# Joining between Boron Nitride Nanocones and Nanotubes

Authors: Nawa Alshammari
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

Different nanostructures of boron nitride have been observed experimentally such as fullerenes, tubes, cones, and graphene. They have received much attention due to their physical, chemical, and electronic properties that lead them to numerous applications in many nanoscale devices. Joining between nanostructures gives rise to new structures with outstanding properties and potential applications for the design of probes for scanning tunneling microscopy and other nanoscale devices, as carriers for drug delivery and liquid separation. This paper utilizes calculus of variations to model the joining between two types of BN nanostructures, namely, BN nanotubes and BN nanocones. Based on the curvature of the join curve, the joining of these structures can be divided into two models. Model I refers to when the join profile includes positive curvature only, and Model II contains both positive and negative curvatures. The main goal here is to formulate the basic underlying structure from which any such small perturbations can be viewed as departures from an ideal model. For this scenario of joining, we successfully present simple models based on joining BN nanotubes to BN nanocones with five different angles of the cone.


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## Body

## 1. Introduction

Boron nitride (BN) nanostructures have been the focus of extensive research area in current years. BN nanostructures are chemical compounds containing atoms of boron and nitrogen. They are similar to carbon nanomaterials in their geometrical structures and vary in their physiochemical properties. In particular, BN in nanoscale materials are important because of their electronic, optical, mechanical, and magnetic properties. Moreover, they have high oxidation resistance, high thermal conductivity, constant wide band gap, and lower toxicity. This makes them a promising candidate for application in different conditions [1]. Similar to carbon nanostructures, BN in nanoscale has different structures such as nanotubes, sheets, fullerenes, and nanocones [2].Boron nitride nanotubes (BNNTs) were first predicated in 1994 and synthesized in 1995. BNNTs can be created by rolling graphene sheet as a cylinder. They have significant attention in recent years due to their different physical properties. For example, they have high thermal conductivity, mechanical strength with an elastic modulus of 1.2 TPa, and wide band gap of about 5.5 eV with exceptional radiation shielding compared to carbon NTs [3, 4]. In addition, they are stable in air and in an inert atmosphere. Based on these fantastic properties, BNNTs have different applications such as in biomedical applications specially in drug delivery and bone scaffolding, electronic and microelectronic mechanical devices, and energy storage [3, 5, 6]. Also, they can be used as alternative to CNTs to help enhance strength of materials [4].BN nanocones are another form of BN nanostructures. BNNCs are discovered in 1994 as cap nanotubes end; then, they are synthesized as free structures by different groups. BNNCs can be made by rolling nanostructure sheet [1, 7]. Their physical, chemical, and electronic features such as chemical oxidation inertness, mechanical toughness, and thermal stability attract researchers’ more attention to explore their potential applications in different aspects [2, 8].Joining nanostructures can enhance the physiochemical and trochemical performance of the joined structures with new applications. In particular, the new produced structures are useful for the design of probes for scanning tunneling microscopy, energy storage, and other electronic devices as carriers for drug delivery [9].Researches in [9], [10–13] minimize the elastic energy which depends on the axial curvature only by using calculus of variations to determine the joining curve between two carbon nanostructures. Following [9–14], this paper extends this model to determine the joining curve between two boron nitride nanostructures: BN nanotubes and BN nanocones. We comment that this model does not take into account the chemical issues, such as positions of atoms and bonds. Finally, using Willmore energy which depends on the axial and rotational curvatures to determine the joining between nanostructures gives rise to similar joining profiles of using elastic energy as studied in [15]. Furthermore, similar techniques have been used and investigated by many researches such as in [16–18] and [19].In the following section, we state the fundamental equations of the calculus of variations to model the joining region between BNNTs and BNNCs. Namely, Model I denotes positive joining curvature while Model II denotes positive and negative joining curvatures. Results and discussion are given in Section3. Section 4 provides the conclusion.


