Study of Size Effect on Ni 60 Nb 40 Amorphous Particles and Thin Films by Molecular Dynamic Simulations

: Ni 60 Nb 40 amorphous particles (APs) and amorphous thin films (ATFs) with various sizes were investigated by molecular dynamic simulations. It is revealed that sample size has effects on both Ni 60 Nb 40 APs and ATFs composed of shell or surface and core components. Ni 60 Nb 40 APs have an average bond length of 2.57 Å with major fivefold-symmetry atomic packing and low bond-orientation orders of Q 6 and Q 4 in both core and shell components. Ni atoms in Ni 60 Nb 40 APs and ATFs prefer to segregate to the shell and surface regions, respectively. Atomic packing structure differences between various-sized Ni 60 Nb 40 APs and ATFs affect their glass transition temperatures T g , i.e., T g decreases as the particle size or the film thickness decreases in Ni 60 Nb 40 APs and ATFs, respectively. Our obtained results for Ni 60 Nb 40 APs and ATFs clearly reveal a size effect on atomic packing and glass transition temperature in low-dimensional metallic glass systems


Introduction
Due to its unique atomic packing structure with no long-range translation or orientation order, metallic glass (MG), a new class of materials, exhibits excellent properties, e.g., high strength, large elastic limit, superior wear and corrosion resistance [1,2].Concerning the structure in MG, although considerable studies have been reported, due to the complicated interactions between elements and topological complexity in multi-components MG with more than three elements [3,4], a complete understanding of this issue is still lacking.Fortunately, the Ni-Nb binary system, which has good glass-forming ability (GFA) and can be prepared in mm-sized MG by copper mold casting, provides prototypes to investigate the correlation of the structure with the properties of MG, and relatively reliable semi-empirical embedded atom method (EAM) potentials of this binary system were recently reported [5].Extensive works via experiments and simulations [6] have been carried out to describe the structure or to reveal the effects of various factors, e.g., the correlation between cooling rate and defect concentration [7] and composition [8].The atomic-level structure of a representative multi-component metallic glass (MG) has been resolved.It was also found that the sample size is an important factor affecting properties of MG.For example, Ref. [9] showed that the yield strength and deformation mechanism in the metallic glass were size-dependent, and Ref. [10] suggest that the transition from brittle to ductile behavior is driven by sample size, while the extent of ductility is driven by the surface state elastic limit [11].It was experimentally revealed by in situ transmission electron microscopy observations that small-sized Ni 60 Nb 40 magnetron-sputtered glassy films confined by a frame exhibited a significant size effect on the elastic strain limits [12].The mechanical behavior of nanoglass is also affected by the grain boundary thickness and the fraction of atoms at interfaces for a given average grain size [13,14].However, few studies have been reported of the size effect on the atomic structure of MG.One possible reason for this could be that free-standing amorphous particles (APs) and amorphous thin films (ATFs)

