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Crystalline nucleation in undercooled liquid nickel
Acta Materialia 124 (2017) 261e267
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Crystalline nucleation in undercooled liquid nickel A. Filipponi a, *, A. Di Cicco b, S. De Panfilis c, P. Giammatteo a, F. Iesari b degli Studi dell'Aquila, I-67100, L'Aquila, Italy Dipartimento di Scienze Fisiche e Chimiche, Universita di Camerino, I-62032, Camerino, MC, Italy Physics Division, School of Science and Technology, Universita c Istituto Italiano di Tecnologia, Centre for Life Nanoscience - IIT@ Sapienza, Viale Regina Elena 291, I-00161, Roma, Italy a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 September 2016 Accepted 31 October 2016
We investigate crystalline nucleation in undercooled liquid Ni by means of x-ray absorption temperature scans in powder samples dispersed in Al2O3. We show that the crystallization process is initially dominated by a heterogeneous nucleation process that induces a purity/size selection effect. The nucleation rate displayed by the residual mass fraction of about 30% (determined down to 364 K below the melting point) is in agreement with empirical data from previous single droplet experiments extrapolated to the deeper undercooling range presently achieved. We investigate the possibility that this ultimate nucleation process is actually homogeneous by computing the rate in the framework of the classical nucleation theory. We adopted a recently implemented kinetic Monte Carlo simulation approach using extrapolated values for the self-diffusion coefficient and empirical or simulated models for the excess free energy. The comparison between measured and calculated rates indicates that the homogeneous nucleation scenario is likely. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Nickel Nucleation X-ray synchrotron radiation Kinetic Monte Carlo simulation
1. Introduction Elemental nickel is a representative 3d transition metal exhibiting a close packed liquid phase above the melting temperature Tm ¼ 1728 K at ambient pressure. Ni, similarly to other elemental or complex liquids, can be appreciably undercooled below Tm into a metastable liquid state. Following Frank's suggestions [1] that the possible icosahedral short range order developing in the undercooled liquid state is responsible for the large degree of undercooling and the successive views on the polytetrahedral structures in liquids [2], several modern experiments were performed to unravel the relevant structure details [3,4]. These experiments, performed using neutron scattering on electromagnetically levitated samples [3] down to 1435 K or X-ray absorption spectroscopy [4] down to 1493 K, agree that a certain degree of icosahedral short range order develops upon undercooling. Large-scale molecular dynamics (MD) simulations, recently performed [5] to model iron melts, were able to provide an atomistic insight into the nucleation process. A major issue regards the characteristics of the crystalline nucleation process limiting the lower accessible temperature. The
* Corresponding author. E-mail address: adriano.fi[email protected] (A. Filipponi). http://dx.doi.org/10.1016/j.actamat.2016.10.076 1359-6454/© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
phenomenon is currently understood in the framework of the classical nucleation theory (CNT) [6]. For volume nucleation phenomena the temperature dependent physical quantity of interest is the nucleation rate per unit mass J(T). Any sample of mass m subject to isothermal or cooling processes under thermodynamic equilibrium will have a survival probability in the molten state at time t equal to
In the ideal case of homogeneous nucleation J(T) ¼ Jh(T) while in the presence of heterogeneous nucleation effects J(T) [ Jh(T) and the probability P (t) will be correspondingly reduced as well as the accessible undercooling T range. Early estimates of the nucleation rate based on the maximum observed undercooling temperature have been superseded by modern approaches based on the inversion of the information on the inhomogeneous Poisson process embodied in Eq. (1) on a set of repeated thermal cycles on the same sample. Early treatments [7e9] based on a model dependent inversion were successively improved adopting model independent approaches [10e14]. In these approaches it is possible to retrieve experimental information on J(T) that is subsequently analyzed to asses the nature of the dominant processes. In a set of
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recent papers [13,15] undercooled liquid Ni samples embedded in a molten glass matrix were subject to repeated thermal cycles in a differential scanning calorimeter (DSC) obtaining J(T) in the 1418e1443 K range. The data interpretation was supported by umbrella sampling Monte Carlo simulations to obtain the excess free energy of a nucleus of size n, DG(n,T), embedded in the surrounding undercooled liquid for the adopted model interaction. The results provide a strong evidence that the samples were actually subject to homogeneous nucleation and, in spite the simulations displayed evident departures from the spherical nucleus shape, the DG(n*,T) barrier height (where n* is the critical nucleus size) was found to follow the temperature dependence predicted by the CNT. In this paper we extend our previous work [4] and focus on the information related to the nucleation rate that can be obtained from x-ray absorption temperature scans [16] based on a previously assessed methodology [17]. The idea behind this approach is that by using a phase sensitive bulk probe, such as the x-ray absorption coefficient at a selected energy in the element absorption edge region, it is possible to measure the sample liquid (and solid) fractions as a function of temperature (and time) during repeated thermal cycles probing (and averaging) simultaneously about 107 droplets. It will be shown that the highest purity fraction of our sample displays a nucleation rate in agreement with previous data extending the undercooling range down to z370 K. In order to investigate the nature of the nucleation process the experimental rates are compared with calculated rates for various models, within the coarse grained CNT scenario, using a recently developed kinetic Monte Carlo (KMC) simulation scheme [18]. The paper is organized as follows: in Sec. 2 we discuss and clarify how the nucleation rate in the 1350e1470 K range was obtained in comparison with previous experiments. In Sec. 3 the newly implemented kinetic Monte Carlo simulation approach [18] was used to compute the Ni nucleation rate within the CNT assumptions or adopting the published DG(n,T) functions. The results are discussed in Sec. 4 and conclusions are drawn in Sec. 5. 2. X-ray absorption temperature scans
way to the Pd case [21]. In the first thermal cycles a sintering of the alumina grains occurs leading to a slight shrinking of the sample pellets but successively the samples become mechanically stable and can be subject to repeated cycles maintaining the same average Ni surface density relevant to the x-ray transmission measurement. The sample temperature was measured with a high-temperature pyrometer pointing on the outer side of the crucible and calibrated using the graphite emissivity and the Ni melting signature at Tm. The maximum absolute uncertainty in the temperature measurement in the range of interest is ±15 K. The sample phase was assessed using both x-ray diffraction measurements, collected with an area detector, and x-ray absorption spectroscopy at the Ni Kedge. The sample absorption coefficient, apart from a smooth irrelevant background associated with the windows, crucible, Al2O3 absorption and ion chamber setting, is related to a ¼ ln(I0/I1) where I0 and I1 are the electron currents in the upstream and downstream ion chambers. Preliminary XAS spectra were collected to verify the contrast between solid (prior to melting) and undercooled liquid phases at Tu x 1493 K as shown in Fig. 1 and the point of highest contrast identified in the edge region at E* ¼ 8.3382 keV.