## 2. Model

In this section, the basic variational equations of the model that is used to join boron nitride nanocone and boron nitride nanotube are formulated. In particular, variational calculus is used to specify the curve adopted by a line smoothly connecting BN nanocone base to a vertical BN nanotube, where the arc length of the joining curve and the defect site at the BN nanocone base are specified. Thus, the distance iny-direction y0 of the join to the cylindrical tube is not specified and it is found as a part of the solution.Using calculus of variations to find the curveyx, with an element of arc length ds, which minimizes the energy functional Jy that is given by
(1)Jy=∫0ℓκ2ds+λ∫0ℓds,where κ is the curvature, λ is a Lagrange multiplier corresponding to the fixed length constraint, and ℓ is the length of the joining curve. The boundaries of the join region are x0 and x1, where at x=x0; we have s=0, and at x=x1; we have s=ℓ. For a curve in two dimensions described as a graph y=yx, we have κ=y¨/1+ẏ23/2, and ds=1+ẏ21/2dx, so that equation (1) becomes
(2)Jy=∫absinγ/2y¨21+ẏ25/2dx+λ∫absinγ/21+ẏ21/2dx,where throughout this paper, dot denotes differentiation with respect to x. Applying the delta variational operator and integration by parts twice, the standard equation can be written as
(3)δJy=Fẏ−ddxFy¨δy+Fy¨δẏabsinγ/2+∫absinγ/2Fy−ddxFẏ+d2dx2Fy¨δydx,where subscripts denote partial derivatives and here the function F is given by
(4)Fẏ,y¨=y¨21+ẏ25/2+λ1+ẏ21/2.By imposing the continuity of the functiony and its derivatives, the boundary conditions at the join to the nanocone can be determined as
(5)ybsinγ2=bcosγ2,ẏbsinγ2=cotγ2.As the height of the nanotubey0 is unknown, we require the natural or alternative boundary condition given by
(6)Fẏ−ddxFy¨x=a=0.The value ofẏ, in Model I, ranges from cotγ/2, at x=bsinγ/2, to ∞ at x=a. Thus, the boundary condition for this model is ẏa=∞. For Model II, ẏ ranges from cotγ/2 to ∞, where it changes sign and then ranges from −∞ down to some finite negative value before turning to −∞. Thus, the boundary condition in the case of Model II is ẏa=−∞. From equation (3) the usual Euler-Lagrange equation for Fx,y,ẏ,y¨, is given by
(7)Fy−ddxFẏ+d2dx2Fy¨=0.Solving this equation and using the above alternative boundary condition,(8)F−y¨Fy¨=−α,where α is an arbitrary constant of integration. Now, substituting (4) into (8), the curvature κ can be written as in [11],
(9)κ=±λ+α1+ẏ21/21/2.


### 2.1. Model I: Positive Curvature

The curvature in Model I is positive along the entire arc lengthℓ as shown in Figure 1(a). Based on that, the positive case from (9) is considered only. By using ẏ=tanθ, equation (9) becomes
(10)κ=λ+αcosθ1/2.Figure 1
Model I: positive curvature and Model II: positive and negative curvatures [11].
(a)
Model I
(b)
Model IIFrom the definition of the curvatureκ=y¨/1+ẏ23/2, making the same substitution for ẏ, and simplifying that by introducing a new parametric variable ϕ which is defined by
(11)cosθ=1−2k2sin2ϕ,where k=γ+α/2α1/2, that gives
(12)dydϕ=2βksinϕ,where β=2/α1/2. Now by integrating this equation and using the boundary conditions at the point bsinγ/2,bcosγ/2 of attachment to the nanocone to find the constant of the integration, we have
(13)yϕ=2βkcosϕ0−cosϕ+bcosγ2,where
(14)ϕ0=sin−11−sinγ/22k21/2,corresponds to θ=π/2−γ/2, at the point bsinγ/2,bcosγ/2. Now if we use the boundary condition of the tube open end at the point a,y0 with ϕt=sin−11/2k, where θ=π/2, we find
(15)y0=2βkcosϕ0−cosϕ+bcosγ2,and in this model θ∈π/2−γ/2,π/2. Similarly, we derive
(16)xϕ=bsinγ2+β2Eϕ,k−Eϕ0,k−Fϕ,k−Fϕ0,k,where Fϕ,k and Eϕ,k denote the usual Legendre incomplete elliptic integrals of the first and second kinds, respectively. Using the boundary condition at the point a,y0 on the open tube end, we have
(17)a−=β2Eϕt,k−Eϕ0,k−Fϕt,k−Fϕ0,k.From the definition of the arc length, we have(18)ℓ=∫bsinγ/2a1+ẏ21/2dx.Upon substitutingẏ=tanθ, changing the parameter to ϕ as in cosθ=1−2k2sin2ϕ, and integrating, we have
(19)ℓ=βFϕt,k−Fϕ0,k.Now, substitute equation (19) into equation (17), we derive
(20)μ=2Eϕt,k−Eϕ0,kFϕt,k−Fϕ0,k−1,where μ=a−bsinγ/2/ℓ, and −1<μ<1. For prescribed values of a, b, γ, and ℓ, equation (20) can be solved numerically to determine the value of k. Then, substituting k into equation (19), the value of β can be determined, and therefore, y0 can be obtained from (15) [11].