Methods
Molecular dynamics simulations of Ni 60 Nb 40 amorphous particles (APs) and amorphous thin films (ATFs) were performed based on the embedded atom method (EAM) potential in the canonical Ensemble (NVT) using the large-scale atomic molecular massively parallel simulator (LAMMPS) code.The initial structure of APs was constructed as follows: (1) creating a cubic zone containing 50, 100, 200, 400, 700, 1000, 4000, and 5000 atoms randomly distributed with a density of about 8.5 g/cm 3 .(2) Locating this cubic zone at the center of a big simulation box with the periodic boundary condition, i.e., the simulation box is 62 Å for 50-1000 atoms, and 80 Å for 4000 and 5000 atoms, to achieve a sufficient vacuum layer and negligible interaction between MGPs to mimic isolated particles.(3) The simulation box was melted and equilibrated at 2200 K for 1 ns, then quenched to 300 K with a cooling rate of 1 × 10 12 K/s.The centered cubic zone turned into a spherical-like particle which almost remains during the melting and the subsequent cooling process.The time step was 1 fs, i.e., 1 × 10 6 configurations were generated for 1 ns and we saved 10,000 configurations at intervals of 100 for data analyses.Ni 60 Nb 40 APs had various atom numbers of 50, 100, 200, 400, 700, 1000, 4000, and 5000 and their corresponding radii of about 5.5 Å, 7.2 Å, 9.5 Å, 11.7 Å, 13.8 Å, 15.8 Å, 25.1 Å, and 27.0 Å.These particles were then put at the center of simulation box which was under the periodic boundary condition with sizes of 62 Å for 50-1000 atoms, and 80 Å for 4000 and 5000 atoms, to achieve a sufficient vacuum layer and negligible interaction between APs.The Ni 60 Nb 40 APs were melted and equilibrated at 2200 K for 1 ns to remove the influence of initial configurations, then quenched to 300 K with a cooling rate of 1 × 10 12 K/s.Ni 60 Nb 40 APs were equilibrated at 300 K for 1 ns and exhibited an amorphous structure as shown in Figure 1.We further analyzed atomic structures of all studied APs with various sizes, in which two components, i.e., core and shell, were divided.In order to obtain reliable structural information, all structure analyses with 2000 configurations were carried out to obtain average values.Secondly, Ni 60 Nb 40 amorphous thin films (ATFs) with different thicknesses were obtained by cutting the bulk box with a size of 56 Å to the given thicknesses from 8.6 Å to 40 Å, and setting two vacuum layers along the thick direction to reach the free-standing condition.After relaxing the thin films for 1 ns at 2200 K, all the systems were quenched down to 300 K with a cooling rate of 1 × 10 12 K/s and relaxed at 300 K for 1 ns.The initial bulk box (used for thin films) was generated in a different way for the particles.We started with a Ni single crystal containing 10,000 atoms with a simulation cell approximately 56 Å × 56 Å × 56 Å in X-, Y-, and Z-directions, respectively, and then randomly substituted the atomic percentage (40%) of Ni atoms by Nb atoms.This mixture was then held at T = 2200 K (a temperature well above the melting point of the Ni 60 Nb 40 alloy) for 1 ns to allow for relaxation.The simulation time step was 1 fs.The purpose of this relaxation was to ensure the initial system has become a structurally homogeneous liquid.The model was then cooled to T = 300 K at a cooling rate of 10 12 K/s and then relaxed for 1 ns at 300 K. Periodic boundary conditions were applied to all three dimensions.The films with different thicknesses were cut from the center part of the bulk box to avoid the composition segregation.Compositions of all thin films were found to be close to the average value of 60 at.% for Ni atoms and 40 at.% for Nb atoms in the bulk with a fluctuation less than 0.5 at.%.The thin films with thicknesses from 8.6 Å to 40 Å were relaxed for 1 ns at 300 K.For each size MGP (or MGTF) investigated, we ran MD trajectories.In order to obtain reliable structural information, all structure analyses were carried out with 2000 configurations to obtain average values.
Metals 2024, 14, x FOR PEER REVIEW 3 of 18 in which two components, i.e., core and shell, were divided.In order to obtain reliable structural information, all structure analyses with 2000 configurations were carried out to obtain average values.Secondly, Ni60Nb40 amorphous thin films (ATFs) with different thicknesses were obtained by cutting the bulk box with a size of 56 Å to the given thicknesses from 8.6 Å to 40 Å, and setting two vacuum layers along the thick direction to reach the free-standing condition.After relaxing the thin films for 1 ns at 2200 K, all the systems were quenched down to 300 K with a cooling rate of 1 × 10 12 K/s and relaxed at 300 K for 1 ns.The initial bulk box (used for thin films) was generated in a different way for the particles.We started with a Ni single crystal containing 10,000 atoms with a simulation cell approximately 56 Å × 56 Å × 56 Å in X-, Y-, and Z-directions, respectively, and then randomly substituted the atomic percentage (40%) of Ni atoms by Nb atoms.This mixture was then held at T = 2200 K (a temperature well above the melting point of the Ni60Nb40 alloy) for 1 ns to allow for relaxation.The simulation time step was 1 fs.The purpose of this relaxation was to ensure the initial system has become a structurally homogeneous liquid.The model was then cooled to T = 300 K at a cooling rate of 10 12 K/s and then relaxed for 1 ns at 300 K. Periodic boundary conditions were applied to all three dimensions.The films with different thicknesses were cut from the center part of the bulk box to avoid the composition segregation.Compositions of all thin films were found to be close to the average value of 60 at.% for Ni atoms and 40 at.% for Nb atoms in the bulk with a fluctuation less than 0.5 at.%.The thin films with thicknesses from 8.6 Å to 40 Å were relaxed for 1 ns at 300 K.For each size MGP (or MGTF) investigated, we ran MD trajectories.In order to obtain reliable structural information, all structure analyses were carried out with 2000 configurations to obtain average values.

