#### Abstract

Boron nitride (BN) nanomaterials such as boron nitride graphenes, boron nitride nanotubes, and boron nitride nanocones are attracting attention among the most promising nanomaterials due to their physical, chemical, and electronic properties when compared to other nanomaterials. BN nanomaterials suggest many exciting potential applications in various fields. Joining between BN nanostructures gives new enhanced structures with outstanding properties and potential applications for design of probes for scanning tunnelling microscopy and other nanoscale devices. This paper uses calculus of variations to model the joining between BN graphene with other BN nanostructures: BNNTs and BNNCs. Furthermore, during the joining between these BN nanostructures, this research examines two models which are depending on the curvature of the join profile. For the first case, Model I refers to when the join profile only includes positive curvature where for the second case, Model II is considered for both positive and negative curvatures. Thus, the purpose of this research is to formulate the basic underlying structure to present simple models based on joining BN graphene to other BN nanostructures.

#### 1. Introduction

Materials in nanoscale have attracted research in different scientific fields because of their chemical, physical, and electronic properties. In particular, carbon in nanoscale has different shapes and structures such as carbon nanotubes, carbon nanocones, carbon nanosheets (graphene), and fullerene.

Due to similarity to carbon nanostructures, boron nitride nanostructures have received specific attention due to their remarkable properties that lead them to a wide range of applications such as developing many nanoscale devices [1, 2]. Hexagonal BN structure involves boron and nitrogen atoms bound by strong covalent bonds. Moreover, many forms of boron nitride in nanoscale have been reported in the previous literature such as boron nitride nanotubes (BNNTs), boron nitride graphene (BN graphene), boron nitride nanocones (BNNCs), and boron nitride fullerene (BN fullerene) [3].

BNNTs were first discovered in 1994 [4] and contain a tubular structure as carbon nanotubes with replacing carbon atoms by boron and nitrogen atoms arranging in a hexagonal lattice. BNNTs have gained significant attention in recent years because of their interesting properties that are not available in other nanomaterials. While they are structurally close to carbon nanotubes, BNNTs have completely different physical properties. For example, BNNTs have mechanical, thermal, electrical, and chemical properties like high tensile stiffness and high thermal conductivity. Also, their electronic properties are independent of their chirality and radii of the tube [4–6].

Another boron nitride nanostructure has attracted much attention, that is, boron nitride graphene. Geometrically, it is similar to carbon nanosheet with boron and nitrogen atoms instead of carbon atoms. BN graphene has physical and chemical distinct properties, such as high-temperature stability, intrinsic electrical insulation, and semiconducting and antioxidation ability. As a result, BN graphene has a promising application in various areas; namely, it is used as a two-dimensional filler, nonwetting coatings, and field emitters [7–9].

Boron nitride nanocones (BNNCs) are important structures of BN nanostructures. These conical structures have been considered in many studies experimentally and theoretically. Their special structure offers suitability for fabrication of nanodevices such as electron field emitters, sensors, nanoindenters, and probe tips. Carbon nanocones (CNCs) can be formed by curling the fan-shaped graphite sheet with a disclination angle of an integer multiple of 60° where BNNCs can be produced by only rolling up the white graphene with a disclination angle of an integer multiple of 120° [10]. In addition, carbon nanocones have five possible structures depending on the number of pentagons, and the conical structure of boron nitride has been studied using transmission electron microscopy and observed their angles to be 84°, 19.2°, and 38.9° [11].

New structures produced from the joining between two nanostructures can enhance the physiochemical and electrochemical performances of the joined materials. For example, carbon nanotubes and carbon nanosheets are useful in nanosensors and nanooscillators, and connections of these carbon nanostructures may also be employed in the same applications. Furthermore, the same applications exist for joining BNNTs and BN graphene [1]. Moreover, the composite structures are useful for the design of probes for scanning tunnelling microscopy, energy storage, and other electronic devices as carriers for drug delivery [2].