2.2. Temperature scans By setting the monochromator energy to E* it is possible to measure the sample absorption while cycling the temperature in the desired range. Examples of these temperature scans are reported in Fig. 2. The sample switches from the low absorption solid phase level to the higher liquid phase absorption level (at this selected energy), following the phase transitions occurring in the scan. The sharp rise of absorption at Tm in the heating ramps marks sample melting and the small temperature interval in which this transition occurs DT z 10 K is due to the temperature spread across the sample area probed by the x-ray beam at Tm. An even better temperature homogeneity (estimated within DT z 2e3 K) occurs in the cooling ramps and at lower temperatures. In the cooling ramps the evident hysteresis indicates that Ni droplets crystallize only in the temperature range between 1450 K and 1370 K
2.1. Experimental details The experiment, described in a previous paper [4], was performed at the BM29 beamline [19,20] of the European Synchrotron Radiation Facility using an energy scanning monochromator equipped with a Si(311) flat crystal pair. The rectangular beam size was defined by the primary slit vertical aperture 0.3 mm and the horizontal aperture of the secondary slits prior to the first ion chamber set to a size of 2.0 mm. Samples suitable for X-ray absorption measurements in transmission mode were prepared from a mixture of sub-micron size Ni powder (99.9% purity) and alumina (Al2O3) powder in a 1:20 mass ratio suspended in ethyl alcohol, subject to ultrasonic mixing and filtered through a poly-carbonate membrane. The homogeneous mixture of the two powders was pressed in 13 mm diameter and about 100 mm thick pellets suitable for the successive thermal treatments. The overall Ni surface density was optimized for an absorption contrast at the edge of about 1.0e1.5 while the amount of alumina was determined by the compromise to limit the background absorption and simultaneously achieve a sufficient dispersion of the Ni powder. The sample slices were inserted into a folded graphite foil (with a negligible x-ray absorption) acting as a resistive heater and providing a uniform temperature field. This crucible was placed in the center of a high-vacuum P z 103 Pa cylindrical Pyrex glass vessel with suitable x-ray windows. Alumina was not found to react with Ni at the temperatures involved in the experiment in a similar
Fig. 1. Ni K-edge x-ray absorption spectra of the liquid (red) and solid (blue) phases at Tu x 1493 K. The difference spectrum da ¼ a[as is reported in the lower panel on a magnified scale. The vertical arrow marks the energy E* corresponding to the maximum contrast. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. Temporal sequence (from top to bottom and according to the arrows) of temperature scans at E ¼ 8.3382 keV. Reproducible hysteresis loops provide a clear indication of the undercooling sample capabilities. The vertical green dashed line marks Tm while the nearly horizontal red dashed lines are the linear fits to the liquid and solid absorption levels. Data points were collected at intervals of about 3.5 s and the cooling rates were 2.1 K/s in the first two loops and 1.1 K/s in the last. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
displaying an evident undercooling. A not negligible mass fraction of Ni droplets reaches a deep undercooling Tm Tu z 350 K. These scans are essential to understand the sample behavior and prepare it in the desired state for the XAFS measurements at constant temperature required for the structural investigation [4]. The molten or crystallized sample state was confirmed by the absence or presence of the f.c.c. Debye-Scherrer rings in the diffraction patterns collected with the area detector. The temperature scans not only provide a qualitative insight on the sample phase transition dynamics but also a quantitative information on the relative mass fraction of molten/crystallized sample in the cooling ramps. This mass fraction can be reliably estimated by the absorption level position between a linear extrapolation of the liquid phase absorption level in the nucleation temperature range and a linear fit of the solid phase absorption level in the heating ramp (shown with red dashed lines in Fig. 2). For this sample and scans the uncertainty in the mass fraction is in the 1% range due to the combined uncertainty in the extrapolations and data noise.
2.3. Experimental nucleation rate The relatively broad temperature region in which crystallization takes place is due to the relatively wide distribution of droplet masses. The larger droplets or those containing stronger heterogeneous nucleants are likely to crystallize earlier than the smallest droplets. With respect to single droplet experiments this strategy has clearly the drawback to require an estimate of the droplet mass distribution, however the advantage is that a single cooling ramp measurement provides the required ensemble statistical average over about 107 droplets subject to the same thermal treatment. Information on the temperature dependence of the nucleation rate can be retrieved in the entire, relatively wide, crystallization temperature range. The data analysis of the molten sample fraction is performed using the previously assessed methodology [17]. The average Ni liquid mass fraction in a cooling experiment is given by a mass weighted average of Eq. (1) over the mass probability density f(m):
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Fig. 3. Log-normal model droplet diameter distribution (solid line). The dashed lines display the fraction of droplets in the liquid state for increasing values of l ¼ 1 106, 1 107, 1 108, 1 109, 1 1010, 1 1011 mg1.
Z By indicating lðt Þ ¼
Z X¼
0
∞
t
0 0 dt the equation J T t
0
m f ðmÞexpð mlÞdm Z ∞ m f ðmÞdm
(3)
0
can be numerically solved to find the l values for each measured X at the corresponding t and T. Then the nucleation rate is simply obtained by the numerical differentiation corresponding to the analytic identity:
JðTÞ ¼
vl vt t¼tðTÞ
(4)
The powder size distribution was modeled as a lognormal with average 〈lnðd=1mmÞ〉x0:45 and standard deviation s x 1, parameters supported by a microscopic inspection of the recovered sample after the measurements. This distribution is illustrated in Fig. 3 together with the progressive effect of the exp(ml) factor for increasing l. The specific nucleation rates were extracted from the three cooling ramps in Fig. 2 and the results are reported in Fig. 4. The data collected with different cooling rates are compatible with a unique monotonic J(T) function. The dashed curve refers to the previous empirical fit to single droplet experiments processed in a DSC [13] based on the function
J ðT Þ ¼ G exp
BT 2 kB ðTm TÞ2
(5)
with G ¼ 3.98 1023 Hz/mg, B ¼ 2.14 104 eV/K, highlighted as a thick line in the region where the measurements were actually taken. It is evident that for T ( 1400 K the present results are in excellent agreement with this previous empirical functional dependence extrapolated to the deeper undercooling range
Fig. 4. Specific nucleation rates obtained in the three successive (, first, + second, third) cycles of the temperature scan. The dashed curve refers to the previous fit to the single drop experiment data [13]. The dotted curve is a fit to the present data for T > 1400 K.
achieved in the present study (dashed part). This comparison provides an insight on the crystalline nucleation process occurring in the present droplet samples. The first droplets begin to crystallize around T z 1470 K with a rate that exceeds by several orders of magnitudes the one of single droplet experiments A fit with the empirical function (5) yields in this case G ¼ 5.0 108 Hz/mg, B ¼ 4.53 105 eV/K, indicating the dominance of a heterogeneous nucleation process. Around T z 1400 K present data clearly display a cross-over towards the functional trend observed in single droplet experiments. The subsequent crystallization for T < 1400 K involves smaller size droplets accounting for a sample mass fraction of about 30%. The present experiment clearly involves a less cleaner environment than the one of single droplet experiments, but the nucleation process itself enforces (see Fig. 3) a selection of small size and higher purity droplets that progressively remain in the liquid state. The present evidence is that about 30% of the Ni mass in our sample behaves in terms of nucleation properties similarly to single droplet experiments that, following previous assignments [13], may be ascribed to the homogeneous nucleation limit. With this interpretation, present data validate previous determinations in a broader undercooling temperature range.
