https://orcid.org/0000-0002-4709-3860
Figures
Abstract
Borophene nanosheets appear in various sizes and shapes, ranging from simple planar structures to complicated polyhedral formations. Due to their unique chemical, optical, and electrical properties, Borophene nanosheets are theoretically and practically attractive and because of their high thermal conductivity, boron nanosheets are suitable for efficient heat transmission applications. In this paper, temperature indices of borophene nanosheets are computed and these indices are employed in QSPR analysis of attributes like Young’s modulus, Shear modulus, and Poisson’s ratio of borophene nanosheets and borophene β12 sheets. The regression model for the F-Temperature index is discovered to be the best fit for shear modulus, the reciprocal product connectivity temperature index is discovered to be fit for Poisson’s ratio and the second hyper temperature index is discovered to be fit for Young’s modulus based on the correlation coefficient.
Citation: Khan AR, Ullah Z, Imran M, Malik SA, Alamoudi LM, Cancan M (2024) Molecular temperature descriptors as a novel approach for QSPR analysis of Borophene nanosheets. PLoS ONE 19(6): e0302157. https://doi.org/10.1371/journal.pone.0302157
Editor: Niravkumar Joshi, Federal University of ABC, BRAZIL
Received: November 15, 2023; Accepted: March 26, 2024; Published: June 18, 2024
Copyright: © 2024 Khan et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All data generated or analyzed during this study are included in this article.
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
1 Introduction
Boron was simultaneously discovered in 1808 by DAVY in England and France by GAY-LUSSAC, and THENARD used separate methods [1]. A unique element called boron shares a periodic table element with its neighbour, carbon, which can form bonds formed by atoms from the same element. This characteristic enables the generation of various boron hydrides and the heteroatom variants of hydrides, collectively called Borophene nanosheets [2, 3]. As the name indicates, Borophene nanosheets have a polyhedral cage-like structure compared to organic compounds, which are made up of chains and rings. Atoms of boron or heteroatoms that comprise this structure are engaged on multiple edges, ranging from the highly brittle quadrilateral Borophene nanosheet to the compounds built out of the octahedral Borophene nanosheet [4, 5]. This stability is due to the delocalization of bonds of the 3-dimensional aromatic structure of Borophene nanosheet compounds, which differs from the 2D aromaticity of organic compounds [6]. The astonishing structural variety of Borophene nanosheets has recently been uncovered thanks to experimental methods combined with theoretical analysis [7]. Despite the well-established two to three-dimensional transition thresholds, the parameters that determine the geometry that these systems acquire are unknown. Doping frequently causes structural changes that result in the lowest energy structure having an utterly different boron skeleton than the cluster without doping [8]. Borophene nanosheets have remarkable properties in antimicrobial therapy, drug design [9], cancer therapy [10], pharmacophores [11], used as broadband membrane carriers [12], for the synthesis of liquid crystals [13] and luminescent materials [14]. The effectiveness of Borophene nanosheet derivatives against common and multi-drug resistant disease strains is identical. They also exhibit anti-biofilm activity and are less likely to result in medication resistance. Dendrimers, coordinated polymers, metal-organic systems, polymer compounds, and non-linear optical materials have all been made using Borophene nanosheets. These clusters have been employed in nanoscience as the molecules’ fundamental components [15].
