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Jun 29, 2012 - Lindemann melting criterion. In the premelting stage (i.e., below a glass transition temperature, Tg), liquid-like atoms occur first in...
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Melting of Simple Monatomic Amorphous Nanoparticles V. V. Hoang* Department of Physics, Institute of Technology, National University of HochiMinh City, 268 Ly Thuong Kiet Street, District 10, HochiMinh City, Vietnam ABSTRACT: Melting of the simple monatomic amorphous spherical nanoparticles has been studied via molecular dynamics (MD) simulation. Initial amorphous nanoparticles have been heated toward a normal liquid state to study a melting process with Lennard-Jones−Gauss interatomic potential [Engel, M.; Trebin, H.-R. Phys. Rev. Lett. 2007, 98, 225505]. Temperature dependence of various thermodynamic quantities of the system is found and discussed. Atomic mechanism of melting is monitored via analysis of the appearance/growth of the liquid-like atoms upon heating. Liquid-like atoms are determined by using the Lindemann melting criterion. In the premelting stage (i.e., below a glass transition temperature, Tg), liquid-like atoms occur first in the surface shell to form a quasi-liquid surface layer. Further heating leads to the formation of a purely liquid skin at the surface of nanoparticles together with a simultaneous occurrence/growth of liquid-like atoms in the remaining glassy matrix. Melting process proceeds further via propagation/growth of liquid-like configuration into the solid core. Liquid-like configuration includes a purely liquid skin, and liquid-like atoms occurred in the liquid−solid interfacial shell of nanoparticles. Total melting occurs at temperature much higher than Tg. Heat capacity of the system exhibits a single peak at around a total melting point. We find a strong thermal hysteresis between amorphous nanoparticles obtained by heating/ cooling.

I. INTRODUCTION Melting of amorphous and crystalline nanoparticles has been under intensive investigation by both experiments and computer simulations for a long time due to their technological importance. However, much attention has been paid to melting of the latter rather than of the former, although the former also has a great potential for applications in various technological areas.1 Indeed, we find a limited number of works related to the melting of amorphous nanoclusters or nanoparticles.2−8 Ion mobility and calorimetry measurements have been used to probe the melting of amorphous Ga clusters containing 29−55 atoms.6 It was found that while most clusters appear to undergo a first-order transition between solid-like and liquid-like phases, a few present a signature of melting without a significant latent heat.6 Melting of amorphous clusters has been also examined by the MD simulations, as they are predicted to melt without a peak in their heat capacity and with an inflection in their volume, the behavior of a second-order phase transition.2,5 In more details, melting-like transition in amorphous Na92 and Na142 is studied via density functional constant energy MD simulations.4 It was found that melting in Na92 proceeds smoothly over a broad temperature interval without any abrupt in the thermal or structural indicators. In contrast, a relatively larger cluster of Na142 exhibits two-step melting; that is, its specific heat displays two peaks: the first one occurs at around 130 K, which may be related to the growth of a highly intrashell atomic mobility, and the second one should be related to the homogeneous melting occurred at around 270 K (see ref 4). These works are related to the melting of amorphous clusters © XXXX American Chemical Society

containing from tens to hundreds atoms. Experimentally, the melting point of amorphous nanoparticles with a size up to hundreds nanometers has been observed via differential scanning calorimetry technique (see, for example, refs 3,7,8). However, limited information of melting of amorphous nanoparticles can be found via such a technique. There is no detailed information at the atomistic level of melting of the mesoscale amorphous nanoparticles. On the other hand, atomic mechanism of melting of amorphous nanoparticles is still unclear. Therefore, it is of great interest to carry out MD simulations in this direction. In the present work, melting of the simple monatomic amorphous spherical nanoparticles containing 4457 atoms is studied by MD simulations. The main issues of our aim here are given as follows: investigations of atomic mechanism of melting, surface melting, and checking the existence of so-called quasi-liquid surface layer at temperature just below Tg. The size dependence of melting is out of the scope of this Article.