3. KMC simulations of the nucleation process In the framework of the coarse grained scenario of the classical nucleation theory the nuclei population Nn(t) (number of crystalline nuclei of size n at time t) evolves with instantaneous events of single atom accretion or loss with size (and temperature) dependent rates p(n,T) and m(n,T). Nuclei are meaningful from a minimum size, assumed to be nm ¼ 10. In order to calculate the nucleation rate we adopted a recently implemented kinetic Monte Carlo simulation approach [18] that allows to simulate the actual stochastic process and determine a number of average properties including the stationary nucleation rate of the simulated sample. As input information the simulation requires only an initial configuration and the process rates which, in turn, depend on both energetic and kinetic properties of the system. In order to cope with total transition rates Nn(t) p(n) and Nn(t) m(n) (differing by several orders of magnitude due to the exponential increase of Nn(t) with decreasing n), the KMC simulation is
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implemented using a hybrid approach. The kinetic processes in the Nn(t) tail (starting from ne for which Nnne(t) < 8) where treated exactly defining the asynchronous time increment t of the simulation, while for n < ne the dynamics was approximated updating the populations with Poissonian random numbers corresponding to the average rates Nn(t) p(n)t and Nn(t) m(n)t for each process involved. The process rates are constrained by the detailed balance requirement so that the backward (atom loss) rates m(nþ1,T) can be obtained from the accretion rates p(n,T) as:
mðn þ 1; TÞ ¼ pðn; TÞebðDGðn;TÞDGðnþ1;TÞÞ :
(6)
DG(n,T) is one fundamental ingredient of the theory and represents the excess free energy associated with the formation of a nucleus of size n in the undercooled liquid at temperature T. The basic CNT model for DG(n,T) includes a volume and a surface term that can be estimated on the basis of Ni macroscopic properties such as atomic weight 58.69, solid density r ¼ 8908 kg/m3, melting temperature Tm ¼ 1728 K, latent heat of fusion l ¼ 17.48 kJ/mol [22]. The surface term can be better estimated using the negentropic [23e25] model for the liquid/crystal interface energy s ¼ s0T/Tm with s0 ¼ 0.302 J/ m2. The known limitations of this approach, i.e. the attempt to model the properties of microscopic nuclei using bulk properties, can be partly overcame by modern computational techniques and in particular those based on the umbrella sampling Monte Carlo method [26,27]. These simulations also account for the non-sharp crystal-liquid interface and non-spherical nuclei shape, that provide a not-negligible correction to CNT prediction [28], performing a suitable ensemble average. Umbrella sampling Monte Carlo simulations for Ni were performed [13] using a well established embedded-atom model (EAM) potential [29]. The melting point of EAM Ni [30] is only 20 K greater than the actual Tm so it is believed to provide a sufficiently reliable model interaction. In Fig. 5 we report the comparison of different DG(n,T) models at five representative temperatures chosen to be at the same undercooling from the respective Tm as the simulated data. As previously remarked [13], the simulation data, that are reliable up to n z N/8, differ from the basic CNT model but the dependence of the (lower) barrier height DG(n*,T) follows the predicted CNT dependence. In order to exploit the Monte Carlo data in our simulation, following previous suggestions [28], we fitted the digitized data using a DillmannMeier [31] functional form (reported as dotted curves in Fig. 5). The flat region around the DG(n*,T) maximum is not perfectly reproduced especially at low undercooling, but we believe these fits are anyway adequate for the present purpose. The second ingredient of the model is an estimate for the accretion rate that, according to previous approaches [6], is assumed to be: pðnÞ ¼
6DðT Þ
d2
2
4n 3
Fig. 5. Excess free energy DG(n,T) of crystalline nuclei in the undercooled liquid as a function of size n. A set of curves are reported for each of the selected five temperatures T. The thick solid curves refer to the basic CNT model. The thin dashed lines refer to a digitized version of the previous data [13] from the largest umbrella sampling Monte Carlo simulations (N ¼ 8788) at the same undercooling with respect to the EAM Ni melting point as the CNT DG(n,T) with respect to Tm. The dotted lines are the corresponding fitted shapes according to the Dillmann-Meier form.
The KMC simulation can be efficiently performed over a wide range of temperature and sample size and for the specific comparison we investigated the range T ¼ 1280e1480 K using both the CNT model (at intervals DT ¼ 25 K) and the fitted DG(n,T) curves to the umbrella sampling MC simulations. An example of the nuclei distribution in a simulation run is reported in Fig. 6. The simulation refers to the umbrella sampling DG(n,T) model at T ¼ 1380 K (green dotted curve in Fig. 5) with a critical nucleus size n* ¼ 412 and a droplet containing about Nat ¼ 8.7 1015 atoms. The green curve reproduces the reference eq equilibrium nuclei distribution Nn xNat exp½DGðn; TÞ=ðkB TÞ. The metastable state is characterized by the stationary solution of the set of kinetic differential equations for the process Nnst , associated
(7) 2
where D(T) is the self-diffusion coefficient, 4 n3 is a reasonable estimate for the number of attachment sites in a nucleus of size n and d represents average travel distance for an atom in the surrounding liquid required to reach the attachment site. The Ni self-diffusion coefficient was measured by recent inelastic neutron scattering experiments [32] and is accurately described by an Arrhenius empirical function D(T) ¼ D0exp(TD/T) with D0 ¼ 77 109 m2/s and TD ¼ 5454 K. The lowest temperature of the actual measurements was 1514 K and we assume that the Arrhenius behavior can be safely extrapolated in the temperature range of interest (150 K below the actual measurement). The attachment rates are calculated using D(T) in Eq. (7) with a typical travel for attachment d ¼ 0.25 nm.
Fig. 6. Snapshot of the KMC simulation showing a typical instantaneous Nn(t) nuclei eq distribution (red histogram) compared with the equilibrium Nn (green curve) and stationary Nnst distributions (blue curve). The video (visible on-line) follows the KMC simulation of the random process and the related distribution evolution for about 1.3 ns.
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with a constant nuclei flux J, reported as a blue curve. Nn(t) are a set of integer natural numbers representing random realizations of Poisson random variables with average Nnst , and corresponding fluctuations. The nuclei distribution in the snapshot includes an overcritical nucleus with n z 1000 that initiated an irreversible growth and a few embryo nuclei in the n z n* region. In the video associated with Fig. 6 (visible on-line) it is possible to follow the KMC simulation for about 1.3 ns during which two nucleation events occur. Individual frames are sampled every 5000 elementary steps t of the actual KMC simulation. Nat, corresponding to a eq z50 mm liquid Ni droplet, was selected in order that Nn ¼ 0:01, for graphical purposes. Simulations with smaller Nat can be performed to reproduce the behavior of the actual sub-micrometric sample droplets in our experiment. An increased computational efficiency for the nucleation rate determination is however obtained with larger Nat values corresponding to Nneq in the range 1e10. Supplementary video related to this article can be found at http://dx.doi.org/10.1016/j.actamat.2016.10.076. The nucleation rates are computed with a statistics of at least 260 nucleation events in the equilibrated runs while over 1200 nucleations events were recorded at the lowest temperatures (where smaller critical sizes n* are involved). The corresponding statistical uncertainty is in the 3e6% range. The results for the nucleation rate are compared with the experimental data in Fig. 7. The green curve reports the fit to the previous experimental data [13] compatible with the low temperature tail of present data. The homogeneous nucleation rates calculated for the CNT DG model clearly display a greater slope coherent with the previously emphasized larger temperature dependence of DG(n*,T). On the contrary, the umbrella sampling Monte Carlo DG model accurately reproduces the experimental slope. The absolute nucleation rates scale with a kinetic pre-factor determined by the elementary rates of Eq. (7). By adopting the same d parameter used for the CNT model the nucleation rates would result largely overestimated falling on the dashed curve (open symbols) of Fig. 7. The entire curve can be however brought in close agreement with all available experimental data with a reduced kinetic pre-factor.
