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Melting dynamics of ice in the mesoscopic regimeMargherita Citronia,b,1, Samuele Fanettia,c, Naomi Falsinia, Paolo Foggia,d,e, and Roberto Binia,b

aLENS–European Laboratory for Non-Linear Spectroscopy, 50019 Sesto, Florence, Italy; bDipartimento di Chimica Ugo Schiff, University of Florence, 50019Sesto, Florence, Italy; cIstituto di Chimica dei Composti Organo Metallici, Consiglio Nazionale delle Ricerche, 50019 Sesto, Florence, Italy; dIstituto Nazionaledi Ottica, Consiglio Nazionale delle Ricerche, 50125 Florence, Italy; and eDipartimento di Chimica, University of Perugia, I-06123 Perugia, Italy

Edited by Russell J. Hemley, The George Washington University, Washington, DC, and approved May 2, 2017 (received for review December 6, 2016)

How does a crystal melt? How long does it take for melt nucleito grow? The melting mechanisms have been addressed by sev-eral theoretical and experimental works, covering a subnanosec-ond time window with sample sizes of tens of nanometers andthus suitable to determine the onset of the process but unable tounveil the following dynamics. On the other hand, macroscopicobservations of phase transitions, with millisecond or longer timeresolution, account for processes occurring at surfaces and timelimited by thermal contact with the environment. Here, we fill thegap between these two extremes, investigating the melting ofice in the entire mesoscopic regime. A bulk ice Ih or ice VI sampleis homogeneously heated by a picosecond infrared pulse, whichdelivers all of the energy necessary for complete melting. The evo-lution of melt/ice interfaces thereafter is monitored by Mie scat-tering with nanosecond resolution, for all of the time needed forthe sample to reequilibrate. The growth of the liquid domains,over distances of micrometers, takes hundreds of nanoseconds,a time orders of magnitude larger than expected from simpleH-bond dynamics.

Mie scattering | temperature jump | superheating | laser heating |anvil cell

Melting of ice is one of the most common occurrences on ourplanet, from polar ices to glaciers, to the morning frost and

the preparation of our drinks. Ices are present as well in extrater-restrial environments, planetary and intersellar, going throughextremely different pressure and temperature conditions, wheremany phase boundaries are encountered. The molecular mecha-nisms of melting and the characteristic timescales of the processare not well elucidated, as for any other phase transition. In fact,great effort is presently being made with state of the art exper-imental techniques to understand the intrinsic kinetics of struc-tural transformations, particularly under dynamic compression,by which phase-transition boundaries up to extreme pressures(P) and temperatures (T) can be accessed (1–6). Melting is theleast hindered of phase transitions, requiring the disruption ofthe crystalline order and the achievement of the local structureof the liquid phase.

The dynamics of melting have been studied up to the presentin the picosecond timescale, observing structural changes on alength scale of few nanometers, in the attempt to explain thefundamental mechanisms of the transition onset. In both simu-lations and experiments, micrometric- or submicrometric-sizedcrystals can be rapidly heated above the melting temperature(Tm), into a superheated state where melting occurs. Metals andwater ice have been the test systems. Simulations have revealeda nucleation and growth mechanism (7–11), and nucleation hasbeen especially addressed, in trying to determine the minimumnumber of atoms/molecules needed to form a critical nucleusfor the transitions, the solid/melt interfacial energy, the tran-sient local structures achieved during the transformation, and therole of defects and surfaces. Experimentally, the sample can berapidly heated by a pulse of infrared light at a wavelength thatthe sample can absorb (T jump). With this “direct laser heat-ing” the sample is heated from within, and transition nuclei canbe formed in the bulk, overcoming the dominance of heteroge-neous nucleation, induced by surfaces or preexisting defects (12).

The temperature increases by the redistribution of the absorbedenergy via the vibrational relaxation over the lattice modes. Thisthermalization process takes ∼20 ps in water ice (13) and canbe slower in non–H-bonded molecular crystals. The structuralchanges following the heating pulse have been monitored in alu-minum (14) and gold (15) by electron diffraction with a samplethickness of 20 nm, where the main signatures of melting com-plete in a few picoseconds. Water ice melting after an ultrafastT jump has been studied by time-resolved infrared spectra (13,16), with a sample thickness of 1.6 µm and over a time windowof 250 ps. Here, the spectral changes attributed by the authors tomelting mostly occur in the first 150 ps after the pulse (τ =37 ps)(16). At 250 ps an incomplete melting was still observed, consis-tent with the amount of energy delivered.