II. CALCULATIONS Initial amorphous nanoparticles at temperature T0 = 0.01, previously obtained by cooling from the melt in ref 9, have been relaxed for 2 × 105 MD time steps at T0 = 0.01 before heating toward a normal liquid state at the same heating/ cooling rate used in ref 9 (γ = 4.836 × 1010 K/s if taking Ar for Received: May 1, 2012 Revised: June 19, 2012

A

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testing). The temperature of the system is linearly increased with time as T = T0 + γt (t is a time of heating) via simple rescaling of atomic velocity after every MD time step until the temperature reached T = 2.0. Identical atoms interact via LJG potential as given below:10 ⎡ (r − 1.47σ )2 ⎤ ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ V (r ) = ε⎢⎜ ⎟ − 2⎜ ⎟ ⎥ − 1.5ε exp⎢ − ⎥ ⎣⎝ r ⎠ ⎝r⎠ ⎦ ⎣ 0.04σ 2 ⎦ (1)

The LJG potential presented in eq 1 is a sum of the LennardJones potential and a Gaussian contribution. It is a double well interaction potential with the second well located at r0 = 1.47σ (see the figure for potential in ref 11). Note that the general form of pair potentials in metals consists of a strongly repulsive core plus a decaying oscillatory Friedel term. Therefore, LJG potential can be understood as such an oscillatory potential, cut off after the second minimum. Competition between two wells leads to the formation of 3D glass with very high local icosahedral order,9,12 and it is an origin of very long-lived simple monatomic glassy model as compared to those with LJ or Dzugutov’s potentials.13−16 The following LJ-reduced units are used in the present work: the length in units of σ, temperature T in units of ε/kB, and time in units of τ0 = σ(m/ ε)1/2. Here, kB is the Boltzmann constant, σ is an atomic diameter, and m is an atomic mass. The Verlet algorithm is employed, and MD time step is dt = 0.001τ0 or 2.44 fs if taking Ar for testing. The cutoff is applied to the LJG potential at r = 2.5σ like that used in ref 9. We employ NVT ensemble simulation for the spherical simulation cell of the radius R = 12σ with an elastic reflection boundary, while the real radius of amorphous nanoparticles is just around 9σ. This means that nanoparticles have a free surface during the simulation procedure (i.e., the system is under constant zero pressure). To improve the statistics, we average the results over two independent runs.

Figure 1. Temperature dependence of (a) potential energy per atom in nanoparticle, thin film,15 and bulk models;16 (b) the Lindemann ratio for nanoparticle, thin film,15 and bulk models;16 (c) meansquared displacements of atoms, the bold line is for T = 0.5; and (d) heat capacity obtained upon heating of nanoparticles.

topic below. On the other hand, potential energy per atom in nanoparticles is higher than that in thin film,15 while the latter is higher than that of the bulk.16 It points out clearly free surface effects (Figure 1a); that is, three-dimentional free surface effects are stronger than those of two-dimensional free surface ones. A similar tendency of free surface effects is found for temperature dependence of the Lindemann ratio (Figure 1b). Note that the Lindemann ratio for the ith atom is given below:17 δi = Δri 2

1/2

/ R̅

(2)

⟨Δr2i ⟩

Here, is the mean-squared displacement (MSD) of the ith atom, and R̅ = 0.91 is an interatomic distance that is taken equal to the position of the first peak in radial distribution function (RDF). For the supercooled and glassy states, R̅ does not change much with temperature, and thus we fix this value for the calculations. MSD in eq 2 is defined after a characteristic time τC = 5τ0 (i.e., after 5000 MD steps of relaxation or 12.2 ps at a given temperature). Figure 1c shows that τC = 5τ0 is located at the end of a plateau regime of MSD for T ≤ Tg; that is, the time is large enough for atoms to overcome a plateau regime to diffuse if atoms are liquid-like such as that employed previously for nanoparticles9 and thin film,15,18 respectively. Note that data for the bulk have been obtained via heating the cubic models containing 2744 atoms under periodic boundary conditions, which were previously obtained in ref 16. The mean Lindemann ratio δL of the system is defined by averaging of δi over all atoms, δL = ∑iδi/N. One can see in Figure 1b that δL has the same temperature dependence as potential energy, indicating a close correlation between the two quantities in the glass-to-liquid transition. Critical value for the Lindemann ratio at T = Tg is δC = 0.30, and it is higher than that found for nanoparticles at the same size obtained by cooling from the melt (i.e., δC = 0.17, see ref 9). Note that initial amorphous nanoparticles at T = 0.01 have been well-relaxed before heating, and it leads to a strong hysteresis between systems obtained by heating/cooling. A purely Lindemann criterion established that melting occurs when a root of MSD is at least 10% (usually around 15%) of the atomic spacing.17,19 Moreover, it is evident that the criterion is also applicable for glasses,20−22 and, indeed,