4. Discussion The comparison between the calculated homogeneous nucleation rates and the experimental data, reported in Fig. 7, is very interesting. In spite of the order of magnitude differences among the various curves, the possibility to report data on the same scale is anyway a promising result indicating that models and data are at least in reasonable agreement. The fact that the experimental rates are lower than the calculated homogeneous nucleation rates provides a support to the interpretation that the best Ni samples have actually reached the homogeneous nucleation limit, while evidently the models with the present choice of parameters tend to overestimate the rates. The present CNT estimate of the nucleation rate matches the experimental data around T z 1430 K but displays a greater temperature slope associated with the larger barrier heights. By adopting the DG model from umbrella sampling MC simulation the reduced DG(n*,T) barrier heights bring the slope in close agreement with the experimental trends, but at the same time increase the absolute J values. A close agreement between experimental and calculated values is obtained combining these DG values with a reduced kinetic rate than the one obtained according to Eq. (7) with d ¼ 0.25 nm. We believe this occurrence is not fortuitous. The argument behind Eq. (7) is to estimate the reciprocal of the average time taken by an atom to travel a distance of the order of the interatomic distance in a diffusion process. It can be argued that in this time lapse the atom will be somewhere on the surface of a sphere of radius d, but actually the attachment will occur only if the atom falls within a narrow region around the attachment site with a typical cross section of the order of p(d/10)2 being d/10 a typical vibrational standard deviation for a high temperature solid. This correction provides a scaling of the order of 1/400 to the p(n) estimate. An additional reduction factor may be obtained from the requirement that the kinetic energy of the atom lays below a certain threshold to avoid a successive escape. Present evidence and all previous arguments stimulate further investigations on improved models to estimate the basic process rates for the coarse grained description of the nucleation phenomenon, possibly based on computer simulation approaches. If the green curve in Fig. 7 is assumed to represent homogeneous nucleation, from the ratio between the fitting B coefficients and assuming that the nucleus has the shape of a spherical sector with contact angle q, it is possible to estimate the barrier height reduction for heterogeneous nucleation as ð2 3 cos q þ cos3 qÞ=4x0:21 corresponding to qx66 . The crossover to the homogeneous nucleation trend takes place around 1400 K and involves a sample mass fraction of about 30%. 5. Conclusions
Fig. 7. Temperature dependence of the stationary homogeneous nucleation rate J of undercooled liquid Ni calculated using the KMC approach with different DG models. Present experimental data are reported with black symbols as in Fig. 4 while the previous fitting curve to single drop DSC experiments is reported as a green line (thick in the measurement region dashed elsewhere). The KMC homogeneous nucleation rates are reported as points (blue curve) for the CNT model and as , or - (with a reduced kinetic prefactor) points (red curves) for the DG model function fitting the umbrella sampling MC data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
We measured the nucleation rate of undercooled liquid Ni droplets in the 1365e1470 K temperature range using x-ray absorption temperature scans on a sub-micrometric Ni powder sample dispersed in Al2O3. The data display a cross-over between two nucleation regimes, the one occurring at higher temperature being clearly heterogeneous. Present data for T(1400 K are in agreement with the extrapolation of J(T) determined in a previous single droplet experiment [13], supporting the interpretation that the green curve in Fig. 7 actually represents an intrinsic property of undercooled liquid Ni, being compatible with experimental data obtained in two independent experiments with different techniques. In order to investigate the nature of the ultimate nucleation process occurring in all investigated Ni samples we computed the homogeneous nucleation rate using the CNT prescriptions or
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analytic DG(n,T) models fitted to the previous umbrella sampling data obtained with the largest simulated samples. The nucleation rates were calculated exploiting the kinetic Monte Carlo approach [18] that offers a wide flexibility in the input data and allows for the comparison of various aspects of the process. The KMC simulations were performed efficiently over the temperature range of interest and their reliability is confirmed by the coincidence of the simulated results with analytic expressions, when available. The comparison between measured and calculated rates provides a strong support to the interpretation that in single Ni droplet experiments in a DSC [13] and in the XAS Ni experiment [4] (in the case of the smallest size droplets) the sample crystallization was actually triggered by homogeneous nucleation. References [1] F.C. Frank, Supercooling of liquids, Proc. R. Soc. Lond. A 215 (1952) 43e46, http://dx.doi.org/10.1098/rspa.1952.0194. [2] F. Spaepen, Five-fold symmetry in liquids, Nature 408 (2000) 781e782, http:// dx.doi.org/10.1038/35048652. [3] T. Schenk, D. Holland-Moritz, V. Simonet, R. Bellissent, D.M. Herlach, Icosahedral short-range order in deeply undercooled metallic melts, Phys. Rev. Lett. 89 (2002) 075507, http://dx.doi.org/10.1103/PhysRevLett.89.075507. [4] A. Di Cicco, F. Iesari, S. De Panfilis, M. Celino, S. Giusepponi, A. Filipponi, Local fivefold symmetry in liquid and undercooled Ni probed by x-ray absorption spectroscopy and computer simulations, Phys. Rev. B 89 (2014) 060102, http://dx.doi.org/10.1103/PhysRevB.89.060102 (R). [5] Y. Shibuta, S. Sakane, T. Takaki, M. Ohno, Submicrometer-scale molecular dynamics simulation of nucleation and solidification from undercooled melt: linkage between empirical interpretation and atomistic nature, Acta Mater. 105 (2016) 328e337, http://dx.doi.org/10.1016/j.actamat.2015.12.033. [6] K.F. Kelton, A.L. Greer, Nucleation in Condensed Matter: Applications in Materials and Biology, Elsevier, Amsterdam, 2010. [7] C.W. Morton, W.H. Hofmeister, R.J. Bayuzick, M.B. Robinson, A statistical approach to understanding nucleation phenomena, Mater. Sci. Eng. A 178 (1994) 209e215, http://dx.doi.org/10.1016/0921-5093(94)90545-2. [8] W.H. Hofmeister, C.W. Morton, R.J. Bayuzick, Monte Carlo testing of the statistical analysis of nucleation data, Acta Mater. 46 (1998) 1903e1908, http:// dx.doi.org/10.1016/S1359-6454(97)00439-4. [9] T. Schenk, D. Holland-Moritz, W. Bender, D.M. Herlach, Statistical analysis of nucleation in undercooled Co-base alloys, J. Non-Cryst. Solids 250e252 (1999) 694e698, http://dx.doi.org/10.1016/S0022-3093(99)00162-3. [10] A. Filipponi, M. Malvestuto, An experimental set-up for the nucleation rate determination in supported undercooled liquid metal droplets, Meas. Sci. Technol. 14 (2003) 875e882, http://dx.doi.org/10.1088/0957-0233/14/6/325. [11] G. Wilde, J.L. Sebright, J.H. Perepezko, Bulk liquid undercooling and nucleation in gold, Acta Mater. 54 (2006) 4759e4769, http://dx.doi.org/10.1016/ j.actamat.2006.06.007. [12] G. Wilde, C. Santhaweesuk, J.L. Sebright, J. Bokeloh, J.H. Perepezko, Kinetics of heterogeneous nucleation on intrinsic nucleants in pure fcc transitionmetals, J. Phys. Condens. Matter 21 (2009) 464113, http://dx.doi.org/10.1088/09538984/21/46/464113. [13] J. Bokeloh, R.E. Rozas, J. Horbach, G. Wilde, Nucleation barriers for the liquidto-crystal transition in Ni: experiment and simulation, Phys. Rev. Lett. 107 (2011) 145701, http://dx.doi.org/10.1103/PhysRevLett.107.145701. [14] A. Filipponi, A. Di Cicco, E. Principi, Crystalline nucleation in undercooled liquids: a bayesian data-analysis approach for a nonhomogeneous Poisson process, Phys. Rev. E 86 (2012) 066701, http://dx.doi.org/10.1103/