Amorphous Particles
The pair distribution function g(r) is a powerful method for describing the structure of amorphous materials.For finite-sized materials, such as particles, g(r) can be obtained via:

Amorphous Particles
The pair distribution function g(r) is a powerful method for describing the structure of amorphous materials.For finite-sized materials, such as particles, g(r) can be obtained via: where N represents the total number of atoms, V 0 is the model volume, S is the surface area of the particle shell.Coordination number depends on the cutoff length in Ni60Nb40 APs.The fi minimum in g(r) was used here as the cutoff length for the first nearest shell.The total the partial coordination number (CN) distributions for the studied Ni60Nb40 APs plotted in Figure 4. Three peaks located at about CN = 6, 9, 12 are found in the total distribution.The CN distribution depends on the size of Ni60Nb40 APs.For sma Ni60Nb40 APs, e.g., 50, 100, and 200 atoms, the peak at CN = 6 decreases with increas size and almost disappears for sizes larger than 200 atoms.Meanwhile, the peak of C 12 largely increases when the size is larger than 200 atoms.To further investigate lo atomic environments in the studied Ni60Nb40 APs, CN distributions in both core and s components were also calculated.In the core component, CN is only distributed in ab    The Honey-Anderson (HA) index [15] is introduced to investigate local structure of Ni60Nb40 APs, as shown in Figure 5. HA has four indices i, j, l, m, and is a method used to characterize the common neighbors of an atom pair.i = 1 means atom A and atom B form a bond, and otherwise i = 2. j denotes the number of common neighbors which form bonds with atom A and atom B. l represents the number of bonds formed among the neighboring atoms.m is a special index to classify the bonds arrangements.It is found that the 1551 HA indexes account for large proportions that equal more than 40% in total, which is characteristic of the fivefold-symmetry structure, implying that icosahedral-like structures are dominant in Ni60Nb40 APs.Meanwhile, the fractions of 1551, 1541, 1661, and The Honey-Anderson (HA) index [15] is introduced to investigate local structure of Ni 60 Nb 40 APs, as shown in Figure 5. HA has four indices i, j, l, m, and is a method used to characterize the common neighbors of an atom pair.i = 1 means atom A and atom B form a bond, and otherwise i = 2. j denotes the number of common neighbors which form bonds with atom A and atom B. l represents the number of bonds formed among the neighboring atoms.m is a special index to classify the bonds arrangements.It is found that the 1551 HA indexes account for large proportions that equal more than 40% in total, which is characteristic of the fivefold-symmetry structure, implying that icosahedral-like structures are dominant in Ni 60 Nb 40 APs.Meanwhile, the fractions of 1551, 1541, 1661, and 1441 indexes slightly increase with size, while the 1431 and 1321 indexes have the opposite trend of decreasing as size increases from 50 to 5000 atoms.In the core component of Ni 60 Nb 40 APs, HA indexes do not depend on size.The 1551 index has a fraction of more than 50%, and the second, third, fourth and fifth highest indexes are 1661, about 20%; 1541, about 10%; 1441, about 8%; and 1431, about 6%, respectively.In the shell component of Ni 60 Nb 40 APs, 1321 and 1431 indexes have obvious high fractions, while they are tiny in the core component, although the 1551 index is still the dominant local atomic packing.Bond-orientation order (BOO) parameters, e.g., Q 6 and Q 4 , are often used to measure atomic cluster symmetries in disordered systems [16][17][18][19].The averaged parameters (BOO) are sensitive to different bond-orientational symmetries, so they can be used to differentiate the local atomic environments.The complex vector (q l ) can be j=1 Y lm rij , where l represents the free integer parameter, m is an integer that runs from −l to l, Y lm are the spherical harmonics, ∧ r ij is the vector from atom i to atom j, and the sum goes over all neighboring atoms N b (i), the number of atom i.One can average the spatially local bond order parameters, Q 4 and Q 6 , according to the following equation: the core component, although the 1551 index is still the dominant local atomic packing.Bond-orientation order (BOO) parameters, e.g., Q6 and Q4, are often used to measure atomic cluster symmetries in disordered systems [16,17,18,19].The averaged parameters (BOO) are sensitive to different bond-orientational symmetries, so they can be used to differentiate the local atomic environments.The complex vector ( l q ) can be defined as   component.Figure 7 depicts cluster size-dependent Ni contents in the shell component in Ni60Nb40 APs, in which the thickness of the shell determined by the minimum after the first peak in total g(r) is about 3.66 Å in Ni60Nb40 APs.The Ni composition in the shell is larger than 60 at.% and increases from 64 at.% for 50-atom APs to 73 at.% in 5000-atom APs, suggesting a segregation of Ni atoms in shell components in larger clusters, which are roughly correlated with Q6 and Q4 values.Q6 and Q4 values in the shell component of Cu64Zr36 APs illustrated in Ref [20] are even larger than those in Ni60Nb40 APs, i.e., atomic packing in the shell component of Cu64Zr36 APs, roughly speaking, has a slightly higher order than that in Ni60Nb40 APs.These results are closely linked with a larger degree of composition segregation in the shell component of Cu64Zr36 APs [20].All the above-mentioned results clearly reveal that particle size indeed affects the atomic structure of Ni60Nb40 APs.The properties of amorphous materials are always closely related with their structures, which motivated us to compare the size effect on properties of such APs.Hence, we further studied the size dependence of the glass transition temperature, Tg, which is closely related to the atomic structure in disordered systems.The glass transition temperature of Ni60Nb40 APs with various sizes was estimated by the intersection of potential energy of low and high temperatures, as plotted in Figure 8a.It is found that Tg increases as the size of the particle increases in Ni60Nb40 APs, i.e., the sample size indeed influences the glass transition temperature of Ni60Nb40 APs.Tg of all the studied Ni60Nb40 APs can be well fitted with a simple equation of Tg = fshellTg shell + fcTg c , where fshell and fc, and Tg shell and Tg c , represent the fractions and glass

