Mathematical modeling of interaction energies between nanoscale objects: A review of nanotechnology applications

Abstract

In many nanotechnology areas, there is often a lack of well-formed conceptual ideas and sophisticated mathematical modeling in the analysis of fundamental issues involved in atomic and molecular interactions of nanostructures. Mathematical modeling can generate important insights into complex processes and reveal optimal parameters or situations that might be difficult or even impossible to discern through either extensive computation or experimentation. We review the use of applied mathematical modeling in order to determine the atomic and molecular interaction energies between nanoscale objects. In particular, we examine the use of the 6-12 Lennard-Jones potential and the continuous approximation, which assumes that discrete atomic interactions can be replaced by average surface or volume atomic densities distributed on or throughout a volume. The considerable benefit of using the Lennard-Jones potential and the continuous approximation is that the interaction energies can often be evaluated analytically, which means that extensive numerical landscapes can be determined virtually instantaneously. Formulae are presented for idealized molecular building blocks, and then, various applications of the formulae are considered, including gigahertz oscillators, hydrogen storage in metal-organic frameworks, water purification, and targeted drug delivery. The modeling approach reviewed here can be applied to a variety of interacting atomic structures and leads to analytical formulae suitable for numerical evaluation.

Introduction

For the past two decades, nanotechnology has been a major focus in science and technology. However, in various areas of physics, chemistry, and biology, both past and current research involving interacting atomic structures are predominantly either experimental or computational in nature. Both experimental work and large-scale computation, perhaps using molecular dynamics simulations, can often be expensive and time-consuming. On the other hand, applied mathematical modeling often produces analytical formulae giving rise to virtually instantaneous numerical data. This can significantly reduce the time taken in the trial-and-error processes leading to applications and which in turn significantly decreases the research cost. Here, applied mathematical modeling in nanotechnology is reviewed, and particularly, the work of the present authors and their colleagues in the use of classical mathematical modeling procedures to investigate the mechanics of interacting nanoscale systems for various applications, including nano-oscillators, metal-organic frameworks (MOFs), molecular selective separation, and drug delivery.

Throughout, the dominant mechanisms behind these nanoscale systems are assumed to arise from atomic and molecular interactions that can be modeled by the 6-12 Lennard-Jones potential function (see equation (4)), and further simplifications are made by adopting the continuous or continuum assumption. This approximation assumes that two interacting molecules can be replaced by two surfaces or two regions, for which the discrete atomic structure is averaged over the surface or the volume with a constant atomic surface density or a constant atomic volume density, respectively. Basically, the continuous assumption gives an average result, and it is much better suited to those situations involving well-defined surfaces with evenly distributed atoms, such as graphene, carbon nanotubes, or carbon fullerenes. In each of these instances, there exists a uniform distribution of atoms, and the continuous approximation might be most accurate. In the case of non-evenly distributed atomic structures, a hybrid approach is adopted, which deals with the isolated atoms individually, and the continuous approximation is adopted for the remainder. For example, a methane molecule CH₄ is assumed to be replaced by a spherical surface of a certain radius with a constant hydrogen atomic surface density, together with a single carbon atom located at the center of the spherical surface.^1,2

In this review, we comment that we do not include the mechanics of dislocations in metallic materials or the use of the Cauchy–Born rule to bridge interactions since the modeling here assumes that there is no deformation of any surface due to the van der Waals interactions. We refer the reader to Van der Giessent and Needleman ³ for a comprehensive study of plastic discrete dislocations and to Biner and Morris ⁴ for a computational simulation of the discrete dislocation method. Furthermore, a review of the Cauchy–Born rule can be found in Ericksen. ⁵

In the following section, both the 6-12 Lennard-Jones potential function and the continuous approximation are introduced. In the section thereafter, analytical expressions are presented for the interaction energies of the basic molecular building blocks, namely, points, lines, planes, rings, spheres, and cylinders, all deduced utilizing the 6-12 Lennard-Jones potential function together with the continuous approximation. In the section on the mechanics of nanostructures, the mechanics of the so-called gigahertz oscillators is reviewed, including the determination of the energy and force distributions of this nanostructured device. The development of a mathematical model of MOFs for gas storage is presented in the section thereafter. In the next section, the modeling approach is reviewed for molecular selectivity and separation for water purification, ion separation, and biomolecule selection. In the targeted drug delivery section, we present a review of applied mathematical modeling for targeted drug delivery. A brief overall summary is presented in the final section of this article.