The knowledge at this picosecond–nanometer scale barelyconnects to the experiences in the macroscopic world. On themacroscopic side, photographic investigations have been per-formed with millisecond resolution to investigate premeltingstructures in colloidal crystals (17), ice nucleation events uponcooling (18), or the rate and topology of ice and hydrate forma-tion under dynamic compression (19, 20).

Here, we use a picosecond T -jump technique and time-resolved Mie scattering to explore the melting and the successiverefreezing on bulk samples of ice Ih and ice VI. The main pecu-liarities of this experiment are the bulk sample thickness (50 µm)and the time sampling: We access a time window of tens of mil-liseconds, which is the time needed to completely restore theinitial conditions, with nanosecond resolution. We observe thescattering at ice/melt interfaces and the propagation of meltingover distances of micrometers, accessing the characteristic timeof the process. As will be shown, the P, T, and V conditions in thisexperiment are quasi-static over the timescale of melting evolu-tion. The kinetics of melting propagation in a bulk crystal arethus explored over the entire mesoscopic regime, connecting themicroscopic and the macroscopic realms.

Significance

The complete unfolding of the melting process, from thenucleation onset to the achievement of a bulk liquid structure,is not known to date for the timescales and the sizes of theheterogeneities involved. Here, we observe in ice the archety-pal H-bonded molecular crystal, the evolution of solid/liquidinterfaces generated by the absorption of a picosecond infra-red pulse. The pulse delivers all of the energy needed for com-plete melting; therefore, the lattice is homogenously heatedfrom within, above its melting temperature. The growth ofliquid domains over distances of micrometers takes hundredsof nanoseconds and is orders of magnitude slower than vibra-tional dynamics.

Author contributions: M.C., S.F., P.F., and R.B. designed research; M.C., S.F., and N.F. per-formed research; M.C., S.F., and N.F. analyzed data; and M.C. and R.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: citroni@lens.unifi.it.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1620039114/-/DCSupplemental.

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The experimental insight into the melting dynamics of iceshas paramount relevance in understanding natural phenomenaand is a fundamental benchmark for theories at any level ofaccuracy. Ices, ubiquitous in terrestrial and extraterrestrial envi-ronments, are characterized by hydrogen-bond networks thatdetermine the great complexity of their physical and chemicalbehavior. The melting line of water borders at least five crys-talline phases (Ih , III, V, VI, and VII) with different structuresand H-bond arrangements. In this work we investigate the melt-ing dynamics in ice Ih and VI, accessed respectively by the useof low temperature and high static pressure, providing an uniqueexperimental study in the mesoscopic domain and opening theway to investigations in different P and T conditions.

ResultsTemperature Jump and Transient Transmission Loss. Ice Ih and iceVI samples 50 µm thick, at fixed temperature (T0) and pres-sure (P0), are “instantaneously” heated by a 15-ps pulse oflight at 1,930 nm (FWHM = 100 nm) (21) with a focus diame-ter of 400 µm. Ice Ih at ambient pressure is held between flu-orite windows in a cryostat at a static temperature T0 rangingbetween 254.2 K and 272.6 K. Ice VI is in a sapphire anvil cell,between sapphire anvils, at room temperature and P0 =1.2 GPaor 1.4 GPa. The light is absorbed by the ice through excitationof a vibrational combination band (ν1 + ν2, ν3 + ν2), as shown inFig. 1A. The incident energy used (3–9 mJ/pulse) is well belowthe estimated ionization limit (9) and below the threshold ofabsorption saturation. In fact, the measured absorbance of thesample under irradiation with the pump is ∼0.55 at all energiesused. This low absorbance guarantees that the sample is uni-formly irradiated along the beam axis and can thus be uniformlyheated across its length. The pump causes an isochoric temper-ature jump (T jump) up to ∼400 K after the pulse, dependingon the amount of delivered energy (Fig. 1B), and the formationof a superheated crystal at T1>Tm . This T jump is achievedthrough redistribution of the vibrational energy through the lat-tice modes (thermalization), completed in ∼20 ps, during whichmelting already starts (13, 16). The estimated pressure change(1.3 MPa·K−1) (13) is 50 MPa during the pulse and rapidlydecreases due to melting itself in ice Ih (9). In this experiment,the energy absorbed in a single pulse exceeds that needed to heatthe sample to Tm and to obtain its complete melting (E> 2 mJfor ice Ih and E> 3 mJ for ice VI; Physical Values Used in theCalculations).