Amorphous Thin Films
Besides the APs discussed above, we also explored size effects on atomic structure and glass transition temperature for Ni60Nb40 amorphous thin films (ATFs) i.e., metallic glassy thin films (MGTFs).The structure configuration images from the simulations for the thin films of different thicknesses are shown in Figure 9. Figure 10 shows pair

Amorphous Thin Films
Besides the APs discussed above, we also explored size effects on atomic structure and glass transition temperature for Ni 60 Nb 40 amorphous thin films (ATFs) i.e., metallic glassy thin films (MGTFs).The structure configuration images from the simulations for the thin films of different thicknesses are shown in Figure 9.       Furthermore, the coordination number, HA index, and BOO parameters for Ni60Nb40 ATFs were also studied.As plotted in Figure 12, CN distribution depends on the thickness of Ni60Nb40 ATFs and two peaks located at about CN= 9, 12 were found in the total CN distribution.For thinner ATFs, e.g., 8.6 Å, 10.7 Å, and 13.1 Å, the peak at CN = 9 decreases

Conclusions
In conclusion, Ni60Nb40 APs with 50 to 5000 atoms and Ni60Nb40 ATFs with a thickness range of 8.6-40 Å were systematically investigated by molecular dynamics simulations.The sample size effects on the atomic structure were clearly revealed in both Ni60Nb40 APs and ATFs.Ni60Nb40 APs have an average bond length of 2.57 Å and atomic packing with

Figure 2 Figure 2 .
Figure 2. Pair distribution functions g(r) for Ni60Nb40 APs with various sizes of core and s components together; illustration of the core-and-shell model in the inset.

Figure 2 .Figure 3 .
Figure 2. Pair distribution functions g(r) for Ni 60 Nb 40 APs with various sizes of core and shell components together; illustration of the core-and-shell model in the inset.

Figure 3 .
Figure 3. Partial pair distribution functions g(r) for Ni 60 Nb 40 APs with various sizes.
are centered at about 12, while CNs are centered at about 15 for Nb atoms.In the shell component, CN distributions of Ni atoms are similar to the CN distributions of the total, indicating that the shell components in Ni60Nb40 APs should have a Ni-rich composition.This is indeed confirmed later.The CN distribution of Nb atoms in the 50-atom Ni60Nb40 MGP peaks at CN = 9, and shifts to a slightly high value of CN = 10-11 for larger APs.