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

Two regions are considered in this model as shown in Figure1(b); the first one involves positive curvature from the point of attachment at the nanocone bsinγ/2,bcosγ/2 up until the critical point xc,yc where the sign of the curvature changed. In this case, we follow the same process as in the last section. The second region starts from the critical point xc,yc to the point of attachment at the nanotube a,y0 where the curvature is negative. The parameter value of θ as defined in the previous section at the critical point is denoted by θc, and from geometrical considerations, we have 0<θc<π/2. The same process is used to the region of positive curvature where this region is bounded by the point of the curvature κ=0; using equation (10), we find
(21)θc=cos−1−λα.Applying the parameter variableϕ as defined by equation (11), we obtain ϕ=π/2, when θ=θc, and from equation (13) and equation (16), we have the parametric equations for yc and xc as
(22)yc=2βkcosϕ0+bcosγ2,(23)xc=bsinγ2+β2Ek−Eϕ0,k−Kk−Fϕ0,k,where β, k, and ϕ0 are defined in the last section. Fϕ0,k and Eϕ0,k are defined as the usual incomplete elliptic integrals of the first and second kinds, respectively, and kk and Ek are complete elliptic integrals of the first and second kinds, respectively.Similarly for the second region, considering the negative sign of equation (9) and integrating, we obtain
(24)yϕ=2βkcosϕ0+cosϕ+bcosγ2,noting that when ϕ=π/2, the constant of integration arises from y=yc, and then we use equation (22) for yc. From the boundary condition at the point of attachment to the nanotube, we have ϕ=ϕt, at the point a,y0, so we have
(25)y0=2βkcosϕ0+cosϕt+bcosγ2.Similarly, we take the negative sign of equation (9) and solving for the parametric form of x, we find
(26)xϕ=bsinγ2+β22Ek−Eϕ,k−Eϕ0,k−2Kk−Fϕ,k−Fϕ0,k.At the pointa,y0 where ϕ=ϕt, we have
(27)a−bsinγ2=β22Ek−Eϕt,k−Eϕ0,k−2Kk−Fϕt,k−Fϕ0,k.From the two regions, we obtain the arc length constraint as(28)ℓ=∫bsinγ/2xc1+ẏ21/2dx+∫xca1+ẏ21/2dx,by using ẏ=tanθ, and changing to the parameter ϕ,
(29)ℓ=β2Kk−Fϕt,k−Fϕ0,k,(30)μ=22Ek−Eϕt,k−Eϕ0,k2Kk−Fϕt,k−Fϕ0,k−1.We can solve equation (30) numerically to find the value of k, and by substituting k into equation (29), we can determine the value of β so that y0 can be determined. Noting that equation (20) coincides with (30) for the value k=1/2, and k=1−sinγ/2/212. For k=1/2, the value of μ is denoted by μ1, and when k=1/2, and k=1−sinγ/2/21/2, the value of μ is denoted by μ2 [11].