In 2015, researchers from a variety of domains were drawn to Borophene, a 2D boron sheet that was successfully synthesized on Ag substrates. It is the lightest 2D material and has similar properties to graphene. Borophene is superconductive and has the highest pressure and external strain among 2D materials. To assess its suitability for use in a variety of disciplines, a great deal of research has been done on its stability, electrical qualities, and chemical and structural complexity [16]. Borophene’s mechanical properties are significant, including low mass density, high strength, and high in-plane stiffness, making it suitable for composite design and flexible nano device fabrication. Its anisotropic structure allows for effective control of magnetic and electronic properties making it a suitable candidate in various applications. Borophene’s polymorphic nature and strong electron-phonon coupling make it a valuable material for future applications due to its resources, low cost, and excellent electrical performance [17, 18]. Despite the wide range of possible uses for Borophene, one of the most difficult tasks that limit its application involves the synthetic methodology and identification of its neoatomic forms with carefully thought-out structure-property connections. Furthermore, in the case of synthetic 2D materials, a variety of parameters, including component elements, processing conditions, and growth substrates, impact the final atomic structure. The assured synthesis of quality specimens and the separation of Borophene from substrates remain difficult and need ongoing theoretical and experimental work to achieve practical applications [19, 20]. With its good metal properties, Borophene has potential in energy-related applications such as lithium batteries and biosensors. Its electrical sensitivities make it suitable for sequencing DNA and gas sensors. However, its NIR absorption needs improvement. Despite extensive research, applications are limited due to low yield and limited nanosheet size [21].
Researchers have characterized the properties of Borophene, a 2-dimensional material, to find its advantages over graphene and other materials. Borophene is a superconductor that is more robust, flexible, and an excellent heat and electrical conductor. Depending on how the voids are arranged and oriented, their properties can be adjusted. Its amazing mechanical capabilities and orientationally adjustable qualities make a lighter, electron-rich Borophene an equivalent contender to graphene. Borophene is also a superior substitute for graphene in composite progress since it has a greater Young’s modulus. Compared to other 2D materials, it is a superior conductor of heat and electricity due to its remarkable anisotropic behaviour [22–24].
Kulli initiated in 2020 a graph’s temperature indices [25]. In chemical graph theory, these indices are used to quantify the chemical characteristics of chemical compounds. These indies are computed for a molecular graph silicate network and a silicate chain network and present valuable results in [26, 27]. Various graph-theoretic techniques and algorithms can be used to analyze the structure and behaviour of the chemical network once it has been converted into a graph [28]. This approach can be used to discover novel chemical reactions or processes and to understand the operation of complex chemical systems [29]. Chemical networks, physical, chemical, and thermal properties, as well as biological and chemical activities, can all be evaluated using chemical graph theory [30]. It is possible to determine a molecule’s chemical, physical, and biological properties using its topological index [31]. Graph theory is extensively employed in modern chemistry, mainly organic chemistry. Mathematical chemistry has demonstrated that polynomials and functions can uncover instructions concealed in the symmetry of chemical graphs. Topological indices can be used to evaluate QSPR, QSAR, and other chemical applications based on theory [32]. Diverse fields of study have investigated topological indices to fathom various types of graphs. In quantitative structure-activity and structure-property interactions, topological indices are employed as numerical descriptions to compare molecules’ biological, physical, and chemical aspects. Numerous researchers have examined various chemical compounds in recent years and computed topological descriptors of different molecular graphs.
Graph indices can be used for various purposes, including chemical record keeping, establishing relationships between structure and activity, property and relationship, and property and relationship [33]. Mathematicians have developed novel concepts with the use of graph indices [34].
We exclusively consider limited, simple, and connected graphs in this work. Assume that K is a graph, where EK represents the set of edges, and VK represents the set of vertices. The degree is determined by the number of vertices surrounding a point u. We advise the individual to search for simple terms and symbols. Fajtlowicz presented the temperature of a graph [35]:
(1)
The first temperature index is defined as below [36]:
(2)
The second temperature index was introduced by Kulli in 2020 in [37],
(3)
Kulli defined the first and second hyper temperature indices in [37],
(4)
(5)
The same study provides more related topological indices [37]. The reciprocal product connectivity index, the product connectivity temperature index, and the sum connectivity temperature index were given, respectively as
(6)
(7)
(8)
Kulli also examined the F-temperature index of a graph K in [37], as given below
(9)
In this article, the above-defined eight temperature indices are constructed by the atom-bonds partition of Borophene nanosheet .