The probe is a monochromatic continuous-wave laser beam(we used λprobe =405 nm, 457.9 nm, 632.8 nm, and 1,064 nm),

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Fig. 1. Experiment outline. (A) FTIR spectra of liquid water (room conditions), ice Ih (263 K), and ice VI (1.2 GPa) superimposed to the emission spectrumof the pump. (B) Estimate of the initial temperature T1 just after the pulse, for an ice Ih sample at initial temperature T0 = 263 K. The upper and lowerlimits are ideal cases of no melting and complete melting, respectively, during the 15 ps of the pulse. (C and D) During the experiment the probe beam iscollected in two possible configurations to detect the transmission plus the scattering at azimuthal angles 0<θ< 1.5◦ (C) or the scattering at 2◦<θ< 7◦

(D). Corresponding time-dependent signals are shown on the left for each configuration (acquired with E = 3.5 mJ, T0 = 269 K, λprobe = 633 nm).

with which we measure the transmission through the sample witha fast Si photodiode (Fig. 1C). The focus of the probe (∼150 µmin diameter) lies inside the area irradiated by the pump. When apump pulse is shot, the probe transmission drops to a minimumin tens or hundreds of nanoseconds, depending on the amountof deposited energy, and then returns to a static value in a time(micro- to milliseconds) strongly dependent on T0. The trans-mission change is due to scattering at ice/melt interfaces, form-ing with the T jump and evolving in time during melting (dueto superheating) and refreezing (due to the thermal contact withthe environment that decreases T to the thermal bath tempera-ture T0). When the transmitted beam is blocked (Fig. 1D), a pos-itive signal due to off-axis scattered light is consistently observed,with a similar time dependence to that of the transmission change.In both configurations the signal amplitude increases as λprobe

decreases, as expected for scattering. As also expected, the trans-mission change is not observed when liquid samples are heatedby the T -jump pulse, except at the highest E values (E ≥ 4.5 mJ)where a strong oscillation of the transmission is observed, likelydue to the formation of vapor bubbles. Otherwise, the trans-mission of the liquid always remains ∼5–10% higher than thestatic transmission of ice, and only a weak (compared with thatobserved in ice) transmission decrease, due to thermal lensing,is observed (Transmission of the Liquid after the T Jump and Fig.S1). The absence of signal from the liquid confirms that the sig-nal observed in ice is not affected by unwanted contributions suchas interferences between pump and probe light on the photodi-ode or optical effects on the cell windows. Thermoreflectance (22,23) does not contribute to our signal because of the absence ofhighly reflecting interfaces, and its transient signal would, how-ever, be concluded in ∼1 ns. The transmission loss due to scat-tering at ice/water interfaces has been used to detect the freezingof water into ice VII under multiple-shock experiments (6, 24),using white light. Here, with monochromatic probes, we can esti-mate the size and concentration of the scattering domains evolv-ing after the pulse. Several fresh samples were studied, to checkthe reproducibility. Each sample was used for many shots, andeach shot probes a single melting/crystallization event in a crys-tal with slightly different quality. The evolution is monitored forall of the time necessary to recover the initial conditions, and themeasurements are repeated at time intervals >2 s to allow com-plete recrystallization.

Complete melting, indicated by an increase in transmissionto the liquid value, was seldom observed despite delivering anexcess of energy to achieve the process. Only by using themaximum energy values with 270 K<T0< 273 K, or piling up

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T -jump pulses with a 1- to 5-Hz rate, did we observe the liquidtransmission value.

Analysis of the Time-Dependent Transmission Loss. The transmis-sion curves were transformed into absorbance, using as referencethe initial transmission of the crystal (Fig. 2A). To understand thetime evolution of the signal, we must note that after the pulse thesample’s temperature T (t) decreases, due to the thermal contactwith the external environment, until it reaches the initial valueT0. As we will see, the thermal conduction through the sampleand the sample’s walls occurs in a longer timescale than the initialmelting, occurring at quasi-constant temperature T1. The rela-tive magnitude of these two timescales depends on the externaltemperature T0 and on the amount of energy absorbed E , whichdetermines T1 (Fig. 1B). In particular, T1 affects both the rate ofthermal diffusion (∝T (t) – T0) and the rate of melting, whichwe may expect to be kinetically driven by the superheating, thushaving a rate ∝ eT(t)−Tm .

The first maximum of the A(t) curves (Fig. 2 A and B) appearsfor E ≥ 2.5 mJ as a sharp peak followed by a fast decrease. ForE ≥ 3.5 mJ there is a second maximum at longer times. Thecurves are accurately fitted by a monoexponential growth in theinitial 500–1,000 ns. The time range in which this exponential lawwell reproduces the curve depends on both E and T0, being theresult of the characteristic times of the two competing phenom-ena described above: melting at T ∼T1 and thermal conduc-tion, the latter reducing the superheating and the melting rate.As can be seen in Fig. 2C, the monoexponential time constantτ1 decreases exponentially with E, as expected, and spans fiveorders of magnitude from 15,000 ns to 5 ns. This time constantthus represents the characteristic time of melt growth, driven bythe superheating of the crystal. For ice Ih , where many measure-ments were performed as a function of T0, we find that τ1 and A1

are independent of T0 for E ≥ 2.5 mJ, confirming that the ther-mal conduction is not relevant in the first 500–600 ns in the T0

range probed (Effect of Thermal Conduction on the Melting Rateand Fig. S2).