## 2.1. Model I: Positive Curvature

The curvature in Model I is positive along the entire arc lengthℓ as shown in Figure 1(a). Based on that, the positive case from (9) is considered only. By using ẏ=tanθ, equation (9) becomes
(10)κ=λ+αcosθ1/2.Figure 1
Model I: positive curvature and Model II: positive and negative curvatures [11].
(a)
Model I
(b)
Model IIFrom the definition of the curvatureκ=y¨/1+ẏ23/2, making the same substitution for ẏ, and simplifying that by introducing a new parametric variable ϕ which is defined by
(11)cosθ=1−2k2sin2ϕ,where k=γ+α/2α1/2, that gives
(12)dydϕ=2βksinϕ,where β=2/α1/2. Now by integrating this equation and using the boundary conditions at the point bsinγ/2,bcosγ/2 of attachment to the nanocone to find the constant of the integration, we have
(13)yϕ=2βkcosϕ0−cosϕ+bcosγ2,where
(14)ϕ0=sin−11−sinγ/22k21/2,corresponds to θ=π/2−γ/2, at the point bsinγ/2,bcosγ/2. Now if we use the boundary condition of the tube open end at the point a,y0 with ϕt=sin−11/2k, where θ=π/2, we find
(15)y0=2βkcosϕ0−cosϕ+bcosγ2,and in this model θ∈π/2−γ/2,π/2. Similarly, we derive
(16)xϕ=bsinγ2+β2Eϕ,k−Eϕ0,k−Fϕ,k−Fϕ0,k,where Fϕ,k and Eϕ,k denote the usual Legendre incomplete elliptic integrals of the first and second kinds, respectively. Using the boundary condition at the point a,y0 on the open tube end, we have
(17)a−=β2Eϕt,k−Eϕ0,k−Fϕt,k−Fϕ0,k.From the definition of the arc length, we have(18)ℓ=∫bsinγ/2a1+ẏ21/2dx.Upon substitutingẏ=tanθ, changing the parameter to ϕ as in cosθ=1−2k2sin2ϕ, and integrating, we have
(19)ℓ=βFϕt,k−Fϕ0,k.Now, substitute equation (19) into equation (17), we derive
(20)μ=2Eϕt,k−Eϕ0,kFϕt,k−Fϕ0,k−1,where μ=a−bsinγ/2/ℓ, and −1<μ<1. For prescribed values of a, b, γ, and ℓ, equation (20) can be solved numerically to determine the value of k. Then, substituting k into equation (19), the value of β can be determined, and therefore, y0 can be obtained from (15) [11].


## 2.2. Model II: Positive and Negative Curvatures

Two regions are considered in this model as shown in Figure1(b); the first one involves positive curvature from the point of attachment at the nanocone bsinγ/2,bcosγ/2 up until the critical point xc,yc where the sign of the curvature changed. In this case, we follow the same process as in the last section. The second region starts from the critical point xc,yc to the point of attachment at the nanotube a,y0 where the curvature is negative. The parameter value of θ as defined in the previous section at the critical point is denoted by θc, and from geometrical considerations, we have 0<θc<π/2. The same process is used to the region of positive curvature where this region is bounded by the point of the curvature κ=0; using equation (10), we find
(21)θc=cos−1−λα.Applying the parameter variableϕ as defined by equation (11), we obtain ϕ=π/2, when θ=θc, and from equation (13) and equation (16), we have the parametric equations for yc and xc as
(22)yc=2βkcosϕ0+bcosγ2,(23)xc=bsinγ2+β2Ek−Eϕ0,k−Kk−Fϕ0,k,where β, k, and ϕ0 are defined in the last section. Fϕ0,k and Eϕ0,k are defined as the usual incomplete elliptic integrals of the first and second kinds, respectively, and kk and Ek are complete elliptic integrals of the first and second kinds, respectively.Similarly for the second region, considering the negative sign of equation (9) and integrating, we obtain
(24)yϕ=2βkcosϕ0+cosϕ+bcosγ2,noting that when ϕ=π/2, the constant of integration arises from y=yc, and then we use equation (22) for yc. From the boundary condition at the point of attachment to the nanotube, we have ϕ=ϕt, at the point a,y0, so we have
(25)y0=2βkcosϕ0+cosϕt+bcosγ2.Similarly, we take the negative sign of equation (9) and solving for the parametric form of x, we find
(26)xϕ=bsinγ2+β22Ek−Eϕ,k−Eϕ0,k−2Kk−Fϕ,k−Fϕ0,k.At the pointa,y0 where ϕ=ϕt, we have
(27)a−bsinγ2=β22Ek−Eϕt,k−Eϕ0,k−2Kk−Fϕt,k−Fϕ0,k.From the two regions, we obtain the arc length constraint as(28)ℓ=∫bsinγ/2xc1+ẏ21/2dx+∫xca1+ẏ21/2dx,by using ẏ=tanθ, and changing to the parameter ϕ,
(29)ℓ=β2Kk−Fϕt,k−Fϕ0,k,(30)μ=22Ek−Eϕt,k−Eϕ0,k2Kk−Fϕt,k−Fϕ0,k−1.We can solve equation (30) numerically to find the value of k, and by substituting k into equation (29), we can determine the value of β so that y0 can be determined. Noting that equation (20) coincides with (30) for the value k=1/2, and k=1−sinγ/2/212. For k=1/2, the value of μ is denoted by μ1, and when k=1/2, and k=1−sinγ/2/21/2, the value of μ is denoted by μ2 [11].