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PhysRevE.86.066701. [15] J. Bokeloh, G. Wilde, R.E. Rozas, R. Benjamin, J. Horbach, Nucleation barriers for the liquid-to-crystal transition in simple metals: experiment vs. simulation, Eur. Phys. J. Spec. Top. 223 (2014) 511e526, http://dx.doi.org/10.1140/ epjst/e2014-02106-2. [16] A. Filipponi, M. Borowski, P.W. Loeffen, S. De Panfilis, A. Di Cicco, F. Sperandini, M. Minicucci, M. Giorgetti, Single energy x-ray absorption detection: a combined electronic and structural local probe for phase transitions in condensed matter, J. Phys. Condens. Matter 10 (1998) 235e253, http://dx.doi.org/ 10.1088/0953-8984/10/1/026. [17] S. De Panfilis, A. Filipponi, Nucleation rate of solidification probed by x-ray absorption temperature scans in undercooled liquid metals, J. Appl. Phys. 88 (2000) 562e570, http://dx.doi.org/10.1063/1.373696. [18] A. Filipponi, P. Giammatteo, Kinetic Monte Carlo simulation of the classical nucleation process, J. Chem. Phys. 145 (2016) 211913, http://dx.doi.org/ 10.1063/1.4962757. [19] A. Filipponi, M. Borowski, D.T. Bowron, S. Ansell, S. De Panfilis, A. Di Cicco, J., An experimental station for advanced research on condensed matter P. Itie under extreme conditions at the esrf - BM29 beamline, Rev. Sci. Instrum. 71 (2000) 2422e2432, http://dx.doi.org/10.1063/1.1150631. [20] O. Mathon, S. Pascarelli, Upgrade of BM29 and ID24 optics. two complementary beamlines for XAS measurements at ESRF, of AIP Conference Pro€hr, H.A. Padmore, J. Arthur (Eds.), Synchrotron ceedings, in: T. Warwick, J. Sto Radiation Instrumentation: Eighth International Conference on Synchrotron Radiation Instrumentation, vol. 705, 2004, pp. 498e501, http://dx.doi.org/ 10.1063/1.1757843. [21] A. Filipponi, A. Di Cicco, S. De Panfilis, Structure of undercooled liquid Pd probed by x-ray absorption spectroscopy, Phys. Rev. Lett. 83 (1999) 560e563, http://dx.doi.org/10.1103/PhysRevLett.83.560. [22] J.M.W. Chase, NIST-JANAF themochemical tables, fourth ed. J. Phys. Chem. Ref. Data Monogr. 9 (1998) 1e1951 https://srd.nist.gov/JPCRD/jpcrdM9.pdf. [23] F. Spaepen, A structural model for the solid-liquid interface in monatomic systems, Acta Metall. 23 (1975) 729e743, http://dx.doi.org/10.1016/00016160(75)90056-5. [24] F. Spaepen, R.B. Meyer, The surface tension in a structural model for the solidliquid interface, Scr. Metall. 10 (1976) 37e43, http://dx.doi.org/10.1016/00369748(76)90323-9. [25] F. Spaepen, R.B. Meyer, The surface tension in a structural model for the solidliquid interface, Scr. Metall. 10 (1976) 257e263, http://dx.doi.org/10.1016/ 0036-9748(76)90374-4. [26] S. Auer, D. Frenkel, Prediction of absolute crystal-nucleation rate in hardsphere colloids, Nat. Lond. 409 (2001) 1020e1023, http://dx.doi.org/ 10.1038/35059035. [27] S. Auer, D. Frenkel, Numerical prediction of absolute crystallization rates in hard-sphere colloids, J. Chem. Phys. 120 (2004) 3015e3029, http://dx.doi.org/ 10.1063/1.1638740. [28] S. Prestipino, A. Laio, E. Tosatti, Systematic improvement of classical nucleation theory, Phys. Rev. Lett. 108 (2012) 225701, http://dx.doi.org/10.1103/ PhysRevLett.108.225701. [29] S.M. Foiles, Application of the embedded-atom method to liquid transition metals, Phys. Rev. B 32 (1985) 3409e3415, http://dx.doi.org/10.1103/ PhysRevB.32.3409. [30] R.E. Rozas, A.D. Demiraǧ, P.G. Toledo, Thermophysical properties of liquid Ni around the melting temperature from molecular dynamics simulation, J. H. 1, J. Chem. Phys. 145 (2016) 064515, http://dx.doi.org/10.1063/1.4960771. [31] A. Dillmann, G.E.A. Meier, A refined droplet approach to the problem of homogeneous nucleation from the vapor phase, J. Chem. Phys. 94 (1991) 3872e3884, http://dx.doi.org/10.1063/1.460663. [32] A. Meyer, S. Stüber, D. Holland-Moritz, O. Heinen, T. Unruh, Determination of self-diffusion coefficients by quasielastic neutron scattering measurements of levitated Ni droplets, Phys. Rev. B 77 (2008) 092201, http://dx.doi.org/ 10.1103/PhysRevB.77.092201.
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Crystalline nucleation in undercooled liquid nickel A. Filipponi a, *, A. Di Cicco b, S. De Panfilis c, P. Giammatteo a, F. Iesari b degli Studi dell'Aquila, I-67100, L'Aquila, Italy Dipartimento di Scienze Fisiche e Chimiche, Universita di Camerino, I-62032, Camerino, MC, Italy Physics Division, School of Science and Technology, Universita c Istituto Italiano di Tecnologia, Centre for Life Nanoscience - IIT@ Sapienza, Viale Regina Elena 291, I-00161, Roma, Italy a
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 September 2016 Accepted 31 October 2016
We investigate crystalline nucleation in undercooled liquid Ni by means of x-ray absorption temperature scans in powder samples dispersed in Al2O3. We show that the crystallization process is initially dominated by a heterogeneous nucleation process that induces a purity/size selection effect. The nucleation rate displayed by the residual mass fraction of about 30% (determined down to 364 K below the melting point) is in agreement with empirical data from previous single droplet experiments extrapolated to the deeper undercooling range presently achieved. We investigate the possibility that this ultimate nucleation process is actually homogeneous by computing the rate in the framework of the classical nucleation theory. We adopted a recently implemented kinetic Monte Carlo simulation approach using extrapolated values for the self-diffusion coefficient and empirical or simulated models for the excess free energy. The comparison between measured and calculated rates indicates that the homogeneous nucleation scenario is likely. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Nickel Nucleation X-ray synchrotron radiation Kinetic Monte Carlo simulation
1. Introduction Elemental nickel is a representative 3d transition metal exhibiting a close packed liquid phase above the melting temperature Tm ¼ 1728 K at ambient pressure. Ni, similarly to other elemental or complex liquids, can be appreciably undercooled below Tm into a metastable liquid state. Following Frank's suggestions [1] that the possible icosahedral short range order developing in the undercooled liquid state is responsible for the large degree of undercooling and the successive views on the polytetrahedral structures in liquids [2], several modern experiments were performed to unravel the relevant structure details [3,4]. These experiments, performed using neutron scattering on electromagnetically levitated samples [3] down to 1435 K or X-ray absorption spectroscopy [4] down to 1493 K, agree that a certain degree of icosahedral short range order develops upon undercooling. Large-scale molecular dynamics (MD) simulations, recently performed [5] to model iron melts, were able to provide an atomistic insight into the nucleation process. A major issue regards the characteristics of the crystalline nucleation process limiting the lower accessible temperature. The
* Corresponding author. E-mail address: adriano.fi[email protected] (A. Filipponi). http://dx.doi.org/10.1016/j.actamat.2016.10.076 1359-6454/© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
phenomenon is currently understood in the framework of the classical nucleation theory (CNT) [6]. For volume nucleation phenomena the temperature dependent physical quantity of interest is the nucleation rate per unit mass J(T). Any sample of mass m subject to isothermal or cooling processes under thermodynamic equilibrium will have a survival probability in the molten state at time t equal to
In the ideal case of homogeneous nucleation J(T) ¼ Jh(T) while in the presence of heterogeneous nucleation effects J(T) [ Jh(T) and the probability P (t) will be correspondingly reduced as well as the accessible undercooling T range. Early estimates of the nucleation rate based on the maximum observed undercooling temperature have been superseded by modern approaches based on the inversion of the information on the inhomogeneous Poisson process embodied in Eq. (1) on a set of repeated thermal cycles on the same sample. Early treatments [7e9] based on a model dependent inversion were successively improved adopting model independent approaches [10e14]. In these approaches it is possible to retrieve experimental information on J(T) that is subsequently analyzed to asses the nature of the dominant processes. In a set of