The values of τ1 are the same for ice Ih and ice VI within errorbars, which represent the shot-to-shot repeatability of the scat-tering signal. The two ices have different crystal structures anddifferent volume changes upon melting. Our measurements sug-gest, however, that the differences in the growth of the melt parti-cles are very similar in the two ices. In the time range investigated(t > 1 ns) as we discuss in the next section, we have no access tothe nucleation dynamics, where probably the largest differencebetween the two ices would appear.

A B C

Fig. 2. Analysis of the experimental curves at short delays. (A, Upper) The transmission curve is transformed into absorbance using the static transmissionIs as reference. (A, Lower) Example of exponential fit up to the first maximum A1. (B) Absorbance curves obtained using different pump energies. (B, Lower)Zoom-in of the same curves in the range 0–2 µs. (C) Average A1 and τ1 parameters resulting from the fit. The values are averaged over≥ 10 pulses, and forice Ih different T0 conditions are included. The error bars represent the standard deviations.

Size Evolution of Molten DomainsCalibration of our Setup and Determination of the Size of the MoltenParticles. To assess our experimental sensitivity to Mie scatter-ing, we used the configuration of Fig. 1C, with the T jump off,to measure the transmission through monodisperse suspensionsof SiO2 nanoparticles in water with diameters D = 20 nm, 80 nm,200 nm, and 400 nm and different concentrations (C), in a quartzcuvette with optical path 10 mm or 0.1 mm (Calibration of theDetection Sensitivity and Fig. S3). The expected transmission loss(reported as absorbance) of each suspension was calculated asA = Csca ∗ C ∗ h , where C is the particle’s concentration, h isthe thickness of the sample, and the Mie scattering cross-sectionCsca is computed using a FORTRAN code based on the programreported by Bohren and Huffman for monodisperse suspensionsof spherical scatterers (25). The input also contains the wave-length λprobe of the probe light, the complex refractive indexesof the particles and surrounding medium, the diameter D of theparticles, and the range of azimuthal angles θ with respect tothe forward direction (Fig. 1 C and D) at which the light is col-lected. The output contains the matrix elements of the scatter-ing cross-section for the single particles, from which the angu-lar distribution of scattered light (for polarized or unpolarizedlight) can be calculated. The angular distribution of the scatteredlight is a property of the single particle and does not dependon the concentration and can thus be used to estimate D for amonodisperse suspension. As shown in Fig. S3, the absorbancevalues measured on the calibration suspensions are in perfectagreement with those expected by the Mie-scattering theory. Forh = 0.1 mm (twice that of our samples) the setup is sensitive toparticle sizes down to D = 200 nm if C ≥ 1 µm−3. For one sus-pension (D = 400 nm, C = 0.01273 µm−3, with λprobe = 632.8 nmand h = 10 mm) we measured the signal in both configurationsof Fig. 1 C and D with the pump off. The ratio of the two signals,which is due to the angular distribution of the scattered light,yields an estimate of the scatterer’s diameter D = 409 nm. Thisresult indicates that we are able, with the ratio of the signals inthe two configurations, to correctly estimate D for a monodis-perse suspension.

For ice Ih at T0 = 269 K, using E = 3.5 mJ and λprobe =632.8 nm, transmission curves were measured in the two con-figurations of Fig. 1 C and D after T -jump pulses as a func-tion of time. In the simplest hypothesis that all of the scatteringdomains are approximately spheres with the same diameter D,the ratio between the two curves gives an estimate of D (as dis-cussed above) as a function of time. This diameter monotonicallyincreases up to ∼1.5 µm in 2.3 ms and then decreases (Fig. 3).

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Fig. 3. Determination of the scatterer’s diameter D(t) in our samples by theangular distribution of scattering, in the approximation of a monodispersesuspension of spherical particles. (A) The ratio of the signals measured as afunction of time in the two configurations of Fig. 1 C and D (ratio 1C/1D).(B) The calculated ratio of the signals detected with the configurations, asa function of D. (C) D(t) in our samples obtained by discretely sampling andcomparing the curves in A and B.