III. RESULTS AND DISCUSSION A. Thermodynamics and Structural Evolution. The temperature dependence of various thermodynamic quantities of nanoparticles upon heating can be seen in Figure 1. One can see that the temperature dependence of potential energy per atom in nanoparticles exhibits a common type like that found for thin film15 and bulk16 (Figure 1a). The linear part of a low temperature region is related to the glassy state, and the starting point of deviation from the linearity can be considered as a glass transition temperature. In the low temperature region, potential energy per atom just slightly increases with temperature because the system remains in the solid state and vibrational motion of atoms around their quasi-equilibrium positions dominates in the system. At T ≥ Tg, contribution of anharmonic motion of atoms is strong enough, leading to massive collapse of a glassy matrix and formation of a purely liquid skin (see our discussion below) such that potential energy starts to deviate from the linearity. The obtained glass transition temperature Tg = 0.52 is close to that obtained by cooling the system from the melt (versus Tg = 0.50 previously found in ref 9), which is also close to Tg found for thin film,15 but it is much smaller than Tg of the bulk.16 A slight difference between values for Tg in the models obtained by heating/ cooling indicates the existence of a hysteresis between amorphous nanoparticles obtained by heating/cooling. However, it also lies within statistical error, and we will return to this B

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follows. First, evolution of radial distribution function (RDF) in nanoparticles upon heating is shown in Figure 2. One can see

the Lindemann criterion of melting has been successfully employed for studying of melting of glassy thin film.15 To monitor the melting process, one should analyze spatiotemporal arrangements of liquid-like atoms that occurred in the models during heating. Liquid-like atoms can be detected via the Lindemann melting criterion; that is, atoms with δi < δC are classified as solid-like and atoms with δi ≥ δC are classified as liquid-like. Note that the critical value for the Lindemann ratio at a melting point of crystalline Ni mesoscale cluster23 is found as large as δC = 0.24 and is close to ours. It is evident that high mobility of atoms located at the surface of nanoparticles obtained by heating leads to a high critical value of the averaged Lindemann ratio as compared to that of the bulk. Note that the Lindeman criterion of melting has been successfully employed for studying melting of crystalline nanoparticles of various materials including Ni and Al (see refs 23, 24, and references therein). However, temperature dependence of the Lindemann ratio has a sudden change at a melting point indicating a firstorder behavior of the transition24 versus a continuous one observed for amorphous nanoparticles in the present work (see Figure 1b). One can see that free surfaces greatly enhance atomic mobility in the systems; that is, the Lindemann ratio in nanoparticles is always larger than that in thin film, while δL in the latter is larger than that in the bulk (Figure 1b) due to the existence of a mobile surface layer. However, it is found that the discrepancy between atomic mobility in the surface shell and that in the interior (the latter exhibits bulk-like dynamics) decreases with decreasing temperature.15 Therefore, δL of both nanoparticles and thin film becomes closer to that of the bulk in the low temperature region (Figure 1b). We also present MSD of atoms in nanoparticles obtained at different temperatures (Figure 1c). Commonly, MSD also exhibits three typical regimes: the ballistic one in the beginning, followed by a plateau, and diffusive regime at longer time. However, MSD of atoms in our nanoparticles has an additional saturation regime of diffusion length of the simulation cell, and it can be seen more clearly at high temperature (Figure 1c). At high temperatures, nanoparticles can expand its size to the calculation cell of the radius RC = 12σ (initial radius of nanoparticles is R0 = 9σ). A similar tendency was found for MSD of atoms in liquid SiO2 nanoparticles25,26 or in simple monatomic nanoparticles obtained by cooling from the melt.9 Moreover, one can see that the MSD curve for T = 0.5 (the bold line) is a bound between two areas of atomic dynamics, that is, dynamics of low and high temperatures. Therefore, glass transition temperature Tg = 0.52 obtained above via temperature dependence of potential energy is correct. An additional important quantity is also calculated, the heat capacity at constant zero pressure, which is approximately calculated via the simple relation: Cp = ΔE/ΔT. Here, ΔE is a discrepancy in total energy per atom upon heating from T1 to T2 with ΔT = T2 − T1 = 0.01 (all quantities are given in the LJ reduced units). As shown in Figure 1d, the heat capacity curve exhibits a single peak at around T = 1.5, which is related to the total melting of nanoparticles. Nonsmooth curve for the heat capacity is related to the approximation of method of determination and to the small number of averaging (i.e., we average results over two independent runs). Single peak curve of the heat capacity will be explained in detail below because we need more information of structure and thermodynamics for the topic. To get more detailed information on melting of amorphous nanoparticles, we calculate various additional quantities as

Figure 2. Temperature dependence of radial distribution function; the dotted line is for T = 0.5, while the bold line is for T = 1.5.