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recent papers [13,15] undercooled liquid Ni samples embedded in a molten glass matrix were subject to repeated thermal cycles in a differential scanning calorimeter (DSC) obtaining J(T) in the 1418e1443 K range. The data interpretation was supported by umbrella sampling Monte Carlo simulations to obtain the excess free energy of a nucleus of size n, DG(n,T), embedded in the surrounding undercooled liquid for the adopted model interaction. The results provide a strong evidence that the samples were actually subject to homogeneous nucleation and, in spite the simulations displayed evident departures from the spherical nucleus shape, the DG(n*,T) barrier height (where n* is the critical nucleus size) was found to follow the temperature dependence predicted by the CNT. In this paper we extend our previous work [4] and focus on the information related to the nucleation rate that can be obtained from x-ray absorption temperature scans [16] based on a previously assessed methodology [17]. The idea behind this approach is that by using a phase sensitive bulk probe, such as the x-ray absorption coefficient at a selected energy in the element absorption edge region, it is possible to measure the sample liquid (and solid) fractions as a function of temperature (and time) during repeated thermal cycles probing (and averaging) simultaneously about 107 droplets. It will be shown that the highest purity fraction of our sample displays a nucleation rate in agreement with previous data extending the undercooling range down to z370 K. In order to investigate the nature of the nucleation process the experimental rates are compared with calculated rates for various models, within the coarse grained CNT scenario, using a recently developed kinetic Monte Carlo (KMC) simulation scheme [18]. The paper is organized as follows: in Sec. 2 we discuss and clarify how the nucleation rate in the 1350e1470 K range was obtained in comparison with previous experiments. In Sec. 3 the newly implemented kinetic Monte Carlo simulation approach [18] was used to compute the Ni nucleation rate within the CNT assumptions or adopting the published DG(n,T) functions. The results are discussed in Sec. 4 and conclusions are drawn in Sec. 5. 2. X-ray absorption temperature scans
way to the Pd case [21]. In the first thermal cycles a sintering of the alumina grains occurs leading to a slight shrinking of the sample pellets but successively the samples become mechanically stable and can be subject to repeated cycles maintaining the same average Ni surface density relevant to the x-ray transmission measurement. The sample temperature was measured with a high-temperature pyrometer pointing on the outer side of the crucible and calibrated using the graphite emissivity and the Ni melting signature at Tm. The maximum absolute uncertainty in the temperature measurement in the range of interest is ±15 K. The sample phase was assessed using both x-ray diffraction measurements, collected with an area detector, and x-ray absorption spectroscopy at the Ni Kedge. The sample absorption coefficient, apart from a smooth irrelevant background associated with the windows, crucible, Al2O3 absorption and ion chamber setting, is related to a ¼ ln(I0/I1) where I0 and I1 are the electron currents in the upstream and downstream ion chambers. Preliminary XAS spectra were collected to verify the contrast between solid (prior to melting) and undercooled liquid phases at Tu x 1493 K as shown in Fig. 1 and the point of highest contrast identified in the edge region at E* ¼ 8.3382 keV.
2.2. Temperature scans By setting the monochromator energy to E* it is possible to measure the sample absorption while cycling the temperature in the desired range. Examples of these temperature scans are reported in Fig. 2. The sample switches from the low absorption solid phase level to the higher liquid phase absorption level (at this selected energy), following the phase transitions occurring in the scan. The sharp rise of absorption at Tm in the heating ramps marks sample melting and the small temperature interval in which this transition occurs DT z 10 K is due to the temperature spread across the sample area probed by the x-ray beam at Tm. An even better temperature homogeneity (estimated within DT z 2e3 K) occurs in the cooling ramps and at lower temperatures. In the cooling ramps the evident hysteresis indicates that Ni droplets crystallize only in the temperature range between 1450 K and 1370 K
2.1. Experimental details The experiment, described in a previous paper [4], was performed at the BM29 beamline [19,20] of the European Synchrotron Radiation Facility using an energy scanning monochromator equipped with a Si(311) flat crystal pair. The rectangular beam size was defined by the primary slit vertical aperture 0.3 mm and the horizontal aperture of the secondary slits prior to the first ion chamber set to a size of 2.0 mm. Samples suitable for X-ray absorption measurements in transmission mode were prepared from a mixture of sub-micron size Ni powder (99.9% purity) and alumina (Al2O3) powder in a 1:20 mass ratio suspended in ethyl alcohol, subject to ultrasonic mixing and filtered through a poly-carbonate membrane. The homogeneous mixture of the two powders was pressed in 13 mm diameter and about 100 mm thick pellets suitable for the successive thermal treatments. The overall Ni surface density was optimized for an absorption contrast at the edge of about 1.0e1.5 while the amount of alumina was determined by the compromise to limit the background absorption and simultaneously achieve a sufficient dispersion of the Ni powder. The sample slices were inserted into a folded graphite foil (with a negligible x-ray absorption) acting as a resistive heater and providing a uniform temperature field. This crucible was placed in the center of a high-vacuum P z 103 Pa cylindrical Pyrex glass vessel with suitable x-ray windows. Alumina was not found to react with Ni at the temperatures involved in the experiment in a similar
Fig. 1. Ni K-edge x-ray absorption spectra of the liquid (red) and solid (blue) phases at Tu x 1493 K. The difference spectrum da ¼ a[as is reported in the lower panel on a magnified scale. The vertical arrow marks the energy E* corresponding to the maximum contrast. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. Temporal sequence (from top to bottom and according to the arrows) of temperature scans at E ¼ 8.3382 keV. Reproducible hysteresis loops provide a clear indication of the undercooling sample capabilities. The vertical green dashed line marks Tm while the nearly horizontal red dashed lines are the linear fits to the liquid and solid absorption levels. Data points were collected at intervals of about 3.5 s and the cooling rates were 2.1 K/s in the first two loops and 1.1 K/s in the last. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
displaying an evident undercooling. A not negligible mass fraction of Ni droplets reaches a deep undercooling Tm Tu z 350 K. These scans are essential to understand the sample behavior and prepare it in the desired state for the XAFS measurements at constant temperature required for the structural investigation [4]. The molten or crystallized sample state was confirmed by the absence or presence of the f.c.c. Debye-Scherrer rings in the diffraction patterns collected with the area detector. The temperature scans not only provide a qualitative insight on the sample phase transition dynamics but also a quantitative information on the relative mass fraction of molten/crystallized sample in the cooling ramps. This mass fraction can be reliably estimated by the absorption level position between a linear extrapolation of the liquid phase absorption level in the nucleation temperature range and a linear fit of the solid phase absorption level in the heating ramp (shown with red dashed lines in Fig. 2). For this sample and scans the uncertainty in the mass fraction is in the 1% range due to the combined uncertainty in the extrapolations and data noise.