Research in [12] minimizes the elastic energy (which depends on the axial curvature only) using calculus of variations to determine the joining area between two carbon nanostructures. The chemical issues are ignored in this model such as the positions of the carbon atoms and bonds. Based on the length constraint of the joining between two nanostructures, the curvature of the join curve can be either positive or both positive and negative. In addition, the study in [13] studies different cases for joining between carbon nanostructures using the same method described above, including the joining of two carbon nanotubes, tube with fullerene, tube with cone, and two fullerenes. Furthermore, this model has been extended to study the joining of some boron nitride nanostructures as in [14].

In this paper, we use the same model to determine the shape of the joining between BN graphene with BNNTs and BNNCs, where the arc length of the curve and the size of the defect in the BN graphene are specified, and the distance of the BNNTs and BNNCs from the BN graphene is not prescribed and is determined as a part of the solution.

Finally, we comment that using Willmore energy which depends on the axial and rotational curvatures to determine the joining between two nanostructures gives rise to similar join profiles of using elastic energy as studied in [15]. Moreover, the mathematical techniques in [16–19] have similar ideas as in this research.

In the following section, we state the fundamental equations of the calculus of variations to model the joining region between BN graphene with BNNTs and BNNCs. In Subsection 2.1, the curvature is assumed to remain positive in the join region, and we denote this case as Model I. Subsection 2.2 presents Model II which is assumed to have two join regions, one of positive curvature and the second of negative curvature. Results and discussion are given in Section 4. Section 4 provides the summary.

#### 2. Model

Here, the basic variational equations of the model that is used to join boron nitride graphene to other boron nitride nanostructures are formulated. In details, variational calculus is used to identify the curve adopted by a line smoothly connecting a BN graphene base to a vertical other boron nitride nanostructure, where the arc length of the joining curve and the defect site at the BN graphene base are specified. Therefore, the distance in the -direction of the joining to the tube or cone is not specified, and it is found as a part of the solution.

We position the first nanostructure (BN graphene) in the -plane with a circular defect of radius centred on the origin. Assuming that a second nanostructure (BNNTs or BNNCs) of radius is located with its axis colinear with the -axis starting from an unknown positive distance above the -plane that is denoted by , the defect and second nanostructure are assumed to be rotationally symmetric about the -axis; the problem can be reduced to two-dimensional -plane. The total prescribed arc length is assumed to connect the defects at the points and We comment that this technique is used first by Cox and Hill in [12] and then by Baowan et al. [2, 13].

Using calculus of variations to find the curve , with an element of arc length , minimizes the energy functional that is given by where is the curvature, is a Lagrange multiplier corresponding to the fixed length constraint, and is the length of the join curve. The boundaries of the join region are and , where at we have and at we have . The curve in two dimensions described as a graph , we have and , then equation (1) becomes where in this paper, dot denotes differentiation with respect to . We have two different models based on the curvature’s sign, and the join curvature remains positive as shown in Figure 1 and both positive and negative curvatures as in Figure 2. For these models, we impose the continuity boundary conditions at the first nanostructure (BN graphene) where ,

To determine the boundary conditions at the second nanostructure (BNNTs or BNNCs), we integrate by parts to derive the standard equation where subscripts denote partial derivatives, and here, the function is given by

In this case, only is prescribed at as is unknown, and at we require the natural or alternative boundary condition given by

For the case of BNNTs, . The value of in Model I ranges from at to at Thus, the boundary condition for this model is For Model II, ranges from to , where it changes sign and then ranges from down to some finite positive value before turning to . Therefore, the boundary condition in the case of Model II is For both models, we have at equation (6) along with the boundary condition for Model I: and for Model II is

Noting that, for the case of BNNCs, is the angle of the nanocone. From equation (4), we have the usual Euler-Lagrange equation for , which is given by

