The phase transition from a crystalline solid to a liquid state

Generally, energy is supplied to the configuration upon heating; the atoms vibrate more vigorously about their lattice positions. This increased kinetic energy gradually overcomes the interatomic forces holding the crystal structure together. When the total energy per atom reaches a critical threshold, the crystalline order collapses, and the configuration transitions to a liquid phase characterized by a much higher degree of atomic mobility. Therefore, to study the phase transition from a crystalline solid to a liquid state, the total energy per atom of each configuration is calculated and presented in Figure 2.

Overall, the total energy landscapes of the free-standing configuration, zigzag h-BN nanoribbon configuration, and armchair h-BN nanoribbon configuration are divided into three parts. In the first part, the total energy increases linearly. In the second part, the total energy increases sharply. In the last part, the total energy shows linear behavior again. In general, this behavior of the total energy landscape indicates that the three configurations exhibit first-order phase transitions from a crystalline solid to a liquid state (Figure 2).

However, in detail, there are differences between the free-standing h-BN configuration and the remaining two configurations. For example, in the first part, the total energy of the free-standing configuration increases gradually (square symbols in Figure 2a), whereas the total energy of the zigzag (pentagon symbols in Figure 2b) and armchair (circle symbols in Figure 2c) h-BN nanoribbons increases and fluctuates. This occurs because the armchair (or zigzag) edge is fixed to form zigzag (or armchair) nanoribbon configurations.

Defects form at the armchair and zigzag edges of h-BN. Compared with zigzag edges, armchair edges are more precarious because of the many dangling bonds in their structure. This leads to differences in the second part: the total energy behavior of the free-standing configuration, zigzag h-BN nanoribbon configuration, and armchair h-BN nanoribbon configuration (Figure 2). The total energy per atom of the free-standing configuration increases sharply at 4550 K (square symbols in Figure 2a). In comparison, this situation occurs in the zigzag h-BN nanoribbon configuration at 4200 K (pentagon symbols in Figure 2b) and in the armchair h-BN nanoribbon configuration at 3700 K (circle symbols in Figure 2c).

The melting point of each configuration can be defined as the temperature at which the total energy per atom reaches a critical threshold sufficient to overcome the interatomic forces holding the solid lattice together. The melting temperature point can also be found on the basis of the heat capacity, which obeys the following formula (2): This temperature point represents the peak of the heat capacity landscape.

where is the total energy per atom and where is the temperature. In this study, the melting points of the free-standing configuration (solid line in Figure 2a), zigzag h-BN nanoribbon configuration (solid line in Figure 2b), and armchair h-BN nanoribbon configuration (solid line in Figure 2c) are 4550 K, 4200 K, and 3700 K, respectively. One can conclude that the unstable structures at the armchair edge strongly affect the melting point of h-BN.

In addition to the phase transition from a crystalline state to a liquid state, the radial distribution function (RDF) is also a powerful tool. The RDF describes the density distribution of particles around a central particle as a function of distance. On the basis of the RDF, we can determine how the probability of finding a particle changes as we move away from a reference particle. In this study, the RDF of the free-standing h-BN configuration is calculated and presented in Figure 3.

In a crystal with the h-BN configuration (50 K), the particles are arranged in a highly ordered, periodic structure. This is reflected in the RDF by sharp, well-defined peaks at specific distances corresponding to the nearest neighbors, next-nearest neighbors, and so on (square symbols in Figure 3). Upon further heating to 4250 K (circle symbols in Figure 3), the peaks decrease sharply, indicating that the atoms are no longer arranged in an orderly manner. This is because the atoms receive energy in the form of heat and vibrate violently, breaking the bonds. At 4550 K, the ordered structure undergoes a significant change as follows: First, in the N‒N case, the first peak decreases and moves toward the origin and splits into three small peaks, whereas the remaining peaks are almost smooth (triangle symbols in Figure 3a). Second, in the B-B case, the first peak decreases and moves toward the origin, whereas the remaining peaks are almost smooth (triangle symbols in Figure 3b). Finally, in the B-N case, the first peak does not shift toward the origin but decreases significantly, whereas the remaining peaks are almost smooth (triangle symbols in Figure 3c). This indicates that the particles are more disordered, and the liquid state is approached at this temperature point (4550 K). Thus, by monitoring the RDF of free-standing h-BN as the temperature increases, the melting point can be determined at 4550 K, which coincides with the melting point defined by the heat capacity (solid lines in Figure 2).