The idealized spherical scatterers in the sample are thus iden-tifiable as molten domains that grow and evolve until T >Tm

and decrease during refreezing when T = Tm . The estimate oftheir dimensions with this method is limited to large delays (t≤0.3 ms) because the configuration of Fig. 1D is affected by astrong reflection of the pump beam. Also, it makes use of datafrom two different pulses, meaning two different ice samples,and of a weak off-axis scattering signal. Thus, the D(t) valuesobtained with this method are quite approximate but are usefulto get an overall insight into our system.

Model of the Melting and Refreezing Sample. The simplest modelcapable of describing our heated sample and its evolution afterexcitation takes as the starting point the sample at ∼1 ns afterthe T jump (the minimum delay experimentally accessed) asan already nucleated monodisperse suspension of water spheresrandomly positioned in an ice matrix at homogenous tempera-ture T1, with initial diameter D1 and initial concentration C1.T1 is set to 360 K (Fig. 1B); D1 = 25 nm, as estimated in refs.13 and 16; and C1 is set to a different value for each simula-tion, ranging between 0.1 µm−3 and 5 µm−3. In the simulation(Simulation Details) we impose that the temperature at the sam-ple’s wall T decreases in time to the final value T0 with an expo-nential law. The drops are set to grow, at each time step, all bythe same amount, depending on T –Tm , and coalesce when theytouch each other. When T <Tm , the drops decrease insteadof growing. At all time steps, the absorbance is calculated on amonodisperse suspension where all of the drops have the aver-age diameter. The rate constants for the temperature decrease,drops’ growth, and drops’ decrease are chosen to reproduce theshape of the curve. The simulation does not provide any indica-tion on the actual timescales of the processes, which are derivedonly by the experiment. The simulation is essential to validatethe approximations made. In fact, despite its simplicity and thesimulation box being quite small (10× 10× 10 µm3), it is ableto reproduce with great detail the experimental curves, as canbe seen by comparing Figs. 2 and 4. In particular, the effectof increasing C1 reproduces perfectly the effect experimentallyobserved by increasing E (Figs. 2B and 4B). When C1 is suf-ficiently large, a critical diameter is reached for which a catas-trophic coalescence occurs, and the resulting A(t) curve has asharp maximum (A1). By further increasing C1 a second maxi-mum due to large drops formation appears. This behavior con-firms that E , as expected, by determining the initial temperatureafter the T jump (T1) also determines the initial concentrationof melting nuclei. The saturation of A1 with E (Fig. 2C) likelyindicates a limit superheating temperature, previously estimated

as ∼330 K (13, 26), above which the number of melting nucleibecomes independent of T.

The model allows us to constrain the scatterers’ concentrationC and obtain an estimate for their diameters D, in the approx-imation of the sample as a monodisperse suspension of waterspheres in ice (Fig. 5A). In fact, the simulation shows that Cis approximately constant before the catastrophic coalescence,occurring at a time tC where the sharp maximum is observed.During coalescence (t ≥ tC ) C and D can be linked by a packingconstraint, such as the simple cubic C = 1/D3. We thus obtain,for the A(t) curves measured with E = 3.5 mJ, D =430± 15 nmand C = 13 µm−3 at tC ∼ 200 ns (Fig. 5B), with very good agree-ment among the λprobe values (Size of the Molten Regions VersusTime and Fig. S4). Moreover, imposing that the concentrationis constant at C(t) = C(tC ) for 0< t < tC , we can deduce D(t)at t < 200 ns (Fig. S4). As shown in Fig. 5B, the melt domainsgrow slowly in time, from ∼200 nm to ∼400 nm in ∼200 ns. Therefreezing starts when the temperature of the sample’s wall, incontact with the environment, has reached Tm = 273 K. Duringfreezing, heat is constantly removed by the thermal bath, whereaswe can assume that the sample remains at Tm . Thus, the freezingrate should be dependent on the rate of heat diffusion throughthe ice/water system to the cell walls, proportional to Tm –T0. Toinvestigate the refreezing rate, we analyzed A(t) curves obtainedin ice Ih with E = 2.5 mJ and λprobe = 632.8 nm, in a series ofmeasurements at different T0 conditions. The decreasing part ofthe A(t) curves was fitted to an exponential decay. To relate thisdecay to the sample’s state, we estimated here an average diam-eter D(t) of the molten domains, decreasing with time, assum-ing C constant in this time range, at a value C2. This value isobtained by imposing the packing constraint C2 = 1/D3

2 , whereD2 is the maximum diameter of the drops (1,500 nm) obtainedfrom the analysis of the angular distribution (Fig. 3). Theresulting V (t)=πD(t)3/6 curves are well fitted to an exponen-tial decay. As expected, the rate constant 1/τ increases linearlywith T0 – Tm (Fig. 5C).

