1. Introduction

Carbon nanostructures have transformed materials science and nanotechnology, providing new opportunity to deal with difficulties across a wide range of scientific and industrial sectors. Since their discovery, carbon nanoparticles (CNPs) have gained massive interest from the scientific community because of their outstanding versatility. Through the numerous carbon-based nanoscale structures, like graphene sheets (CGSs), nanotubes (CNTs), nanocones (CNCs), nanotori (CNTRs), and fullerenes (CNFs), their exceptional properties including high electrical and thermal conductivities, unusual flexibility, and remarkable tensile strength, have allowed for revolutionary developments in energy storage, biotechnology, sensing, actuation systems, and microelectronics [1–4]. Latest studies focus in increasing applicability of nanostructures in numerous areas by energy effectiveness method as in [5] and [6]. The above characteristics may increase applications in preceding technologies as gigahertz oscillators, gas reduction, gene delivery methods, and biosensing devices [7–13]. Furthermore, the improvement of computational modeling proves the consuming of the energy and capacity of these nanostructures [14].

CNTRs, a special class of toroidal-shaped carbon-based nanostructures, have become an interested structure in the scientific community because of their unique electrical and magnetic characteristics. The process of CNTRs fabrication might be achieved by joining sliced nanotube segments, reshaping fullerene structures or joining open-ended CNTs [12, 15, 16]. In the aspect of magnetic, these distinguished materials have special manner, for example as a response to changes in magnetic flux, they provide a fixed current [17–20]. Additionally, they are attractive options for biomolecular design and complex drug delivery systems due to their potential application as charge-free caged structures [21, 22].

The merging of carbon nanostructures states a significant frontier in nanotechnology, authorizing the production of hybrid systems with enhanced properties [23, 24]. Merged nanostructures have confirmed enhanced electrochemical and physicochemical performance in contrast to their individual components, giving rise to breakthroughs in nanosensors, nanooscillators, energy storage devices, and drug delivery systems [25–29]. Energy optimization methods have improvement progress that emphasizes the applications of the hybrid structures in nanotechnology and beyond [30]. As a result of the growth and developments in this topic, researches have studied different techniques in synthesizing carbon nanostructures with conformation on theoretical modeling and experimental aspects.

To identify the ideal curves in connection cases, elastic energy minimization technique is used to determine the connection area between two-dimensional (2D) nanostructures, including CNTs with (CGSs, CNCs, and CNFs), CNFs with (CNFs and CGSs), and two CNCs [29, 31]. Different method applied to merge nanostructures is the Willmore energy method. Enhancing the equilibrium configurations for conjoining nanostructures has been proven by utilizing Willmore energy as a geometric optimization standard [32]. This method assumes that, as a general case of elastic energy, both rotational and axial curvatures are considered to determine the surface conjunction between two nanostructures in three dimensional (3D). Particularly, different structures have been merged based on this approach, such as CNTs with CNFs, CNTs with carbon nanotorus, and two CNFs [33–37]. This study pioneers in the consideration of carbon-based CNTRs connection by employing energy minimization aforementioned models. To model the geometry of the hybrid CNTRs, this approach combines the two curvatures. The axial curvature considers the compression or elongation throughout the produced configuration length. While the bending of torsion is considers by the rotational curvature. Using both curvatures, the proposed model promises fixed configurations with minimized energy that includes difference in segment radii and types of curvatures. The importance of this model can appear in CNTRs smooth and realistic designing which can be applied in drug delivery and molecular design. The purpose of this study is to provide models for improving carbon nanostructures. The potential contribution of the results is enhancing the development of the next-generation biomolecular designs and drug delivery systems. The growing interest in analytical and computational approaches for designing stable geometries is highlighted by recent studies that have investigated curvature-driven energy minimization in carbon nanostructures, including simulations of nanotorus stability and variations of the Willmore energy model for example [25, 32]. By include both axial and rotational curvatures, our study goes one step farther in this direction and provides a single model for hybrid structures.

The remainder of this study is organized as follows: The details of the mathematical model of the merging energies are presented in Section 2. The results of these methods are discussed in Section 3, and the conclusions are presented in Section 4.