Lennard-Jones atomic interaction potential and the continuous approach

For two separate non-bonded molecular structures, the interaction energy E can be evaluated either directly using a discrete atom–atom formulation or approximately using the continuous approach. Thus, the non-bonded interaction energy may be obtained either as a summation of the individual interaction energies between each atomic pair, namely

$$E={\displaystyle \sum _i{\displaystyle \sum _j\mathrm{\Phi }({\rho }_{ij})}}$$

(1)

where $\mathrm{\Phi }({\rho }_{\mathrm{ij}})$ is the potential function for atoms i and j located a distance ${\rho }_{\mathrm{ij}}$ apart on two distinct molecular structures, assuming that each atom on the two molecules has a well-defined coordinate position. Alternatively, the continuous approximation assumes that the atoms are uniformly distributed over the entire surface of the molecule, and the double summation in equation (1) is replaced by a double integral over the surface of each molecule, thus

$$E={\eta }_1{\eta }_2{\displaystyle \int }{\displaystyle \int \mathrm{\Phi }(\rho )d{S}_{1}d{S}_2}$$

(2)

where ${\eta }_1$ and ${\eta }_2$ represent the mean surface densities of atoms on the two interacting molecules, and $\rho$ is the distance between the two typical surface elements $d{S}_1$ and $d{S}_2$ located, respectively, on the two interacting molecules. Note that the mean atomic surface density is determined by dividing a number of atoms which make up the molecule by the surface area of the molecule. The continuous approximation is rather like taking the average or mean behavior, and in the limit of a large number of atoms, the continuous approximation approaches the energy arising from the discrete model.

The hybrid discrete–continuous approach applies to the modeling of irregularly shaped molecules, such as drugs, and constitutes an alternative approximation to determine the interaction energy. The hybrid approach is represented by elements of both equations (1) and (2) and can be effective when a symmetrical molecule is interacting with a molecule comprising asymmetrically located atoms. In this case, the interaction energy is given as follows

$$E={\displaystyle \sum _i\eta {\displaystyle \int \mathrm{\Phi }({\rho }_i)dS}}$$

(3)

where $\eta$ is the surface density of atoms on the symmetrical molecule, and ${\rho }_i$ is the distance between a typical surface element dS on the continuously modeled molecule and atom i in the molecule which is modeled as discrete. Again, $\mathrm{\Phi }({\rho }_i)$ is the potential function, and the energy is obtained by summing overall atoms in the drug or the molecule which is represented discretely.

The continuous approach is an important approximation, and Girifalco et al. ⁶ state that

From a physical point of view the discrete atom-atom model is not necessarily preferable to the continuum model. The discrete model assumes that each atom is the center of a spherically symmetric electron distribution while the continuum model assumes that the electron distribution is uniform over the surface. Both of these assumptions are incorrect and a case can even be made that the continuum model is closer to reality than a set of discrete Lennard-Jones centers.

One such example is a ${\mathrm{C}}_{60}$ fullerene, in which the molecule rotates freely at high temperatures so that the continuous distribution averages out the effect. Qian et al. ⁷ suggest that the continuous approach is more accurate for the case where the “C nuclei do not lie exactly in the center of the electron distribution, as is the case for carbon nanotubes.” However, one of the constraints of the continuous approach is that the shape of the molecule must be reasonably well defined in order to evaluate the integral analytically, and therefore, the continuous approach is mostly applicable to highly symmetrical structures, such as cylinders, spheres, and cones. Hodak and Girifalco ⁸ point out that for nanotubes, the continuous approach ignores the effect of chirality, so that effectively nanotubes are only characterized by their diameters. For the graphite-based and C₆₀-based potentials, Girifalco et al. ⁶ state that calculations using the continuous and discrete approximations give similar results, such that the difference between equilibrium distances for the atom–atom interactions is less than 2%. Hilder and Hill ⁹ undertake a detailed comparison of the continuous approach, the discrete atom–atom formulation and a hybrid discrete–continuous formulation, for a range of molecular interactions involving a carbon nanotube, including interactions with another carbon nanotube and the three fullerenes ${\mathrm{C}}_{60}$ , ${\mathrm{C}}_{70}$ , and ${\mathrm{C}}_{80}$ . In the hybrid approach, only one of the interacting molecules is discretized, while the other is considered to be continuous. The hybrid discrete–continuous formulation enables non-regular-shaped molecules to be described and is particularly useful for drug delivery systems which employ carbon nanotubes as carriers and discussed subsequently. The Hilder and Hill ⁹ investigation obtains estimates of the anticipated percentage errors which may occur between the various approaches in a specific application. Although, it is shown that the interaction energies for the three approaches can differ on average by at most 10%, while the forces can differ by at most 5%, with the exception of the ${\mathrm{C}}_{80}$ fullerene. For the ${\mathrm{C}}_{80}$ fullerene, while the intermolecular forces and the suction energies are shown to be in reasonable overall agreement, the pointwise energies may be significantly different. This is perhaps due to the differences in modeling the geometry of the ${\mathrm{C}}_{80}$ fullerene, noting that the suction energies involve integrals of the energy, and therefore, any error or discrepancy in the pointwise energy tends to be smoothed out to give reasonable overall agreement for the former quantities.