ConclusionsTime-resolved Mie scattering has been successfully used to mon-itor the melting and the successive recrystallization of water icesIh and VI with a nanosecond time resolution. The melting isinduced by an ultrafast IR pulse resonant with a vibrationalcombination band and its dynamics are probed from a few

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Fig. 5. Evolution of the ice/water system during melting and refreezing.(A) Cartoon of the proposed model. Liquid domains, approximated asspheres, grow at constant concentration C1 until coalescence occurs. Aftera coalescence period with simple cubic or other packing, when T = 273 K,the domains start decreasing in volume at fixed concentration C2. (B) Thesystem’s evolution for E = 3.5 mJ and T0 = 269 K, with the approximationof simple cubic packing during coalescence. (B, Inset) Diameter of the scat-tering domains for t< 200 ns (E = 3.5 mJ). The shaded area contains valuesobtained with different packing hypotheses. (C) The volume of the scatter-ing domains as a function of time during refreezing, for ice Ih at differenttemperatures of the thermal bath T0, obtained as described in the text andreproduced as exponential decays. (C, Lower) The time constant of theseexponential decays depends linearly on 1/∆T (∆T = T0 – Tm).

nanoseconds after the heating pulse, not having therefore accessto the nucleation of the melt seeds. The latter form after thefast relaxation of the absorbed energy (tens of picoseconds) athotspots created by or localized around absorbing molecules.The entire growth dynamics of these melting seeds, from a fewnanoseconds to tens of milliseconds, are monitored in the meso-scopic regime up to the successive and complete recrystalliza-tion. The experimental data, including the most peculiar fea-tures, are excellently reproduced by a simple phenomenologicalmodel in which a monodisperse distribution of liquid sphericalparticles grows, rearranges during coalescence, and decreases itsvolume when the temperature reaches again the melting tem-perature. Despite the strong approximations used (nucleationlaw, shape and size homogeneity) we are able to provide a pre-cise time evolution of the droplets’ dimensions that nicely con-nects the previous knowledge of the initial 250 ps (9, 13, 16,27) to the macroscopic domain. The melting implies the relax-ation of the superheated lattice at ice/water interfaces. However,the temperature at these interfaces decreases in time due tomelting itself, which is endothermic, and thus the growth slowsdown as it proceeds, as also reproduced in calculations for thesubnanosecond time window (9). The heat exchange with theenvironment also contributes to slowing down the growth bydecreasing the sample’s temperature and eventually reverses theprocess; however, this has a much slower rate (micro- to mil-liseconds) and for E≥ 2.5 mJ does not affect the sample’s behav-ior in the initial microseconds as demonstrated by the indepen-dency of τ1 on T0. Thus, in this timescale (1 ns–1 µs) the relevantdynamics are those of the lattice relaxation to its equilibrium liq-uid structure at T >Tm . For our sensitivity, ice Ih and ice VImelt with the same dynamics in the 1-ns to 1-µs time range. The

specific arrangement of the crystal structure is more likely relatedto the nucleation kinetics, which are not accessed here. We havethus unveiled fundamental aspects of the dynamics of melting ofwater ices, in a time domain previously unexplored. The phasetransitions of water rule many natural phenomena. Moreover,water is the archetypal H-bonded system and is a benchmark fortheory and simulations that can take advantage of this knowledgein the mesoscopic domain and in a timescale inaccessible to manycomputational methods. The experimental approach used in thiswork paves the way to a plethora of dynamic studies concern-ing the formation and growth of heterogeneities in bulk mate-rials, including solid-state reactivity and formation of clathratehydrates, topics of current great interest for understanding theprocesses at the Earth’s interior.

Materials and MethodsIce Ih and ice VI are prepared using water from Aldrich, HPLC grade, fil-tered with MillexVV Durapore PVDF syringe filters (pore size 0.1 µm). Forice Ih, the water is loaded in a room-pressure cell with CaF2 windows and apolytetrafluoroethylene (Teflon) spacer with 50 µm thickness and 25.4 mmdiameter, which is inserted in a Peltier cryostat and cooled to 250 K. Thetemperature is then increased to the desired T0 value (254.2–272.6 K). For iceVI, liquid water is loaded in a membrane sapphire anvil cell (SAC) equippedwith z-cut, low fluorescence sapphire anvils (Almax Easylab) and a copper-beryllium gasket. The sample is 50 µm in thickness and 500 µm in diameter.The sample is compressed to 1.5 GPa at room temperature, and then thepressure is lowered to the desired P0 value (1.2 GPa or 1.4 GPa). The fluo-rescence of a ruby chip 5 µm in diameter is used as a pressure gauge (28).The presence of ice VI in the SAC is checked by measuring the FTIR spectrumwith a Bruker spectrometer IFS120 HR (29).