that at low temperature RDF exhibits a glassy behavior; that is, it has a splitting of the second peak like that found for simple monatomic and metallic glasses (see ref 16 and references therein). Splitting of the second peak in RDF is related to the existence of a strong local icosahedral order in the superooled liquids and glasses.9,15,16,18,27 Furthermore, splitting of the second peak in RDF becomes weaker with increasing temperature, and at around T = 1.5 splitting of second peak in RDF almost disappears. It indicates that the system completely transforms into a normal liquid, and this point can be considered as a total melting point (Ttm = 1.5) of amorphous nanoparticles, and it confirms that the peak in the heat capacity at around Ttm = 1.5 is indeed related to the total melting of nanoparticles (see Figure 1d). Similar evolution of RDF upon heating has been found for various monatomic glasses including glassy thin film with LJG interatomic potential.15,28 On the other hand, one can see that upon heating from glassy state our nanoparticles transform into normal liquid via an intermediate supercooled liquid state without crystallization of glass (see evolution of RDF and MSD in Figure 2 and Figure 1c). A similar situation is analyzed in detail via evolution of the self-intermediate scattering function for the thin film models.15 Note that we can have various scenarios of glass to liquid transition upon heating of glasses depending on the heating rate employed, that is, glass→liquid or glass→crystal→liquid transitions (see ref 29 and references therein). If transition of our amorphous nanoparticles into liquid is also accompanied by crystallization, additional peaks should occur in RDF. Note that crystallization of the monatomic LJ glass occurs if the heating rate is slow enough; that is, additional peaks of the crystalline structure occur in RDF, and further heating leads to the melting of obtained crystal.29 As noted above, the glassy state of the system with LJG interatomic potential is very stable against crystallization due to a high fraction of local icosahedral order because this order is incompatible with the global crystallographic symmetry.16 Therefore, the same glass→liquid transition via an intermediate supercooled liquid state should occur in our amorphous nanoparticles if a lower heating rate is employed. It is clear that one can easily monitor various scenarios of melting of simple amorphous nanoparticles via varying heating rate if the LJ interatomic potential is used instead of LJG because the glass with LJ interatomic potential is rather unstable against crystallization as described above.29 C

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specific atomic mechanism of melting of amorphous nanoparticles as described below. One more thing we would like to discuss here is a layering of radial density profile; that is, it contains orderly high and low density layers (Figure 3). It seems that layering of nanoparticles obtained by cooling from the melt9 is stronger than that of nanoparticles obtained by heating from glassy state in the present work. Indeed, layering in the density profile in our nanoparticles becomes weaker with radial distance (Figure 3), and it is unlike that found for nanoparticles obtained by cooling from the melt.9 Layering of liquids and glasses with free surfaces was found for various materials30,31 including amorphous nanoparticles.1,9 The origin of layering is still unclear. However, layering may have an impact on melting of the disordered clusters.4 We also calculate other interesting quantities. As shown in Figure 4a, atomic mobility becomes closer to that of the bulk

Furthermore, radial density profile and radial atomic displacement distribution (add) in nanoparticles obtained at different temperatures are presented in Figure 3. Density profile

Figure 3. Radial density profile and atomic displacement distribution in models obtained at different temperatures. Thickness of the surface layer (or thickness of the mobile surface layer), that is, d0, can be determined as presented in (d) (the dotted line serves as a guide for eyes).

at a given temperature is calculated by partitioning nanoparticles into spherical shells of the thickness 0.2σ, and then we divide the number of atoms in each spherical shell by the volume of a given shell. In contrast, add is found via dividing a total displacement of all atoms in the shell by the number of atoms in a given shell; that is, add corresponds to the displacement of atoms in the shell after a characteristic time τC = 5τ0. One can see that radial density profile exhibits surface and interior behaviors; that is, in the interior, density almost fluctuates around a high constant value. However, in the surface shell, it strongly decreases to zero (Figure 3). The point at which density starts to strongly decrease can be considered as the bound between surface shell and interior. Therefore, the thickness of the surface shell can be determined as presented in Figure 3d (denoted as d0). On the other hand, in the interior, density also has a tendency to slightly decrease with the radial distance, especially in the region close to the surface shell, and it is unlike that observed in nanoparticles obtained by cooling from the melt.9 It may lead to the significant changes in structure and thermodynamics of the system obtained by heating as compared to that obtained by cooling (see our discussions below). Indeed, add has a tendency to increase with the radial distance from the center of nanoparticles, and increment is more pronounced in the surface shell (Figure 3). It is unlike that observed in nanoparticles obtained by cooling from the melt;9 that is, add remains constant at a small value in the interior, and in the surface shell it increases with the distance from the interior leading to the formation of a mobile surface layer.9 Although there is a mobile surface layer of nanoparticles as presented in Figure 3, there is almost no clear separation between atomic mobility in the surface shell and that in the interior; that is, the change in atomic mobility from the interior to the surface shell is rather smooth. It may lead to a