However, since our function is independent of , integrating equation (9) gives where is an arbitrary constant. From the natural boundary condition in equation (6), we find that so we obtain

Using the definition of the full derivative, we have

Since , from equation (11), we obtain by integrating with respect to , we get where is an arbitrary constant. Now, we substitute equation (5) into equation (14) and obtain

As a result, we can write the curvature as

##### 2.1. Model I: Positive Curvature

The curvature in Model I is positive through the arc length as shown in Figures 3 and 4. As a result, we only consider the positive case from equation (16). Using , equation (8) becomes

If , and using the same substitution for , we obtain

By introducing the constant and a new parametric variable which is defined by then, equation (18) can be written as and on introducing another constant , we obtain

By the integration and the boundary condition at the point we obtain

We note that equation (22) can be considered as a parametric equation for in terms of the parameter and denote the value of at the point where with , and for Model I,

Similarly, we derive and in terms of the variable , we have

Integrating (24) gives where and denote the usual Legendre incomplete elliptic integrals of the first and second kinds, respectively. Using equations (22) and (25), we obtain

From the definition of the arc length, we have

Upon substituting , changing the parameter to as in and integrating, we have

Now, we define a dimensionless parameter which can be shown to be where and For prescribed values of , , and , equation (29) can be solved numerically to determine the value of . Then, substituting into equation (28), the value of can be determined, and therefore, can be obtained from equation (28).

##### 2.2. Model II: Positive and Negative Curvatures

For the first region, the curvature is positive at up until the point , where the curvature changes sign to be negative until the point is reached as in Figures 5 and 6. At , and by solving equation (17), we have noting that from geometrical considerations, we have . By making the substitution used in equation (19) for , we have . By substituting into equations (22) and (25), we can calculate and , noting that and are the complete elliptic integrals of the first and second kinds, respectively. In the second joining region which is between and , we take the negative sign of equation (16), and following the same method used in Model I, we obtain

As a result, from the boundary conditions at the point , we know that By substitution into (32), we find

In this model, the arc length is determined in two parts, which is given by

As a result, we find a dimensionless parameter for this case to be

Also, for prescribed values of and we can solve equation (36) numerically to find the value of , and by substituting into equation (35), we can determine the value of so that can be determined from equation (34) [12]. Here, we note that equation (29) coincides with (36) for the value and the value of at this point is

#### 3. Results

In this section, we investigate the numerical solutions for equations (29) and (37) when they are characterized by the nondimensional parameter subjecting to the constraint as shown in [12]. Figure 7 shows that the three distinct regions are evident: the first one is where the parameter is negative that is corresponding to Model I; the second region is which also corresponds to Model I; and the third region is which corresponds to Model II.

Results for the joining between BN graphene and BN nanotube for both models I and II are shown in Figures 8 and 9, respectively. Furthermore, Figures 10 and 11 illustrate the join profile for BN graphene and BN nanocone with both models.

#### 4. Conclusion

In this paper, conventional applied mathematical modelling is considered to identify an approximate analytical solution to the shape of a surface joining BN graphene to other BN nanostructures. The resulted improved nanostructures could be useful for the design of probes for scanning tunnelling microscopy and other nanoscale devices. Using calculus of variations, this research minimizes the elastic energy of the joining curve which relates to minimization of the covalent bond energy. In particular, the joining is considered based on two different models that depend on the sign of the curvature: Model I refers to positive curvature only and Model II refers to positive and negative curvatures. The main purpose of this work is to formulate the axially symmetric model to have a reference basis for the comparison of real physical structures. As a result, these models lead to significative approximations to complex structures that are formulated by these certain BN nanostructures, BN graphene with two different BN nanostructures, BNNTs and BNNCs. Finally, other different BN nanostructures can be joined using the same model, and this is an area for future work.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author declares no competing interests.

#### Authors’ Contributions

N.A. wrote the main manuscript text, prepared all figures, and reviewed the manuscript.