## 3. Numerical Results

In this section, we investigate the numerical solution for Model I and Model II when they are characterized by the nondimensional parameterμ=a−bsinγ/2/ℓ. Figure 2 shows the relation between μ=a−bsinγ/2/ℓ, where −1<μ<1, and B=1/k2. There are two main regions; the first region is determined when μ<μ0, and the value of B<2, where μ0 is the asymptotic value for μ when k tends to be zero, given by [11]
(31)μ0=1+21−2cosωln2−1/tanω/2,where ω=π−γ/4. This region may be divided into three subregions. The first subregion is when μ3<μ<μ0, where μ3 is the asymptotic value of μ when k tends to be ∞ [11]
(32)μ3=2γ1−cosγ2.Figure 2
Relation between the parametersμ=a−bsinγ/2/ℓ, and B=1/k2.The value of the parameterB in this subregion is negative with a negative value of α an imaginary modulus k for the elliptic functions. The second subregion is μ1<μ<μ3, where the value of the parameter B corresponds to 0<B≤2, with a positive value of α and real k. The third subregion is −1<μ<μ1, where the value of the parameter B is between 1<B≤2, and positive value for α.The second region is whenμ>μ0, where the value of B>2/1−sinγ/2. This region also involves two subregions; the first subregion is μ0<μ<μ2, which corresponds to Model I, and the second subregion is μ2<μ<1, which corresponds to Model II. The values of the parameter B for these two subregions are positive with a negative value of α.Based on the above results, we join nanocones to nanotubes for Model I and Model II as in Figures3 and 4, respectively. In particular, here, we assume that heights of the cone are equal and the cone radii are found from r=htanγ/2. The five possible nanocones have fixed arc length which assumed to be ℓ=3.Figure 3
Plot of the joiny=yx, for Model I for nanocones with apex angles of 180∘, 112.9∘, 83.6∘, 60∘, 38.9∘, and 19.2∘, ℓ=3, and B=−10.Figure 4
Plot of the joiny=yx, for Model II for nanocones with apex angles of 180∘, 112.9∘, 83.6∘, 60∘, 38.9∘, and 19.2∘, ℓ=3, and B=1.5.


## 4. Conclusion

This paper uses conventional applied mathematical modelling in an essentially discrete problem of determining the profiles of the joins between boron nitride nanotubes and boron nitride nanocones. These new combined nanostructures may be useful for the design of probes for scanning tunneling microscopy and other nanoscale devices. In particular, calculus of variations is used to minimize the elastic energy for the joining curve which leads to minimize the covalent bond energy. During this joining, two models are considered depending on the sign of the curvature: Model I refers to the positive curvature only, and Model II refers to both positive and negative curvatures. By considering these two models, we join two boron nitride nanostructures which are nanotubes and nanocones. Because of the real physical composite structures which have undulations included from pentagons in the cone, this system is assumed to be axially symmetric and that is the problem that may be reduced to two dimensions. The main purpose here is to formulate the axially symmetric model to have a reference basis for the comparison of real physical structures where it is believed that the undulations are small. As a result, these models lead to significative approximations to complex structures.



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