser exposed specimens after the XFEL data collection is not viable either. We have designed the membrane to have the window-to-window distance of 380 μm, far more distant than a laser footprint of 100 μm, further to prevent the exposure of the same particle by the IR pulses illuminating nearby. This data collection protocol is strictly preserved to ascertain that the specimens are exposed only once by the IR laser and XFEL pulses. Each single pulse diffraction pattern was recorded on the multiport charge-coupled device detector with a pixel size of 50 μm by 50 μm (). The total detector area with 2048 by 2048 array was used at SACLA and 1024 by 1024 at PAL-XFEL.

Image reconstruction from phase retrieval of the coherent diffraction patterns

Images of single Bi NPs were obtained by performing numerical phase retrievals of the measured single-pulse coherent diffraction patterns. The numerical iteration of the phase retrieval starts from the randomly guessed phase. Experimentally measured diffraction intensity is used as an amplitude modulus after taking the square roots of the background noise–subtracted intensity. It is then inverse Fourier transformed to obtain an image in real space by assigning random phase initially. This image is further inspected by imposing empirical constraints such as the boundary of the image, non-negativity of the image density, etc. The constrained image is Fourier transformed to obtain a diffraction pattern with the amplitude modulus and phase. Each time, the amplitude is replaced by the experimentally measured values but keeping the obtained phase, which is then inverse Fourier transformed to get an image again. This process is iterated for thousands of times to converge an image of real specimen.

We used the guided hybrid input-output algorithm for the numerical phase retrievals (). Random seeds of 16 sets were used with 3000 iterations for each random seed. The reconstruction ran for the sixth generation with good convergence among the images. Five best images were averaged to represent the image of each single-pulse diffraction data. Good convergence in phase retrieval was obtained for the image resolution better than 10 nm, evidenced by the calculated phase retrieval transfer function (see the Supplementary Materials) ().

Atomic polyhedron modeling and comparison with the experimental data

The atomic polyhedron models were constructed out of a sphere with close-packed face-centered cubic structure by cutting out surfaces normal to the major crystallographic directions. By adjusting the size of the facets manually with visual inspection, developmental stages of the faceted polyhedron were represented; a polyhedron with more developed facets has larger areas of flat surfaces. To simulate the NPs at later stages of melting (i.e., after ~80 ps), the polyhedron was further modified to have lower and inhomogeneous density values. Last, the orientations of the atomic polyhedron model were found to match with the two-dimensional images of projected densities and diffraction patterns from experiments.

Observed symmetries in the diffraction patterns resulted from random orientations of the faceted Bi NPs relative to the incident x-rays. Initially, the spherical Bi NPs in random orientations were dispersed on the Si3N4 membrane. Hence, the facets developed along crystallographic directions upon laser illumination are in arbitrary orientations relative to the incident x-ray. The projection density of the particle along the [111] direction of threefold axis yields the sixfold symmetric diffraction patterns (Fig. 1D and movie S1), and similarly, projections along the two- or fourfold axis yield the fourfold symmetric pattern (fig. S5 and movie S2).

Atomistic TTMD simulations

We performed the TTMD simulations for fs-IR laser excitation of Bi NP at a laser fluence of 140 mJ cm−2 (high fluence) and 17 mJ cm−2 (low fluence) using the embedded atom method (EAM) interatomic potential that was parameterized for the PES at electronic ground state. Instead of explicitly incorporating the excited-state PES, our calculations have been focused on identifying simulation conditions for the occurrence of faceting via abrupt, nonequilibrium heating of the electrons and subsequent rapid transfer of their excessive kinetic energy to the ions. The simulation results were compared with those performed under steady-state conditions, which confirmed that the surface faceting was forbidden in the near-equilibrium process. As evidenced by the results of our simulations performed at 10 different initial conditions, the barrier crossing was achieved via nonequilibrium thermal kinetics driven by high-fluence fs-IR laser pulse excitation.

A Bi NP was constructed by cutting out a sphere of 20-nm diameter from the bulk Bi lattice (). The unsupported NP was first equilibrated at 300 K for 2 ps using the MD simulation with the EAM interatomic potential and a time step of 1.4 fs (). The TTMD simulation was then carried out with a pre-equilibrated NP. It was assumed that the localized surface plasmon excitation was negligible, and thus, the NP absorbed energy from the fs-IR laser irradiation isotropically. For the TTMD simulations, the method developed by Ihm et al. () was used, in which the atomistic MD is coupled with the following finite difference equation of electron heat transport

Here, Fabs = F(1 − R), τ, lp, and r0 are the absorbed fluence, pulse width, absorption length of the incident laser, and the intact NP radius, respectively. For the simulation under high laser fluence, an incident fluence of 140 mJ cm−2 (F), a refractive index of 0.97 (R), and a pulse width of 50 fs (τ) were used, which are consistent with the experimental conditions. For the low-fluence case, F = 17 mJ cm−2 was used, which is slightly lower than the one used in the experiment. The penetration depth was scaled according to the NP diameter. The electronic heat capacity (Ce), thermal conductivity (Ke), and the electron-phonon coupling constant (Ge) were constructed to reflect the electron excitations. The Ke and Ge were obtained following Giret et al. () as well as Arnaud and Giret (), respectively. The Ce was chosen such that it increased linearly at low temperatures and reached the ideal gas at high temperatures. As the NP size used in simulations was smaller than those of the experiments, the shape transformation took place earlier in simulations than in experiments. Therefore, the time evolution in simulations was scaled such that the simulated structure at 120 ps under high laser fluence was consistent with that of the experiment.

Surface curvature calculation

The surface curvature at a given point has continuous distribution of its values depending on the direction. For our estimation of the surface curvature, we first calculated the curvature values along directions to have maximum and minimum curvature at a given point, which is called principal curvatures. We then obtained the mean value of the two principal values to identify the surface curvature. The numerical values of the mean curvatures were estimated using Avizo (Thermo Fisher Scientific). Using the color map, we displayed the surface curvature with the values scaled to an ideal sphere of the same radius (Fig. 2, A and B).

Free energy calculation

The per-atom Gibbs free energy was obtained as G = H − TS, where the enthalpy H consisted of the kinetic energy, potential energy, and PV, and S was the atomic configurational entropy (). The interatomic potentials of Yan et al. () and Belashchenko and Ostrovskii () were used for free energy calculations at below and above melting temperature, respectively. The free energy comparison between the spherical and polyhedral NPs at 0 K was performed with a spherical NP of 20 nm in diameter and a polyhedral NP of compatible size that was manually generated by cutting out the {100}, {110}, and {111} planes (fig. S9). The relative free energy of nonequilibrium kinetics–driven with respect to the equilibrium kinetics–driven structure was ΔG = Gneq − Geq. Gneq was obtained from TTMD-simulated structures at high fluence, and Geq was from the structures generated by slowly heating the lattice from 300 K to the target temperature in steps of 50 K, with additional relaxation for 10 ps at the end of each step. The average values obtained from 10 independent simulations were used for free energy comparison (fig. S14).

Order parameter calculation

The order parameter of the individual atoms in the NP, as shown in fig. S12, was calculated similarly in () and defined as . This order parameter gives the measure of the deviation of each atom from the perfect hexagonal crystalline order and ranges between zero and one. Six q vectors were selected, which satisfy e = 1 for the vectors ξ connecting near neighbors in the perfect crystalline lattice with Nn = 6.