Figure 4 .
Figure 4.The coordination number distributions of the total sample, core, and shell components, and partial coordination number distributions for Ni60Nb40 APs with various sizes.

Figure 4 .
Figure 4.The coordination number distributions of the total sample, core, and shell components, and partial coordination number distributions for Ni 60 Nb 40 APs with various sizes.

Figure 5 .
Figure 5.The HA index for total sample, core, and shell components for Ni 60 Nb 40 APs with various sizes.

Figure 6
Figure 6 depicts Q 6 and Q 4 values for Ni 60 Nb 40 APs with various sizes.It is found that Q 6 and Q 4 values in the shell component are larger than those in the core component in this system.In total particles, both parameters have values of the averages in both shell and core components.These results indicate more ordering in the shell component than in the core component.This might be linked with composition segregation in the shell component.Figure 7 depicts cluster size-dependent Ni contents in the shell component in Ni 60 Nb 40 APs, in which the thickness of the shell determined by the minimum after the first peak in total g(r) is about 3.66 Å in Ni 60 Nb 40 APs.The Ni composition in the shell is larger than 60 at.% and increases from 64 at.% for 50-atom APs to 73 at.% in 5000-atom APs, suggesting a segregation of Ni atoms in shell components in larger clusters, which are roughly correlated with Q 6 and Q 4 values.Q 6 and Q 4 values in the shell component of

Figure 7
Figure 6 depicts Q 6 and Q 4 values for Ni 60 Nb 40 APs with various sizes.It is found that Q 6 and Q 4 values in the shell component are larger than those in the core component in this system.In total particles, both parameters have values of the averages in both shell and core components.These results indicate more ordering in the shell component than in the core component.This might be linked with composition segregation in the shell component.Figure 7 depicts cluster size-dependent Ni contents in the shell component in Ni 60 Nb 40 APs, in which the thickness of the shell determined by the minimum after the first peak in total g(r) is about 3.66 Å in Ni 60 Nb 40 APs.The Ni composition in the shell is larger than 60 at.% and increases from 64 at.% for 50-atom APs to 73 at.% in 5000-atom APs, suggesting a segregation of Ni atoms in shell components in larger clusters, which are roughly correlated with Q 6 and Q 4 values.Q 6 and Q 4 values in the shell component of

Figure 6 .
Figure 6.The Q6-Q4 distribution of total, core, and shell components for Ni60Nb40 APs with various sizes.Numbers of atoms are marked.

Figure 6 .
Figure 6.The Q 6 -Q 4 distribution of total, core, and shell components for Ni 60 Nb 40 APs with various sizes.Numbers of atoms are marked.Metals 2024, 14, x FOR PEER REVIEW 10 of 18

Figure 7 .
Figure 7. Ni contents in the shell of Ni60Nb40 APs with various sizes.The dash line represents the initial composition of APs.

Figure 7 . 18 Figure 8 .
Figure 7. Ni contents in the shell of Ni 60 Nb 40 APs with various sizes.The dash line represents the initial composition of APs.All the above-mentioned results clearly reveal that particle size indeed affects the atomic structure of Ni 60 Nb 40 APs.The properties of amorphous materials are always closely related with their structures, which motivated us to compare the size effect on properties of such APs.Hence, we further studied the size dependence of the glass transition temperature, T g , which is closely related to the atomic structure in disordered systems.The glass transition temperature of Ni 60 Nb 40 APs with various sizes was estimated by the inter-