In particular, for the case of B‒N bonds (Figure 3c), the first peak in the RDF corresponds to the nearest neighbor of a particle. For free-standing h-BN, in the crystal state, the number of nearest neighbors of the B atom corresponding to the first peak in the RDF graph is three N atoms (see the inset in Figure 4), often called the coordination number of the B atom. At the melting point (4550 K), the first peak of the B-N RDF is at approximately 20% (triangle symbols in Figure 3c), indicating that the coordination number of three remains in the liquid state. To verify this prediction, the coordination number of the B atoms at the melting point (4550 K) is considered. The number of N atoms closest to the B atoms is calculated and presented in Figure 4. The results show that, as expected, a coordination number of three is still present (3%) at the melting temperature (4550 K). The remaining coordination numbers are two, one, and zero, of which the coordination number of zero is the most abundant at 61%, demonstrating the significant disruption of B‒N bonds (Figure 4).

To contribute to a comprehensive understanding of the h-BN melting process, the coordination number of B atoms at the melting point of the armchair and zigzag h-BN nanoribbons is calculated and presented in Figure 5. The results are as follows:

The formation of B atom coordination numbers is significantly influenced by edge type (armchair or zigzag), particularly in the case of armchair edges under nonperiodic conditions, to form zigzag h-BN nanoribbons. At the melting temperature, most B atoms in the zigzag h-BN nanoribbon exhibit coordination numbers of one or two, whereas most B atoms in the armchair h-BN nanoribbon and the free-standing h-BN have a coordination number of zero.

The atomic mechanism of the melting process

Unlike bulk materials, 2D materials exhibit reduced dimensionality, leading to unique properties and challenges in defining a precise melting criterion. While not definitive, the Lindemann criterion is often used as a starting point for understanding melting in 2D materials. This criterion suggests that melting occurs when the root-mean-square displacement of atoms exceeds a critical fraction of the interatomic spacing. However, its application to 2D systems requires careful consideration because of its reduced dimensionality and potential for anisotropic behavior 36, 37, 38, 39, 40. The formula of the Lindermann criterion is as follows:

where is the density of particles at , is the position of the atom and where the sum over j runs over the n atoms closest to the atom i. In this study, the Lindermann criterion is calculated for the three closest atoms to the i-th atom (Figure 6).

In the temperature range below 4270 K, the value of remains constant, indicating that the configuration is in a crystalline state. When the temperature exceeds 4270 K, the Lindermann criterion increases abruptly, indicating the breakdown of the crystal structure. Consequently, the Lindermann criterion at 4270 K is considered the critical Lindermann criterion value (0.0326). Note that the above Lindermann criterion was calculated for graphene, a structure similar to h-BN 41.

On the basis of the critical Lindermann criterion value (0.0326), atoms can be classified as solid-like or liquid-like. Atoms with Lindermann criterion values less than the critical criterion value (0.0326) are solid-like, whereas those with Lindermann criterion values greater than the critical criterion value (0.0326) are liquid-like. The appearance/growth of the liquid-like atoms upon heating was subsequently calculated and is presented in Figure 7.

The results indicate that at temperatures below 4270 K, a small number of liquid-like atoms exist due to edge defects. Upon further heating to the melting point (4550 K), the number of liquid-like atoms increases substantially, reaching 83.45% (Figure 7). Even in the liquid state (above 4550 K), the number of liquid-like atoms does not reach 100%, suggesting the continued presence of solid-like atoms. This can be verified by examining the angular distribution. If solid atoms persist in the solid state, the angular distribution at 120° would likely remain. Consequently, we examine the angular distribution of the free-standing h-BN in the liquid state (4700 K), as shown in Figure 8. The results show that the angular distribution ranges from 35° to 180°. The distribution ratio is approximately 120° and then 80°. The distribution ratio at 120° is approximately 1%, which is a tiny percentage indicating that the model has distorted structures (Figure 8).

In addition, visualization of the free-standing h-BN upon heating at different temperatures is presented in Figure 9. First, the free-standing h-BN configuration originates in its crystalline structure with a well-organized hexagonal pattern at 50 K (Figure 9a). At 4250 K, some disordered bonds between B-B and N‒N appear in the configuration, indicating that the atoms vibrate and disrupt the bonds and potential to maintain an orderly structural arrangement. However, the freestanding h-BN configuration remains packed hexagonally at this temperature (Figure 9b). Upon further heating from 4250 K to 4550 K, almost the entire crystal model collapsed, resulting in a phase transition from the crystalline state to the liquid state at 4550 K (Figure 9c).