Figure 4. (a) Atomic displacement distributions in nanoparticle obtained at T = 1.0 as compared to that of the bulk; (b) temperature dependence of the thickness of surface layer in nanoparticles obtained upon heating; (c) temperature dependence of the averaged thickness of surface layer in nanoparticles and thin film obtained upon heating; data for thin film are taken from ref 15; and (d) atomic displacement distributions in nanoparticles obtained by heating/cooling at T = 0.5.

with decreasing radial distance. However, atomic mobility in the interior of our nanoparticles does not exhibit bulk-like dynamics as thought in the past, as it has a tendency to increase with the radial distance from the bulk-like value in the center of nanoparticles. Note that atomic mobility in the interior of nanoparticles obtained by cooling from the melt exhibits clearly bulk-like dynamics9 like that found for atomic mobility in thin film with LJG potential.15 Surface layer, which can be approximately considered as a mobile surface layer, may play an important role in the melting of amorphous nanoparticles (Figure 3). Therefore, temperature dependence of its thickness is of great interest. For thin film with the same LJG interatomic potential, the thickness of a mobile surface layer grows with temperature,15 and the same situation is found for nanoparticles in the present work (Figure 4b and c). The thickness of surface layer of amorphous nanoparticles should be different from that for a flat surface because of geometrical effects (and/or capillary effects in the high temperature region). Indeed, we find that the thickness of a mobile surface layer of nanoparticles is always smaller than D

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that of thin film (see Figure 4c). In contrast, it was suggested that liquid-skin thickness of crystalline nanoparticles in the premelting stage should be different from that for a flat surface; that is, liquid-skin thickness on a nanocrystal nanoparticle might increase over the value for a flat surface to reduce the solid−liquid interfacial area or the total interfacial energy in general.32 Note that there is no sharp front of the liquid skin on the surfaces of amorphous nanoparticles or that of the amorphous thin film models.15 Therefore, it is difficult to study temperature dependence of the thickness of a purely liquid skin to compare. On the other hand, we also find that add values in nanoparticles obtained by heating/cooling are different from each other (Figure 4d). One can see that add in nanoparticles obtained by cooling from the melt exhibits clearly interior and surface behaviors; that is, in the interior add is rather constant, and then it increases with the radial distance in the surface shell. In contrast, separation between surface and interior dynamics in nanoparticles obtained by heating is not so clear; that is, add increases with radial distance almost monotonously from the center, although the increment in the surface is more intensive as compared to that in the interior as discussed above. The temperature dependence of add can be seen in Figure 5, and some points can be drawn: (i) Indeed, glass transition

Figure 6. Coordination number distributions in nanoparticles obtained by heating/cooling as compared to that of the bulk obtained at the same temperature T = 0.5.

see Figure 6), and it leads to a higher atomic mobility in the former as found above (see Figure 4d). (ii) It is clear that the surface of nanoparticles has a more porous structure than that of the core and that of the bulk, and it was found that liquid-like atoms often occur in the nonclosed packed atomic arrangement regions in the glassy matrix.15,28,29 Therefore, melting of amorphous nanoparticles should initiate in the surface shell, and then propagates into the core. (iii) The structure of the core of amorphous nanoparticles is close to that of the bulk as was found and discussed.9 Differences in structure and dynamics of nanoparticles obtained by heating/cooling point out a strong hysteresis between them. Note that thermal hysteresis is commonly found for glasses (see ref 33 and references therein). B. Atomic Mechanism of Melting. To study the atomic mechanism of melting of amorphous nanoparticles, we present spatiotemporal arrangements of liquid-like atoms occurred in nanoparticles during the heating process. The temperature dependence of the fraction of liquid-like atoms (i.e., NL/N) and the size of the largest liquid-like cluster (i.e., Smax/N) occurred during the heating process (N is the total number of atoms in the model) can be found in Figure 7. One can see that liquidlike atoms occur first at temperature far below the glass transition, and their number grows fast with temperature. Temperature at which a significant amount of liquid-like atoms occurs in the glassy matrix can be considered as a limit of