2. Modeling Approach

2.1. Structure of Torus

“Torus equation is identified as $$, which may be transformed as, it yields x(ϕ, θ) = (R + a cosϕ)cosθ, y(ϕ, θ) = (R + a cosϕ)sinθ, and z(ϕ, θ) = a sinϕ, where θ is the polar angle and ϕ is the azimuthal angle in the x-axis and xy-plane. Moreover, R refers to the major radius of the torus and a refers to the minor radius” [25] (P. 4), as shown in Figure 1.

Therefore, its cylindrical form formulated as,

$$ ()

2.2. First Model: Energy Minimisation in Terms of Axial Curvature

This case formulates the connecting areas among nanostructures by calculus of variations. The lower torus had radii of a₁ as a minor and R₁ as a major centered at the origin. Furthermore, the upper torus had radii of a₂ as a minor and R₂ as a major, which collinear with y-axis, and y₀ is the starting unspecified distance overhead the xz-plane. In particular, L is the distance determined in the solution between the two structures in the y-direction. Assuming that the CNTRs are positioned symmetrically among axis of y, y₀ is positioned on the 2D xyplane. Referring to Figure 2.

A connecting curve with a defined arc length l connects the structures of the lower and upper tori at (a₁cosψ₁, a₁sinψ₁) and (a₂cosψ₂, L + a₂sinψ₂), respectively. Given that y = y(x) identical to the arc length ds, that is, the curve minimizes J₁[y] as

$$

where κ₁ is the axial curvature, κ₂ is the rotational curvature (which is assumed here to be zero), the curve length is l and λ indicates the Lagrange multiplier introduced to enforce the length constraint of the curve. Following the curvature energy framework outlined in [25],

$$ ()

which derived from the Euler–Lagrange equation with same parameter definitions as above and α is a constant is indicated by structural properties.

2.3. Second Model: Energy Minimisation in Terms of Axial and Rotational Curvatures

The previous model supposed the deformation based on perfect elasticity where the elastic energy considers the effect of the axial curvature only. Correspondingly, this case declares the effect of the rotational curvature in addition to the axial. The function of this case was utilised to find the joining surface among nanoscale forms. This is defined as the mean curvature H as summation of κ₁ and κ₂ as in,

$$

with Lagrange multiplier λ identical to the area constraint, and the area element dμ. As the catenoid can be considered as minimiser for this case, a part of it might be included in the connection scenario, as S = {(x, y, z) : x = r cosθ, y = r sinθ, z = f(r)}, Figure 3 shows an example of a catenoid surface.

“The mean curvature is expressed as follows:

$$

[25] (P. 3) If we assume the mean curvature equals to zero, this provides an absolute minimizer of this energy. Therefore, the solution is given as

$$

with arbitrary constants B₁ and B₂. Thus,

$$ ()

This function describes the catenoid as a connecting surface between the two nanostructures. A positive value indicates the up side of the catenoid, and a negative value denotes the lower catenoid part, as detailed in [36] which ensures C¹-smooth transitions and energy efficiency in the assembled structure.

The assumptions of this model are linear elasticity, moderate curvature, and fixed boundary conditions. Thus, they make it appropriate for ideal cases while it may be not suitable for powerful boundary, major deformations, and extreme curvature.

3. Results and Discussion

3.1. Results of First Model

First model identifies the connecting curve between the CNTRs. When y^′ ranged from (a₁ cosψ₁) to (a₂ cosψ₂), such that (−cotψ₁) to (−cotψ₂), the B.C. was y^′(a₂ cosψ₂) = −cotψ₂. Based on this, the joint curvature, in this situation, remains positive along the arc lengthl; therefore, positive sign of Equation (2) is examined only.