2 Borophene nanosheet 
Borophene nanosheets are three-dimensional formations comprised of boron atoms that are small in size. They appear in various sizes and shapes, ranging from simple planar structures to complicated polyhedral formations. Due to their unique chemical, optical, and electrical properties, Borophene nanosheets are theoretically and practically attractive. Two-dimensional Borophene nanosheet, composed of Borophene nanosheets, has recently received greater interest [38]. Due to their high thermal conductivity, boron nanosheets are suitable for efficient heat transmission applications. Due to their triangular arrangement of the boron atoms, strong covalent bonding occurs, providing the sheet tremendous strength and stiffness. Borophene nanosheets have unique electronic properties because of their 2D structure and boron atoms’ electrical configuration, making them useful in electronic and optoelectronic devices. Because of the sheet’s closely bound boron atoms, it is relatively secure and impenetrable to chemical reactions [39]. Various industries, like electronics, optoelectronics, and energy storage, may utilize boron for their products [40–42]. The Borophene nanosheet, borophene’s physical and chemical characteristics, and its application are given in [43–50]. Borophene nanosheet γ-sheet for the l = 6, m = 3 is sketched in Fig 1 and for l = 2, m = 3 is drawn in Fig 1. Here, in Borophene nanosheet , there are nine types of atom-bonds based on the valency of every atom of
, which are (3 ∼ 3), (3 ∼ 4), (3 ∼ 5), (3 ∼ 6), (4 ∼ 4), (4 ∼ 5), (4 ∼ 6), (5 ∼ 5) and (5 ∼ 6) in
. Tables 1 and 2 represent atom bond partitions based on valency. The atom-bonds partition of
given as: The sum of all the atoms and the bonds of
:
Using edge partition given in Tables 1 and 2 along with Eq (1), we get Tables 3–5 for
.
2.1 Temperature indices for Borophene nanosheet 
In this section, we present some computed results.
- 1. Let
be a Borophene nanosheet. Then the first temperature index of
is
.
Tables 3–5 along with Eq (2), implies
- 2. Let
be a Borophene nanosheet. Then the second temperature index is
.
Tables 3–5 along with Eq (3), gives
After simplification, we get
(10)
- 3. Let
be Borophene nanosheet. Then the first hyper temperature index is
.
From Tables 3–5 along with Eq (4), we obtain
- 4. Let
be Borophene nanosheet. Then the second hyper temperature index is
From Tables 3–5 along with Eq (5), we attain
- 5. Let
be Borophene nanosheet. Then the sum connectivity temperature index is
From Tables 3–5 along with Eq (8), we get
- 6. Let
be Borophene nanosheet. Then the product connectivity temperature index is
From Tables 3–5 along with Eq (7), we obtain
- 7. Let
be Borophene nanosheet. Then the reciprocal product temperature index is
From Tables 3–5 along with Eq (6), we get
- 8. Let
be Borophene nanosheet. Then the F-temperature index is
From Tables 3–5 along with Eq (9), we have
2.2 Comparative analysis of temperature indices for 
Numerical and graphical comparison of computed temperature indices of for l = m is presented in this section. As the value of l or m, increases gradually, the value of temperatures indices HT2, ST and PT increases gradually while the numerical value of temperature indices T1, T2, HT1, RPT and FT decreases gradually. These changes are also represented in Table 6 and Fig 2.
3 Borophene β12-sheets 
Fig 3 is a graph borophene β12-sheet of for the particular values of l = 2 and m = 3. Here, in Borophene β12-sheet , there are eleven types of atom-bonds based on the valency of every atom of
, which are (2 ∼ 2), (2 ∼ 4), (3 ∼ 3), (3 ∼ 4), (3 ∼ 5), (3 ∼ 6), (4 ∼ 4), (4 ∼ 5), (4 ∼ 6), (5 ∼ 5) and (5 ∼ 6) in
. Based on valency, Tables 7 and 8 provide the partition of the atom-bonds of
shown as:
The total number of atoms and atom-bonds :
By using Eq (1) and above partition of
, as described in Tables 7 and 8.
3.1 Temperature indices for Borophene β12-sheet 
- 9. Let
be a Borophene β12-sheet. Then the first temperature index of
is
.