The T-jump source, described in ref. 21, is an optical parametric genera-tor and amplifier pumped by a mode-locked Nd:Yag laser (EKSPLA PL2143 A;20-ps pulses, maximum energy 55 mJ/pulse, maximum repetition rate 10 Hz).The source produces 15-ps pulses tunable in the range 1,700–2,300 nm(FWHM = 100 nm), with a maximum efficiency at 1,930 nm where it can pro-vide up to 15 mJ/pulse. The pulse at 1,930 nm is focused to have an irradiateddiameter of∼400 µm. The absorbance of the sample under irradiation withthe T-jump source was determined by measuring the transmission with apyroelectric energy meter (Gentec-Eo QE25LP-S-MB), using as reference theempty cell. The absorbance is 0.55 ± 0.30 for both types of sample and isconstant in all of the T-jump energy ranges used, indicating that satura-tion of absorption is not reached even at the highest T-jump energy. Theenergy needed to heat the irradiated volume by 1 K is 11.5 µJ for ice Ihand 22.6 µJ for ice VI (30, 31). The energy needed to completely melt theirradiated volume at the melting point is 1.92 mJ for ice Ih and 2.90 mJ forice VI (32). Thus, the energy absorbed by the ice in a single pulse is suf-ficient to completely melt the irradiated volume and heat it of some tensof kelvins. The probe lasers are as follows: a laser diode, Thorlabs CPS 405(405 nm); an Ar+ laser, Coherent Sabre Innova 90 (457.9 nm); a He-Ne laser,Melles Griot 25-LHR-691 (632.8 nm); and an Nd:YAG laser, Lightwave 126(1064 nm), with ∼5–10 mW on the sample. The independence of the trans-mittance on the probe power was always verified. The probe laser is focusedon the sample to a focus diameter of ∼150 µm, and the transmission (Fig.1C) or scattering (Fig. 1D) is collected and focused on a UV-enhanced Siphotodiode with cutoff frequency fc = 60 MHz (Hamamatsu SI722-02) oran APD photodiode with fc = 3 GHz (First Sensors). For transmission (con-figuration of Fig. 1C) a diaphragm with aperture 2–8 mm (depending onthe spatial shape and divergence of the probe beam) is placed close to thelens to collect light at a small angle (0<θ< 1.5◦). For scattering (configu-ration of Fig. 1D) the diaphragm is removed, a beam stopper is placed tostop the directly transmitted beam, and the scattered light is collected inthe whole angle (2<θ< 7◦) accepted by the collecting lens (25.4 mm indiameter). The photodiode signal is monitored and recorded on a Rohde–Schwarz 2-GHz oscilloscope (RS RT1024). The scattering signal as in Fig. 1Dwas measured only in ice Ih at T0 = 269 K and E = 3.5 mJ (energy absorbed bythe sample). After each pump pulse, we can see by eye a different speckleproduced by the static scattering of the probe beam by the sample, consis-tent with melting and refreezing. Window materials with different thermalconductivity (sapphire and fluorite) were used on ice Ih to check whetherthermal diffusion through their surface could be the rate-limiting pro-cess. The results are independent of the window’s material, consistent withthermal diffusion in ice/water (thermal conductivity: k∼ 0.6 WK−1·m−1

for water, k∼ 2.2 WK−1·m−1 for ice) being slower than through the

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windows (k∼ 9 WK−1·m−1 for CaF2, k∼ 25 WK−1·m−1 for sapphire) or atthe window/sample interfaces (values for interfacial thermal resistance notavailable).

ACKNOWLEDGMENTS. This work was supported by the Deep Carbon Obser-vatory initiative (Extreme Physics and Chemistry of Carbon: Forms, Transfor-

mations, and Movements in Planetary Interiors, from the Alfred P. SloanFoundation); by the Grant Futuro in Ricerca 2010 RBFR109ZHQ fundedby the Italian Ministero dell’Istruzione, Universita, Ricerca under the pro-gram Fondo Italiano per la Ricerca di Base (FIRB); and Fondazione Cassa diRisparmio di Firenze under the project “Chimica Ultraveloce ad AltissimaPressione.”

1. Briggs R, et al. (2017) Ultrafast x-ray diffraction studies of the phase transitions andequation of state of scandium shock compressed to 82 GPa. Phys Rev Lett 118:025501.

2. Milathianaki D, et al. (2013) Femtosecond visualization of lattice dynamics in shock-compressed matter. Science 342:220–223.

3. Gorman MG, et al. (2015) Direct observation of melting in shock-compressed bismuthwith femtosecond x-ray diffraction. Phys Rev Lett 115:095701.