The continuum or continuous approximation has been successfully applied to a number of systems, including the interaction energy between nanostructures of various types and shapes, namely, carbon fullerenes,^6,10,11 carbon nanotubes,^6,12–21 carbon nanotube bundles,^22–24 carbon nanotori,^25–30 carbon nanocones,^31–34 carbon nanostacked cups, ³⁵ fullerene–nanotube,^8,36–46 and TiO₂ nanotubes.^47–49 Moreover, this method has also been used in systems involving proteins and enzymes,^50–52 DNA,^52–55 lipid bilayer and lipid nanotube,^56–58 water molecule,^59–64 benzene,^2,65–69 methane,^2,3,70–75 ions,^75–79 and gas storage and porous aromatic frameworks.^80–88

The Lennard-Jones potential function $\mathrm{\Phi }(\rho )$ which accounts for the interaction of two non-bonded atoms can be written in the following form

$$\mathrm{\Phi }\left(\rho \right)=-\frac{A}{{\rho }^6}+\frac{B}{{\rho }^{12}}=4\epsilon \left[-{\left(\frac{\sigma }{\rho }\right)}^6+{\left(\frac{\sigma }{\rho }\right)}^{12}\right]$$

(4)

where $A=4\epsilon {\sigma }^6$ and $B=4\epsilon {\sigma }^{12}$ are positive constants which are referred to as the Lennard-Jones constants. They are empirically determined and correspond to the constants of attraction and repulsion, respectively. Furthermore, $\sigma$ is the van der Waals diameter, and $\epsilon$ denotes the energy well depth. The equilibrium distance ${\rho }_0$ is given by ${\rho }_0=2^{1/6}\sigma =[(2B)/A{]}^{1/6}$ , where $\epsilon =A^2/(4B)$ , as shown in Figure 1. Moreover, when experimental information on particular atomic interactions is lacking, it is possible to use the so-called empirical combining laws or mixing rules, ⁸⁹ which have no theoretical basis but are nevertheless used in many calculations. Thus, if the parameters $\epsilon$ and $\sigma$ are known for the self-interactions of two distinct atomic species designated by $1$ and $2$ , then the parameters for atomic species $1$ interacting with atomic species $2$ are assumed to be given by the geometric and arithmetic means, namely, ${\epsilon }_{12}=({\epsilon }_1{\epsilon }_2{)}^{1/2}$ and ${\sigma }_{12}=({\sigma }_1+{\sigma }_2)/2$ . Following the work by Mayo et al. ⁹⁰ and Rappe et al., ⁹¹ some illustrative numerical values for the Lennard-Jones constants are given in Table 1.

Figure 1.

Lennard-Jones potential.

Table 1.

Numerical values for the Lennard-Jones constants taken from Mayo et al. ⁹⁰

Site–site	$\sigma$ (Å)	$\epsilon$ (kcal/mol)	A (eVÅ6)	B (eVÅ12)
H	3.1950	0.0152	1.4023	745.8187
O	3.4046	0.0957	12.9264	10,065.7103
N	3.6621	0.0774	16.1917	19,527.3227
C	3.8983	0.0951	28.9469	50,795.2337
B	4.0200	0.0950	34.7736	73,379.6427
P	4.1500	0.3200	141.7777	362,131.6551
Si	4.2700	0.3100	162.9665	493,896.0409
Ti	4.5400	0.0550	41.7703	182,882.5525
Fe	4.5400	0.0550	41.7703	182,882.5525
Zn	4.5400	0.0550	41.7703	182,882.5525

When the Lennard-Jones potential function $\mathrm{\Phi }(\rho )$ is used in the context of the integral formulation of equation (2), we observe that the attractive term ${\rho }^{-6}$ and the repulsive term ${\rho }^{-12}$ can be separated and integrated independently. Furthermore, the two terms only vary in the coefficients A and B and the magnitude of the index, applying to the distance variable $\rho$ . Accordingly, for convenience, the Lennard-Jones potential function $\mathrm{\Phi }(\rho )$ is expressed in the following form

$$\mathrm{\Phi }\left(\rho \right)=-A{I}_3\left(\rho \right)+B{I}_6\left(\rho \right)$$