Figure 8 .
Figure 8.(a) Determination of glass transition temperature T g via an intersection of low-and hightemperature dependence of the potential energy for Ni 60 Nb 40 APs with various sizes.(b) Fitting of glass transition temperature data for Ni 60 Nb 40 APs with various sizes.
Figure 10 shows pair distribution function g(r) curves of Ni 60 Nb 40 ATFs at 300 K with different thicknesses from 8.6 Å to 40 Å.The sharp first peak and the split second peak in g(r) curves of films demonstrate that all studied thin films are fully amorphous.Ni 60 Nb 40 ATFs have sharp and narrow first peaks in g(r).For the second peak within 4 to 6 Å, the relative intensity of the shoulder peak for Ni 60 Nb 40 ATFs becomes weak, indicating that the medium range order might be low in Ni 60 Nb 40 ATFs.Similar to Ni 60 Nb 40 APs, Ni 60 Nb 40 ATFs can also be composed of two regions, i.e., the core region and surface layer region.Figure 11 demonstrates the composition variations along the film thicknesses in Ni 60 Nb 40 ATFs with a thickness of 40 Å.The element segregation again appears in the surface layer of the Ni 60 Nb 40 MGTFs, which is also observed in Ni 60 Nb 40 ATFs with other thicknesses.The fraction of Ni atoms is found to be increased in the surface region, where Ni content is about 75~90%, higher than its nominal content of 60%.In contrast, the compositions in the core regions are close to their nominal values in Ni 60 Nb 40 ATFs.Metals 2024, 14, x FOR PEER REVIEW 12 of 18demonstrates the composition variations along the film thicknesses in Ni60Nb40 ATFs with a thickness of 40 Å.The element segregation again appears in the surface layer of the Ni60Nb40 MGTFs, which is also observed in Ni60Nb40 ATFs with other thicknesses.The fraction of Ni atoms is found to be increased in the surface region, where Ni content is about 75~90%, higher than its nominal content of 60%.In contrast, the compositions in the core regions are close to their nominal values in Ni60Nb40 ATFs.

Figure 10 .
Figure 10.Pair distribution functions g(r) for Ni60Nb40 ATFs with various thicknesses.

Figure 9 .
Figure 9. Structure configurations of 8.6 Å, 10.7 Å, 13.1 Å, 15.4 Å, 17.8 Å, 20 Å, 25 Å, and 30 Å thick amorphous thin films, respectively.Furthermore, the coordination number, HA index, and BOO parameters for Ni 60 Nb 40 ATFs were also studied.As plotted in Figure 12, CN distribution depends on the thickness of Ni 60 Nb 40 ATFs and two peaks located at about CN = 9, 12 were found in the total CN distribution.For thinner ATFs, e.g., 8.6 Å, 10.7 Å, and 13.1 Å, the peak at CN = 9 decreases with increasing thickness and the peak of CN = 12 increases when the thickness is larger than 15.4 Å. CN distributions in both core and surface components were also calculated.In the core component, CN is distributed within 11-16 and has a peak at CN = 12.In the shell component for Ni 60 Nb 40 ATFs, CN has a peak located at CN = 9.This lower CN in the surface component can be explained by the surface effect as explained by the AP findings.Ni-and Nb-centered CN distributions are plotted in Figure 12.In the core component, CNs of Ni atoms are centered at about 12, while CNs are centered at about 14 for Nb atoms.In the surface component, CN distributions of Ni atoms are 9 and those for Nb atoms are 11.As shown in Figure 13, 1551 HA indexes account for large proportions, more than 40% in total, implying that icosahedral-like structures are dominant in Ni 60 Nb 40 ATFs, as in Ni 60 Nb 40 APs in Figure 5.Meanwhile, the fractions of 1551, 1541, 1661, and 1321 indexes slightly increase with thickness, while the 1421 and 1311 indexes have the opposite trend of decreasing as thickness increases from 8.6 to 40 Å.In the core component of Ni 60 Nb 40 ATFs,

Figure 10 .
Figure 10.Pair distribution functions g(r) for Ni60Nb40 ATFs with various thicknesses.Figure 10.Pair distribution functions g(r) for Ni 60 Nb 40 ATFs with various thicknesses.

Figure 12 .
Figure 12.The coordination number distributions of the total sample, core, and surface components, and partial coordination number distributions for Ni60Nb40 ATFs with various thickness.