Tables 9–12 along with Eq (2), gives
- 10. Let
be a Borophene β12-sheet. Then the second temperature index is
.
Tables 9–12 along with Eq (3), implies
- 11. Let
be Borophene β12-sheet. Then the first hyper temperature index is
.
From Tables 9–12 along with Eq (4), we obtain
- 12. Let
be Borophene β12-sheet. Then the second hyper temperature index is
From Tables 9–12 along with Eq (5), we get
- 13. Let
be Borophene β12-sheet. Then the sum connectivity temperature index is
.
From Tables 9–12 along with Eq (8), we have
- 14. Let
be Borophene β12-sheet. Then the product connectivity temperature index is
From Tables 9–12 along with Eq (7), we obtain
From Tables 9–12 along with Eq (6), we attain
- 16. Let
be Borophene β12-sheet. Then the F-temperature index is
.
From Tables 9–12 along with Eq (9), we obtain
3.2 Comparative analysis of temperature indices for 
In this section, we perform a numerical and graphical comparison of temperature indices for n = 2, 3, 4, …, 12 of a Borophene β12-sheet . As we increase the value of l or m, the value of temperatures indices T1, T1, T2, HT1, RPT and FT decreased gradually while the numerical values of the temperature indices HT2, ST and PT increased gradually. These changes are also represented in Table 13 and Fig 4.
4 Quantitative structure-activity property relationship model
Topological descriptors are numerical depictions of chemical compounds with information about the structure’s topology. Various descriptors have been learned and are often employed in QSPR to predict chemical characteristics and biological activities. These descriptors include physicochemical, constitutional, electrostatic, geometrical, and topological properties. The process begins with an appropriate molecular topological descriptor and concludes with an inference, hypothesis, or prediction about the molecule’s behaviour, characteristics, and attributes. High-quality experimental data must be available and accurate for this investigation to be successful. Determining a suitable topological descriptor for modeling in the analysis is the essential step in the QSPR/QSAR process. Since there is no consensus regarding the ideal molecular description, all conceivable descriptors are chosen. The capacity to predict a chemical compound’s properties, activities, and behaviour based on the molecular structure of related compounds whose properties, activities, and characteristics have already been assessed is a vital use of QSPR/QSAR models. Numerical values of the properties of borophene nanosheets are given in Table 14.
4.1 Linear regression model
Regression analysis aims to ascertain the significance and strength of the correlations via a numerical value provided by the topological index. Regression models come in a variety of forms, and which one you pick will depend on the kind of data you have and the issue you’re attempting to solve. The following are some popular varieties of regression models: exponential and logarithmic regression models, linear and polynomial regression models such as quadratic and cubic regression models, etc. We employ the linear regression models for a few properties of the borophene nanosheet obtained in this section. For the analysis, the model below is taken into account.
Where Y and K represent the property and structural descriptors. The analysis considers characteristics like Young’s modulus, Poisson’s ratio, and shear modulus. Regression analysis was done using the analytical features of Matlab. The regression model for the F-Temperature index is discovered to be the best fit for shear modulus based on the correlation coefficient (R), and the regression equation can be written as:
The regression formula for shear modulus for the first hyper temperature index can be written as:
The regression model for the reciprocal product connectivity temperature index is discovered to be fit for Poisson’s ratio based on the correlation coefficient (R), and the regression equation can be written as:
The regression model for the second hyper temperature index is discovered to be fit for Young’s modulus based on the correlation coefficient (R), and the regression equation can be written as:
Figs 5–11 depict regression models that show the relationship between attributes and their best-fitted descriptions. However, it can be concluded that the above-mentioned attributes of various boron variations can be anticipated using regression equations and calculated topological descriptors.
5 Conclusion
In this study, we computed degree-based temperature indices. We employed these indices to perform linear regression analysis to predict three physiochemical properties of borophene nanosheets: shear modulus, Young’s modulus, and Poisson’s ratio. We concluded that the linear regression analysis provides the following best-fit models with R2 value for shear modulus:
Young’s modulus:
and Poisson’s ratio:
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