Figure 5. Temperature dependence of atomic displacement distributions in nanoparticles. From bottom to top for temperature ranging from T = 0.1 to T = 1.8 with the increment ΔT = 0.1, the bold line is for T = 0.5 and the dotted line is for T = 1.5.

temperature Tg = 0.52 determined above is correct because the curves below this temperature are related to the low mobility of a glassy state. (ii) One can see that Ttm = 1.5 is the lowest bound of the high mobility region of a liquid state, and indeed it is a total melting point of the system. (iii) Mobility of atoms in the surface shell is always higher than that in the core; upon heating, atomic mobility in the surface shell increases fast and high mobility surface (liquid-like region) propagates into the solid core. A similar tendency of root of MSD of atoms was found for crystalline Ni nanoclusters.23 To highlight the situation, we show in Figure 6 coordination number distributions in nanoparticles obtained by heating/ cooling as compared to that of the bulk at the same temperature T = 0.5. Some points can be drawn as follows: (i) It is clear that the structure of nanoparticles obtained by heating is significantly different from that obtained by cooling from the melt; that is, surface and core of the former exhibit more porous structure than those of the latter (fraction of under coordinated sites in the former is larger than that in the latter, and the situation is more pronounced in the surface shell,

Figure 7. Temperature dependence of fraction of liquid-like atoms (NL/N) and size of the largest liquid-like cluster (Smax/N) that occurred during the heating process (N is the total number of atoms in the model). The inset shows the temperature dependence of the fraction of atoms remaining solid-like (NS/N) during the heating process as compared to that obtained by cooling from the melt. E

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On the other hand, we find that liquid-like atoms have a tendency to form clusters; that is, almost all liquid-like atoms that occurred during the heating process aggregate in clusters (see the curve for Smax/N). However, in the high temperature region, liquid-like atoms do not form a single cluster, which percolates throughout the system unlike that found for melting of thin film15 or bulk LJ system.28 For nanoparticles heated to high temperature (after Ttm = 1.5), due to high mobility a significant amount of atoms in the surface layer has a tendency to disperse away from the main part of nanoparticles; that is, they do not link to a single cluster in the system (see Figure 8). More details of the mechanism of melting of amorphous nanoparticles can be seen via 3D visualization of the occurrence of liquid-like atoms during the heating process (Figure 8). It is clear that liquid-like atoms occur first in the surface shell (Figure 8a), and liquid-like configuration grows/enhances inward to the core. However, there is no sharp front of a liquid-like configuration (Figure 8a−c). Liquid-like atoms often occur locally in the nonclose-packed atomic arrangement regions,28,29 that is, mainly in the surface shell, and further heating stimulates the occurrence of liquid-like atoms in the next inner shells of nanoparticles together with an enhancement of liquid-like population in the outer part of nanoparticles. By this way, liquid-like configuration propagates into the core. As described above via radial density profile, atomic density smoothly decreases with radial distance with an exception of a more intensive decrement in the surface shell. Therefore, liquid-like atoms should occur/grow in the opposite direction with opposite intensity as compared to that of the density (Figure 8). However, it is interesting to discuss the radial distributions of the solid-like and liquid-like atoms during the heating process, which offer a more detailed picture of melting in the system (Figure 9). One can see that liquid-like

thermal stability of glassy matrix because collapse of glassy matrix is started and it is denoted as Tlt. At T = Tg, the fraction of liquid-like atoms in the system reaches 0.37, which is much larger than that found in melting of thin film15 but close to that found for melting of the bulk LJ glass.28 Heating from around T = 0.5 to T = 0.7 just leads to the fluctuation of NL/N around the value of 0.37, indicating a balanced competition between a new formed liquid phase and the remaining solid core. Further heating stimulates growth of a new formed liquid phase, leading to monotonous increasing of liquid-like atoms in the system, and total melting is reached at around Ttm = 1.5. This means that there are three characteristic temperatures for melting of amorphous nanoparticles, that is, Tlt < Tg < Ttm, like those found for melting of glassy thin film.15 Recent MD simulations reveal that, although the number of solid-like atoms in glasses decreases upon heating, their fraction still dominates in the system at a glass transition temperature, that is, from around 70% in the bulk LJ glass28 to around 80% in the LJG glass with free surfaces15 versus 63% in our nanoparticles. However, while solid-like atoms in the bulk LJ glass distribute throughout the system, they have a tendency to concentrate in the interior of the thin film15 and nanoparticles (for nanoparticles, see Figure 8). Moreover, unlike thin film, the central area of nanoparticles