Suppose tan θ = y^′ in Equation (2). Then, the axial curvature κ₁ = (λ + α cosθ)^1/2. The curvature is defined as $$ and y^′ is considered.

$$

and

$$

Now we define a new parametric variable ϕ where cosθ = 1 − 2k² sin²⁡ϕ. Next if we integrate and apply the B.C. at attachment point (a₁ cosψ₁, a₁ sinψ₁), we find

$$

where ψ₁ is the initial angle of the connection with the lower torus and $$. However, from the B.C. at point (a₂ cosψ₂, L + a₂ sinψ₂) and $$, we obtain

$$ ()

where θ ∈ [ψ₁ − π/2, ψ₂ − π/2]. Now we will use the same steps as above for x,

$$

while F(ϕ, k) is the first type and E(ϕ, k) is the second type of typical Legendre incomplete elliptical integrals. Using the B.C. on the upper side of the torus at (a₂cosψ₁, L + a₂sinψ₂), we obtain

$$ ()

In the form of arc-length, we find that

$$

when y^′ = tanθ

$$ ()

Substituting Equation (6) in Equation (5) gives

$$ ()

with μ = (a₁ cosψ₁ − a₂ cosψ₂)/l. Equation (7) is solved numerically after determining a₁, a₂, ψ₁, and ψ₂ which then define the modulus k. As a result of this Equation (6) yields the angle β, and the length L is computed using Equation (4). These relationships are visualized in Figure 4, which explain the computational process used to derive a consistent catenoid profile based on the model. While not an experimental verification, this figure reflects a numerical construction of the surface that follows directly from the theoretical model.

For stability and reliability of the numerical solution, we use Maple fsolve function with different initial estimates with the correct range of k between 0 and 1. As a result, the findings demonstrated uniform convergence. Furthermore, a sensitivity analysis was conducted by changing the input parameters a₁, a₂, ψ₁, ψ₂, and L by a maximum of ±5%. The robustness of the numerical solution was confirmed by the ensuing fluctuations in β and L being within acceptable boundaries.

3.2. Results of Second Model

Here, we modelled the connecting curve to connect the nanostructures (CNTRs) using part of the catenoid. From Equation (3), the catenoid curve equation has the form

$$ ()

where B₁ and B₂ are the constants. In our model, the lower side of the catenoid joins the lower nanotorus while the upper side of the catenoid joins the upper nanotorus, refer to Figure 5. Moreover, throughout the joining curve and at point (r_c, 0), the gradient is ∞; therefore, we can write $$ which gives

$$ ()

At point (r₁, z₁), the negative sign of catenoid Equations (8) and (1) represent this case, from Equation (1) we have

$$ ()

where D₁ is the constant defined from the torus location among axis of z. Consequently, we obtain

$$

Using Equation (9), gives

$$ ()

When we match the gradient, $$, and from Equation (9), we have

$$

which yields

$$ ()

However, at point (r₂, z₂), we consider the positive sign of Equation (8) for the upper torus; then, Equation (1) can be expressed as

$$ ()

where D₂ is the constant defined from the torus location over the negative axis of z. Again, by applying the same calculations as above, we find that $$, which yields

$$ ()

Substituting Equation (9) into Equation (8), Equation (12) into Equation (10), and Equation (14) into Equation (13) with the specific values for r_c, R₁, a₁, R₂,and a₂, the connecting profile utilising the second model is found, as shown in Figure 6.

The outcomes of this research confirm that minimizing bending energy produces stable structures for hybrid CNTRs, mostly next to junctions with the variation of curvatures. While the using of the above curvatures shows the influence of the configuration stability by longitudinal and radial bending. This differ from the previous studies which suppose constant curvature or depend only on Willmore energy [25]. This approach considers actual geometries and provides elasticity in constructing nanostructures with mechanical features control. Our dual-curvature formulation captures more realistic geometrical transitions in hybrid CNTRs, especially when curvature discontinuities or varying radii are present, than recent models that only focus on uniform curvature or Willmore energy [25, 32]. The literature used the merging process of nanostructures depending on axial curvature only with symmetric transitions. However, the proposed model here incorporates both axial and rotational curvatures which provides more flexibility and actual of the produced structure.”

4. Conclusion

Conflicts of Interest

The author declares no conflicts of interest.

Author Contributions

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

Funding

The author did not receive support from any organization for the submitted work.