2.3. Experimental nucleation rate The relatively broad temperature region in which crystallization takes place is due to the relatively wide distribution of droplet masses. The larger droplets or those containing stronger heterogeneous nucleants are likely to crystallize earlier than the smallest droplets. With respect to single droplet experiments this strategy has clearly the drawback to require an estimate of the droplet mass distribution, however the advantage is that a single cooling ramp measurement provides the required ensemble statistical average over about 107 droplets subject to the same thermal treatment. Information on the temperature dependence of the nucleation rate can be retrieved in the entire, relatively wide, crystallization temperature range. The data analysis of the molten sample fraction is performed using the previously assessed methodology [17]. The average Ni liquid mass fraction in a cooling experiment is given by a mass weighted average of Eq. (1) over the mass probability density f(m):
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Fig. 3. Log-normal model droplet diameter distribution (solid line). The dashed lines display the fraction of droplets in the liquid state for increasing values of l ¼ 1 106, 1 107, 1 108, 1 109, 1 1010, 1 1011 mg1.
Z By indicating lðt Þ ¼
Z X¼
0
∞
t
0 0 dt the equation J T t
0
m f ðmÞexpð mlÞdm Z ∞ m f ðmÞdm
(3)
0
can be numerically solved to find the l values for each measured X at the corresponding t and T. Then the nucleation rate is simply obtained by the numerical differentiation corresponding to the analytic identity:
JðTÞ ¼
vl vt t¼tðTÞ
(4)
The powder size distribution was modeled as a lognormal with average 〈lnðd=1mmÞ〉x0:45 and standard deviation s x 1, parameters supported by a microscopic inspection of the recovered sample after the measurements. This distribution is illustrated in Fig. 3 together with the progressive effect of the exp(ml) factor for increasing l. The specific nucleation rates were extracted from the three cooling ramps in Fig. 2 and the results are reported in Fig. 4. The data collected with different cooling rates are compatible with a unique monotonic J(T) function. The dashed curve refers to the previous empirical fit to single droplet experiments processed in a DSC [13] based on the function
J ðT Þ ¼ G exp
BT 2 kB ðTm TÞ2
(5)
with G ¼ 3.98 1023 Hz/mg, B ¼ 2.14 104 eV/K, highlighted as a thick line in the region where the measurements were actually taken. It is evident that for T ( 1400 K the present results are in excellent agreement with this previous empirical functional dependence extrapolated to the deeper undercooling range
Fig. 4. Specific nucleation rates obtained in the three successive (, first, + second, third) cycles of the temperature scan. The dashed curve refers to the previous fit to the single drop experiment data [13]. The dotted curve is a fit to the present data for T > 1400 K.
achieved in the present study (dashed part). This comparison provides an insight on the crystalline nucleation process occurring in the present droplet samples. The first droplets begin to crystallize around T z 1470 K with a rate that exceeds by several orders of magnitudes the one of single droplet experiments A fit with the empirical function (5) yields in this case G ¼ 5.0 108 Hz/mg, B ¼ 4.53 105 eV/K, indicating the dominance of a heterogeneous nucleation process. Around T z 1400 K present data clearly display a cross-over towards the functional trend observed in single droplet experiments. The subsequent crystallization for T < 1400 K involves smaller size droplets accounting for a sample mass fraction of about 30%. The present experiment clearly involves a less cleaner environment than the one of single droplet experiments, but the nucleation process itself enforces (see Fig. 3) a selection of small size and higher purity droplets that progressively remain in the liquid state. The present evidence is that about 30% of the Ni mass in our sample behaves in terms of nucleation properties similarly to single droplet experiments that, following previous assignments [13], may be ascribed to the homogeneous nucleation limit. With this interpretation, present data validate previous determinations in a broader undercooling temperature range.
3. KMC simulations of the nucleation process In the framework of the coarse grained scenario of the classical nucleation theory the nuclei population Nn(t) (number of crystalline nuclei of size n at time t) evolves with instantaneous events of single atom accretion or loss with size (and temperature) dependent rates p(n,T) and m(n,T). Nuclei are meaningful from a minimum size, assumed to be nm ¼ 10. In order to calculate the nucleation rate we adopted a recently implemented kinetic Monte Carlo simulation approach [18] that allows to simulate the actual stochastic process and determine a number of average properties including the stationary nucleation rate of the simulated sample. As input information the simulation requires only an initial configuration and the process rates which, in turn, depend on both energetic and kinetic properties of the system. In order to cope with total transition rates Nn(t) p(n) and Nn(t) m(n) (differing by several orders of magnitude due to the exponential increase of Nn(t) with decreasing n), the KMC simulation is
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implemented using a hybrid approach. The kinetic processes in the Nn(t) tail (starting from ne for which Nnne(t) < 8) where treated exactly defining the asynchronous time increment t of the simulation, while for n < ne the dynamics was approximated updating the populations with Poissonian random numbers corresponding to the average rates Nn(t) p(n)t and Nn(t) m(n)t for each process involved. The process rates are constrained by the detailed balance requirement so that the backward (atom loss) rates m(nþ1,T) can be obtained from the accretion rates p(n,T) as:
mðn þ 1; TÞ ¼ pðn; TÞebðDGðn;TÞDGðnþ1;TÞÞ :
(6)
DG(n,T) is one fundamental ingredient of the theory and represents the excess free energy associated with the formation of a nucleus of size n in the undercooled liquid at temperature T. The basic CNT model for DG(n,T) includes a volume and a surface term that can be estimated on the basis of Ni macroscopic properties such as atomic weight 58.69, solid density r ¼ 8908 kg/m3, melting temperature Tm ¼ 1728 K, latent heat of fusion l ¼ 17.48 kJ/mol [22]. The surface term can be better estimated using the negentropic [23e25] model for the liquid/crystal interface energy s ¼ s0T/Tm with s0 ¼ 0.302 J/ m2. The known limitations of this approach, i.e. the attempt to model the properties of microscopic nuclei using bulk properties, can be partly overcame by modern computational techniques and in particular those based on the umbrella sampling Monte Carlo method [26,27]. These simulations also account for the non-sharp crystal-liquid interface and non-spherical nuclei shape, that provide a not-negligible correction to CNT prediction [28], performing a suitable ensemble average. Umbrella sampling Monte Carlo simulations for Ni were performed [13] using a well established embedded-atom model (EAM) potential [29]. The melting point of EAM Ni [30] is only 20 K greater than the actual Tm so it is believed to provide a sufficiently reliable model interaction. In Fig. 5 we report the comparison of different DG(n,T) models at five representative temperatures chosen to be at the same undercooling from the respective Tm as the simulated data. As previously remarked [13], the simulation data, that are reliable up to n z N/8, differ from the basic CNT model but the dependence of the (lower) barrier height DG(n*,T) follows the predicted CNT dependence. In order to exploit the Monte Carlo data in our simulation, following previous suggestions [28], we fitted the digitized data using a DillmannMeier [31] functional form (reported as dotted curves in Fig. 5). The flat region around the DG(n*,T) maximum is not perfectly reproduced especially at low undercooling, but we believe these fits are anyway adequate for the present purpose. The second ingredient of the model is an estimate for the accretion rate that, according to previous approaches [6], is assumed to be: pðnÞ ¼
6DðT Þ
d2
2
4n 3
Fig. 5. Excess free energy DG(n,T) of crystalline nuclei in the undercooled liquid as a function of size n. A set of curves are reported for each of the selected five temperatures T. The thick solid curves refer to the basic CNT model. The thin dashed lines refer to a digitized version of the previous data [13] from the largest umbrella sampling Monte Carlo simulations (N ¼ 8788) at the same undercooling with respect to the EAM Ni melting point as the CNT DG(n,T) with respect to Tm. The dotted lines are the corresponding fitted shapes according to the Dillmann-Meier form.