4. Anzellini S, Dewaele A, Mezouar M, Loubeyre P, Morard G (2013) Melting of ironat earth’s inner core boundary based on fast x-ray diffraction. Science 340:464–466.

5. Gleason AE, et al. (2015) Ultrafast visualization of crystallization and grain growth inshock-compressed SiO2. Nat Commun 6:8191.

6. Dolan DH, Knudson MD, Hall CA, Deeney C (2007) A metastable limit for compressedliquid water. Nat Phys 3:339–342.

7. Samanta A, Tuckerman ME, Yu TQ, Weinan E (2014) Microscopic mechanisms of equi-librium melting of a solid. Science 346:729–732.

8. Forsblom M, Grimvall G (2005) How superheated crystals melt. Nat Mater 4:388–390.9. Caleman C, van der Spoel D (2008) Picosecond melting of ice by an infrared laser

pulse: A simulation study. Angew Chem Int Ed Engl 47:1417–1420.10. Jin ZH, Gumbsch P, Lu K, Ma E (2001) Melting mechanisms at the limit of superheating.

Phys Rev Lett 87:055703.11. Luo SN, Ahrens TJ (2003) Superheating systematics of crystalline solids. Appl Phys Lett

82:1836–1838.12. Mei Q, Lu K (2007) Melting and superheating of crystalline solids: From bulk to

nanocrystals. Prog Mater Sci 52:1175–1262.13. Iglev H, Schmeisser M, Simeonidis K, Thaller A, Laubereau A (2006) Ultrafast super-

heating and melting of bulk ice. Nature 439:183–186.14. Siwick BJ, Dwyer JR, Jordan RE, Miller RJD (2003) An atomic-level view of melting

using femtosecond electron diffraction. Science 302:1382–1385.15. Musumeci P, Moody JT, Scoby CM, Gutierrez MS, Westfall M (2010) Laser-induced

melting of a single crystal gold sample by time-resolved ultrafast relativistic electrondiffraction. Appl Phys Lett 97:063502.

16. Schmeisser M, Iglev H, Laubereau A (2007) Bulk melting of ice at the limit of super-heating. J Phys Chem B 111:11271–11275.

17. Alsayed AM, Islam MF, Zhang J, Collings PJ, Yodh AG (2005) Premelting at defectswithin bulk colloidal crystals. Science 309:1207–1210.

18. Murray BJ, et al. (2010) Kinetics of the homogeneous freezing of water. Phys ChemChem Phys 12:10380–10387.

19. Lee GW, Evans WJ, Yoo CS (2007) Dynamic pressure-induced dendritic and shock crys-tal growth of ice VI. Proc Natl Acad Sci USA 104:9178–9181.

20. Chen JY, Yoo CS (2012) Formation and phase transitions of methane hydrates underdynamic loadings: Compression rate dependent kinetics. J Chem Phys 136:114513.

21. Citroni M, et al. (2014) Picosecond optical parametric generator and amplifier forlarge temperature-jump. Opt Express 22:30047–30052.

22. Ge Z, Cahill DG, Braun PV (2006) Thermal conductance of hydrophilic and hydropho-bic interfaces. Phys Rev Lett 96:186101.

23. Chen B, Hsieh WP, Cahill DG, Trinkle DR, Li J (2011) Thermal conductivity of com-pressed H2O to 22 GPa: A test of the Leibfried-Schlomann equation. Phys Rev B83:132301.

24. Dolan D, Gupta Y (2003) Time-dependent freezing of water under dynamic compres-sion. Chem Phys Lett 374:608–612.

25. Bohren CF, Huffman DR (1983) Absorption and Scattering of Light by Small Particles(Wiley, New York).

26. Schmeisser M, Iglev H, Laubereau A (2007) Maximum superheating of bulk ice. ChemPhys Lett 442:171–175.

27. Seifert G, Patzlaff T, Graener H (2002) Size dependent ultrafast cooling of waterdroplets in microemulsions by picosecond infrared spectroscopy. Phys Rev Lett88:147402.

28. Syassen K (2008) Ruby under pressure. High Press Res 28:75–126.29. Bini R, Ballerini R, Pratesi G, Jodl HJ (1997) Experimental setup for Fourier transform

infrared spectroscopy studies in condensed matter at high pressure and low temper-atures. Rev Sci Instrum 68:3154–3160.

30. Tchijov V (2004) Heat capacity of high-pressure ice polymorphs. J Phys Chem Solids65:851–854.

31. Bezacier L, et al. (2014) Equations of state of ice VI and ice VII at high pressure andhigh temperature. J Chem Phys 141:104505.

32. Hobbs PV (2010) Ice Physics (Oxford Univ Press, Oxford).

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