Figure 8. 3D visualization of occurrence of the liquid-like atoms in nanoparticle during the heating process.

at a glass transition temperature remains almost purely solid (Figure 8b), indicating a stronger heterogeneous behavior of melting of a three-dimensional free surface glass. We also show the temperature dependence of a fraction of atoms remaining solid-like in the system upon heating from glassy state in the inset of Figure 7, and indeed solid-like atoms almost disappear at around Ttm = 1.5, indicating that this point is a total melting of nanoparticles. It points out again that the peak in the heat capacity at around Ttm = 1.5 is related to the total melting of nanoparticles (see Figure 1d). The temperature dependence of the fraction of atoms remaining solid-like in nanoparticles upon heating is also compared to that obtained by cooling from the melt (i.e., NS/N, see the inset of Figure 7). One can see that two heating/cooling curves for NS/N show a hysteresis like that found and discussed for various quantities.

Figure 9. Radial distributions of solid-like and liquid-like atoms in nanoparticles obtained at different temperatures upon heating.

atoms occur first in the surface shell of nanoparticles to form a quasi-liquid surface shell containing both solid-like and liquidlike atoms (Figure 9a). This stage continues up to a glass transition temperature (at T < Tg), and it can be considered as a premelting of amorphous nanoparticles. Further heating (at T ≥ Tg) leads to the formation of a purely liquid skin at the F

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nanoparticles (at around T ≥ Tg) together with the quasi-liquid interfacial region, that is, the region separating a purely liquid skin and the remaining solid core, in which the concentration of liquid-like atoms smoothly decreases from the surface liquid skin into the core of the nanoparticles (see Figure 9). Melting proceeds further by growth of surface liquid skin and growth/ enhancement of a quasi-liquid interfacial region toward the remaining solid core of nanoparticles. Note that, although large crystalline Cs nanoclusters also exhibit surface and core melting, the heat capacity has only a single peak located at the homogeneous melting (i.e., melting of the core) based on the recent MD simulations of melting of Cs nanoclusters.38 Therefore, the nature of multipeak phenomenon observed for the heat capacity of melting of clusters of some materials is still unclear. On the other hand, although melting of simple clusters with LJ potential has been studied intensively (see refs 34 , 35 , 39, and references therein), much attention has been paid to the relatively small clusters containing up to hundreds of atoms. In particular, LJ clusters containing from 13 to 147 atoms were found to undergo a transition like melting for the bulk materials.39 No work related to the melting of mesoscale amorphous nanoparticles with LJ potential, that is, the systems are close to our LJG ones, has been found to compare to our results. Note that melting of small crystalline particles has been often described by the following three models: (i) homogeneous melting model without the occurrence of a liquid skin (see ref 40 and references therein); (ii) liquid-skin melting model that supports the occurrence of a liquid skin on surface as premelting stage and melting proceeds via propagation of the liquid skin into the core;41−43 and (iii) liquid nucleation and growth model with unstable liquid skin.44−46 One can see that melting of the simple monatomic amorphous nanoparticles does not follow melting models just described, although it shares some trends proposed by the liquid-skin melting model.