The KMC simulation can be efficiently performed over a wide range of temperature and sample size and for the specific comparison we investigated the range T ¼ 1280e1480 K using both the CNT model (at intervals DT ¼ 25 K) and the fitted DG(n,T) curves to the umbrella sampling MC simulations. An example of the nuclei distribution in a simulation run is reported in Fig. 6. The simulation refers to the umbrella sampling DG(n,T) model at T ¼ 1380 K (green dotted curve in Fig. 5) with a critical nucleus size n* ¼ 412 and a droplet containing about Nat ¼ 8.7 1015 atoms. The green curve reproduces the reference eq equilibrium nuclei distribution Nn xNat exp½DGðn; TÞ=ðkB TÞ. The metastable state is characterized by the stationary solution of the set of kinetic differential equations for the process Nnst , associated
(7) 2
where D(T) is the self-diffusion coefficient, 4 n3 is a reasonable estimate for the number of attachment sites in a nucleus of size n and d represents average travel distance for an atom in the surrounding liquid required to reach the attachment site. The Ni self-diffusion coefficient was measured by recent inelastic neutron scattering experiments [32] and is accurately described by an Arrhenius empirical function D(T) ¼ D0exp(TD/T) with D0 ¼ 77 109 m2/s and TD ¼ 5454 K. The lowest temperature of the actual measurements was 1514 K and we assume that the Arrhenius behavior can be safely extrapolated in the temperature range of interest (150 K below the actual measurement). The attachment rates are calculated using D(T) in Eq. (7) with a typical travel for attachment d ¼ 0.25 nm.
Fig. 6. Snapshot of the KMC simulation showing a typical instantaneous Nn(t) nuclei eq distribution (red histogram) compared with the equilibrium Nn (green curve) and stationary Nnst distributions (blue curve). The video (visible on-line) follows the KMC simulation of the random process and the related distribution evolution for about 1.3 ns.
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with a constant nuclei flux J, reported as a blue curve. Nn(t) are a set of integer natural numbers representing random realizations of Poisson random variables with average Nnst , and corresponding fluctuations. The nuclei distribution in the snapshot includes an overcritical nucleus with n z 1000 that initiated an irreversible growth and a few embryo nuclei in the n z n* region. In the video associated with Fig. 6 (visible on-line) it is possible to follow the KMC simulation for about 1.3 ns during which two nucleation events occur. Individual frames are sampled every 5000 elementary steps t of the actual KMC simulation. Nat, corresponding to a eq z50 mm liquid Ni droplet, was selected in order that Nn ¼ 0:01, for graphical purposes. Simulations with smaller Nat can be performed to reproduce the behavior of the actual sub-micrometric sample droplets in our experiment. An increased computational efficiency for the nucleation rate determination is however obtained with larger Nat values corresponding to Nneq in the range 1e10. Supplementary video related to this article can be found at http://dx.doi.org/10.1016/j.actamat.2016.10.076. The nucleation rates are computed with a statistics of at least 260 nucleation events in the equilibrated runs while over 1200 nucleations events were recorded at the lowest temperatures (where smaller critical sizes n* are involved). The corresponding statistical uncertainty is in the 3e6% range. The results for the nucleation rate are compared with the experimental data in Fig. 7. The green curve reports the fit to the previous experimental data [13] compatible with the low temperature tail of present data. The homogeneous nucleation rates calculated for the CNT DG model clearly display a greater slope coherent with the previously emphasized larger temperature dependence of DG(n*,T). On the contrary, the umbrella sampling Monte Carlo DG model accurately reproduces the experimental slope. The absolute nucleation rates scale with a kinetic pre-factor determined by the elementary rates of Eq. (7). By adopting the same d parameter used for the CNT model the nucleation rates would result largely overestimated falling on the dashed curve (open symbols) of Fig. 7. The entire curve can be however brought in close agreement with all available experimental data with a reduced kinetic pre-factor.
4. Discussion The comparison between the calculated homogeneous nucleation rates and the experimental data, reported in Fig. 7, is very interesting. In spite of the order of magnitude differences among the various curves, the possibility to report data on the same scale is anyway a promising result indicating that models and data are at least in reasonable agreement. The fact that the experimental rates are lower than the calculated homogeneous nucleation rates provides a support to the interpretation that the best Ni samples have actually reached the homogeneous nucleation limit, while evidently the models with the present choice of parameters tend to overestimate the rates. The present CNT estimate of the nucleation rate matches the experimental data around T z 1430 K but displays a greater temperature slope associated with the larger barrier heights. By adopting the DG model from umbrella sampling MC simulation the reduced DG(n*,T) barrier heights bring the slope in close agreement with the experimental trends, but at the same time increase the absolute J values. A close agreement between experimental and calculated values is obtained combining these DG values with a reduced kinetic rate than the one obtained according to Eq. (7) with d ¼ 0.25 nm. We believe this occurrence is not fortuitous. The argument behind Eq. (7) is to estimate the reciprocal of the average time taken by an atom to travel a distance of the order of the interatomic distance in a diffusion process. It can be argued that in this time lapse the atom will be somewhere on the surface of a sphere of radius d, but actually the attachment will occur only if the atom falls within a narrow region around the attachment site with a typical cross section of the order of p(d/10)2 being d/10 a typical vibrational standard deviation for a high temperature solid. This correction provides a scaling of the order of 1/400 to the p(n) estimate. An additional reduction factor may be obtained from the requirement that the kinetic energy of the atom lays below a certain threshold to avoid a successive escape. Present evidence and all previous arguments stimulate further investigations on improved models to estimate the basic process rates for the coarse grained description of the nucleation phenomenon, possibly based on computer simulation approaches. If the green curve in Fig. 7 is assumed to represent homogeneous nucleation, from the ratio between the fitting B coefficients and assuming that the nucleus has the shape of a spherical sector with contact angle q, it is possible to estimate the barrier height reduction for heterogeneous nucleation as ð2 3 cos q þ cos3 qÞ=4x0:21 corresponding to qx66 . The crossover to the homogeneous nucleation trend takes place around 1400 K and involves a sample mass fraction of about 30%. 5. Conclusions
Fig. 7. Temperature dependence of the stationary homogeneous nucleation rate J of undercooled liquid Ni calculated using the KMC approach with different DG models. Present experimental data are reported with black symbols as in Fig. 4 while the previous fitting curve to single drop DSC experiments is reported as a green line (thick in the measurement region dashed elsewhere). The KMC homogeneous nucleation rates are reported as points (blue curve) for the CNT model and as , or - (with a reduced kinetic prefactor) points (red curves) for the DG model function fitting the umbrella sampling MC data. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
We measured the nucleation rate of undercooled liquid Ni droplets in the 1365e1470 K temperature range using x-ray absorption temperature scans on a sub-micrometric Ni powder sample dispersed in Al2O3. The data display a cross-over between two nucleation regimes, the one occurring at higher temperature being clearly heterogeneous. Present data for T(1400 K are in agreement with the extrapolation of J(T) determined in a previous single droplet experiment [13], supporting the interpretation that the green curve in Fig. 7 actually represents an intrinsic property of undercooled liquid Ni, being compatible with experimental data obtained in two independent experiments with different techniques. In order to investigate the nature of the ultimate nucleation process occurring in all investigated Ni samples we computed the homogeneous nucleation rate using the CNT prescriptions or
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