Figure 12 . 15 Figure 13 .
Figure 12.The coordination number distributions of the total sample, core, and surface components, and partial coordination number distributions for Ni 60 Nb 40 ATFs with various thickness.The average atomic internal energy of Ni 60 Nb 40 ATFs with various thicknesses as a function of temperature during cooling was studied, from which the glass transition temperature T g for ATFs was estimated by an intersection of potential energy of low and high temperatures, shown in Figure 15a.It was found again that T g decreases substantially as the film thickness decreases in Ni 60 Nb 40 ATFs, similar to the results observed in Ni 60 Nb 40 APs in Figure 8a.The glass transition temperature as a function of film thickness can be well fitted with a simple equation (model II), i.e., T g (d) = T g (∞)(1 − C/d), as shown in Figure 15b, where T g (∞) and C represent two fitting parameters, and d is the film thickness.The obtained T g (∞) and C values are 975 K and 1.6 for Ni 60 Nb 40 ATFs, respectively.In fact, T g of all the studied Ni 60 Nb 40 ATFs can also be fitted with the model I of T g = f surface T g surface + f c T g c , where f surface and f c , and T g surface and T g c , represent the fractions and glass transition temperatures of the surface layer and core components, respectively.The surface thicknesses were determined by the minimum after the first peak in the total g(r) curves of Ni 60 Nb 40 ATFs.T g c and T g surface of the studied Ni 60 Nb 40 ATFs were found to be about 978 K and 906 K, respectively, while T g c and T g shell of the studied Ni 60 Nb 40 APs in Figure 8b were found to be about 1090 K and 910 K, respectively.These results reveal that the Ni 60 Nb 40 APs and ATFs are both composed of the surface (shell) and core components, while their corresponding T g values are slightly different, most likely resulting from different atomic packing in both components for Ni 60 Nb 40 APs and ATFs.The

Figure 13 .
Figure 13.The HA index for total sample, core, and surface components for Ni 60 Nb 40 ATFs with various thickness.

Figure 14 .
Figure 14.The Q6-Q4 distribution of total, core, and surface components for Ni60Nb40 ATFs with various thickness.Thickness of films are marked.The average atomic internal energy of Ni60Nb40 ATFs with various thicknesses as a function of temperature during cooling was studied, from which the glass transition temperature Tg for ATFs was estimated by an intersection of potential energy of low and high temperatures, shown in Figure 15a.It was found again that Tg decreases substantially as the film thickness decreases in Ni60Nb40 ATFs, similar to the results observed in Ni60Nb40 APs in Figure 8a.The glass transition temperature as a function of film thickness can be well fitted with a simple equation (model II), i.e., Tg(d) = Tg(∞)(1 − C/d), as shown in Figure 15b, where Tg (∞) and C represent two fitting parameters, and d is the film thickness.The obtained Tg(∞) and C values are 975 K and 1.6 for Ni60Nb40 ATFs, respectively.In fact, Tg of all the studied Ni60Nb40 ATFs can also be fitted with the model I of Tg = fsurfaceTg surface + fcTg c, where fsurface and fc, and Tg surface and Tg c , represent the fractions and glass transition temperatures of the surface layer and core components, respectively.The surface thicknesses were determined by the minimum after the first peak in the total g(r) curves of Ni60Nb40 ATFs.Tg c and Tg surface of the studied Ni60Nb40 ATFs were found to be about 978 K and 906 K, respectively, while Tg c and Tg shell of the studied Ni60Nb40 APs in Figure8bwere found to be about 1090 K and 910 K, respectively.These results reveal that the Ni60Nb40 APs and ATFs are both composed of the surface (shell) and core components, while their corresponding Tg values are slightly different, most likely resulting from different atomic packing in both components for Ni60Nb40 APs and ATFs.The different shapes of the samples, spherical particles and flat thin films, and different element segregation, could cause the different size effects on the atomic packing and Tg of Ni60Nb40 APs and ATFs.

Figure 14 . 18 Figure 15 .
Figure 14.The Q 6 -Q 4 distribution of total, core, and surface components for Ni 60 Nb 40 ATFs with various thickness.Thickness of films are marked.Metals 2024, 14, x FOR PEER REVIEW 17 of 18

Figure 15 .
Figure 15.(a) Determination of glass transition temperature T g via an intersection of low-and high-temperature dependence of the potential energy for Ni 60 Nb 40 ATFs with different thicknesses.(b) Fitting of glass transition temperature T g for Ni 60 Nb 40 ATFs with various thickness.The two models used give almost the same quality of fittings.

Table 1 .
The first peak positions of total g(r) for Ni60Nb40 amorphous particles (APs) and thin fi (ATFs).

Table 1 .
The first peak positions of total g(r) for Ni 60 Nb 40 amorphous particles (APs) and thin films (ATFs).