surface together with growth/enhancement of liquid-like atoms in the inner part toward the center of nanoparticles (Figure 9b). Because of the appearance of a purely liquid skin, there is a balanced competition between a new formed liquid phase and the remaining solid core over a narrow temperature range. It leads to the formation of a plateau in temperature dependence of fraction of the liquid-like atoms for temperature ranging from around T = 0.5 to T = 0.7 as discussed above (see Figure 9b and c, Figure 7). Further heating above T = 0.7 stimulates a strong growth of liquid-like atoms leading to the formation an entirely liquid phase at much higher temperature (Figure 9d). We stop here for discussion about the heat capacity presented in Figure 1d. It is clear that we find a single peak capacity curve, and the peak located at around Ttm = 1.5 is related to the total melting in nanoparticles; that is, it involves diffusive motion of all atoms across the whole nanoparticle volume like that found for a disordered Na142 cluster.4 There are no data for the melting of amorphous nanoparticles at a similar mesoscale like that studied in the present work to discuss. However, it was found that the heat capacity of a disordered Na142 cluster (in which atoms are believed to be distributed in two distinct shells: surface and inner shells) exhibits two peaks corresponding to two steps of melting: the small peak located at around 130 K characterized by the development of high intrashell atomic mobility, and the second one located at around 270 K related to the total melting.4 In contrast, for a smaller cluster, that is, Na92, melting proceeds smoothly over a large temperature range without any abrupt change in the heat capacity.4 While ordered clusters melt with a substantial peak in their heat capacity due to the latent heat, amorphous clusters are believed to be melted without a peak in their heat capacity and with an inflection in their volume; that is, it behaves as a second-order phase transition.6 It is believed that when cluster size is large enough, clusters should consist of two distinct parts (surface and core), and melting of clusters should undergo in two steps related to the surface and core melting.34,35 It corresponds to two peaks in the heat capacity curve: the low temperature peak (i.e., the smaller one) is thought to be related to the surface melting, and the higher temperature peak (i.e., the broader one) is thought to be related to the melting of a core of clusters.34,35 Moreover, there is a critical radius of nanoparticles below which a liquid surface layer (i.e., surface melting) is generally not observed.36 For Au cluster, the critical size corresponds to the cluster containing around 350 atoms.37 Nanoparticles containing 4457 atoms adopted in the present work are large enough to exhibit two distinct parts: surface and core as presented and discussed above. Unlike crystalline nanoparticles with disordered surface shell and core of a perfect crystalline structure, amorphous nanoparticles have a full disordered structure in that the surface exhibits a porous structure, while the core behaves as a moreclose packed atomic arrangement in that the change in structure from the surface into the core is rather smooth. It was found that melting of glasses initiates from the local nonclose-packed atomic arrangement regions,15,28,29 and although these regions mainly distribute in the surface shell, they also disperse in the inner parts of amorphous nanoparticles in that their concentration decreases with decreasing radial distance. Therefore, surface melting does not separately occur from melting of the core; that is, melting of amorphous nanoparticles is a continuous process leading to the single peak curve of the heat capacity (Figure 1d). Instead, a purely liquid skin occurs only in the second stage of melting of amorphous

IV. CONCLUSIONS We have studied melting of the mesoscale simple monatomic amorphous nanoparticles by MD simulations, and some conclusions can be given as follows: (1) Melting of the mesoscale simple monatomic amorphous nanoparticles consists of two stages: premelting one for T < Tg and “superheating” one for Tg ≤ T ≤ Ttm. In the premelting stage, liquid-like atoms occur in the surface shell to form a quasi-liquid interfacial region, that is, the mixed phase containing both liquid-like and solid-like atoms. At around T = Tg, a purely liquid skin of nanoparticles is forming at the surface. After that there is a balanced competition between a new-formed liquid skin and the remaining solid core for a short temperature range. Further heating leads to an intensive growth of liquid-like configuration; that is, the surface liquid skin and quasi-liquid interfacial region grow/move toward the remaining solid core of nanoparticles. Total melting is reached at temperature much higher than the glass transition one to form an entirely liquid phase. Because systems with LJG interatomic potential are adequate for metallic glasses,16 the same atomic mechanism of melting as described above can be suggested at least for melting of amorphous nanoparticles of metals and alloys. (2) Although our mesoscale amorphous nanoparticles exhibit two distinct parts, surface with a porous structure and G

dx.doi.org/10.1021/jp304211n | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

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core with a close-packed atomic arrangement, melting occurs rather continuously, and the heat capacity has a single peak related to the total melting of nanoparticles. Because of amorphous structure and the change in atomic density between surface shell and core of nanoparticles is rather smooth, there is no clear surface melting unlike that commonly observed for melting of crystalline nanoparticles at a similar size. In the premelting stage, liquid skin of nanoparticles does not occur. Instead, liquid-like atoms appear in the surface shell to form a quasi-liquid shell containing both liquidlike and solid-like atoms. This quasi-liquid shell enhances/grows, and liquid skin occurs just in the following stage of melting. (3) Liquid-like atoms occur locally in the nonclose-packed atomic arrangement regions due to a local instability. These regions distribute mainly in the surface shell, and their concentration decreases smoothly with decreasing radial distance. As described previously,18,28,29 nonclose packed atomic arrangement regions in the glassy matrix are thermally less stable, and it is easy for atoms located in these regions to escape from their position to diffuse, that is, to become liquid-like, and by this way melting of amorphous systems is started. (4) We find a strong thermal hysteresis between amorphous nanoparticles obtained by heating/cooling. It seems that thermal hysteresis of amorphous nanoparticles with free surfaces is stronger than that observed in thin film or in the bulk glasses.33

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS I am thankful for financial support from the Vietnam National University of HochiMinh City under Grant B2011-20-04TĐ. I used VMD software (Illinois University) for 3D visualization of the atomic configurations in this Article.

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REFERENCES

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