(5)

where $I_n(\rho )={\rho }^{-2n}$ , and in the following section, integrals of the following form

$$I={\displaystyle \underset{S_1}{\int }}{\displaystyle \underset{S_2}{\int }{I}_n\left(\rho \right)d{S}_{2}d{S}_1}$$

(6)

must be evaluated. In many instances, integrals of this type can be given explicitly in terms of the hypergeometric function $F(a,b;c;z)$ which is a standard function of mathematical analysis that can be readily evaluated from algebraic packages such as Maple and MATLAB. There are many important results relating to the hypergeometric function, and we refer the reader to Erdélyi et al. ⁹² and Bailey, ⁹³ but the principal formula required for the determination of interaction energies is the integral representation

$$F(a,b;c;z)=\frac{\mathrm{\Gamma }(c)}{\mathrm{\Gamma }(b)\mathrm{\Gamma }(c-b)}{\displaystyle \underset{0}{\overset{1}{\int }}{t}^{b-1}{(1-t)}^{c-b-1}{(1-tz)}^{-a}dt}$$

(7)

provided that $\Re (c)>\Re (b)>0$ and $|\arg (1-z)|<\pi$ . ⁹²

Analytical expressions for idealized molecular building blocks

In this section, the approach adopted by Thornton and colleaugues^80–82 and Lim et al. ⁸⁷ is summarized using idealized building blocks to represent the interactions of both simple and more complicated geometries of nanostructures yielding simple and elegant analytical models. First, the analytical representations of the van der Waals interaction between an atom and the building blocks, which are represented by standard geometrical shapes such as points, lines, planes, rings, spheres, and cylinders are determined. At first sight, such a dramatically simplified modeling approach may seem geometrically severe, but in many situations, it has been shown to provide the major contribution to the interaction energy of the actual structure.

Interaction of two atomic points

Given the coordinates of two atoms, $\mathbf{P}=(x_p,y_p,z_p)$ and $\mathbf{Q}=(x_q,y_q,z_q)$ , the Lennard-Jones potential between the two atoms can be obtained by substituting the parameter $\rho$ into equation (4) which is the distance between the two atoms and is given as follows

$${\rho }^2={(x_q-x_p)}^2+{(y_q-y_p)}^2+{(z_q-z_p)}^2$$

Interaction of atomic point with atomic line

The perpendicular (closest) distance between an atomic point and an atomic line is denoted by $\delta$ . The line parametrically by $\mathcal{L}(p)=(p,0)a$ and the point $\mathbf{P}=(0,\delta )$ are defined, as illustrated in Figure 2(a). Note that the line element is given by dp, and therefore, the integral of interest is given as follows

$$I={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\left(p^2+{\delta }^2\right)}^{-n}dp}$$

Figure 2.

Interaction of atoms with idealized building blocks: (a) point with line, (b) point with plane, (c) point with ring, (d) point with spherical surface, and (e) point with infinitely long right-cylindrical surface.

On making a change of variable and substituting $p=\delta \tan \psi$ , the integral becomes as follows

$$I={\delta }^{1-2n}{\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}{\cos }^{2n-2}\psi d\psi }$$

(8)

which can then be evaluated using

$${\displaystyle \underset{0}{\overset{\pi /2}{\int }}{\sin }^p\theta {\cos }^q\theta d\theta =\frac{1}{2}B\left(\frac{p+1}{2},\frac{q+1}{2}\right)}$$

(9)

to obtain

$$I={\delta }^{1-2n}B\left(n-1/2,1/2\right)$$

(10)

Interaction of atomic point with atomic plane

This situation is relevant to modeling nanostructures as it corresponds to the case of an individual atom interacting with a graphene sheet. Again, the perpendicular spacing between the point and the plane is assumed to be $\delta$ , and therefore, the plane $\mathcal{P}(p,q)=(p,q,0)$ and the point $\mathbf{P}=(0,0,\delta )$ are defined, as shown in Figure 2(b). In this case, the area element of the plane is given by dpdq, and therefore, the integral required to evaluate I is given as follows

$$I={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\left(p^2+q^2+{\delta }^2\right)}^{-n}dpdq}$$

(11)

The substitution $p=\sqrt{q^2+{\delta }^2}\tan \psi$ is made and proceeded as in the previous section to produce the following

$$I=\mathrm{B}\left(n-\frac{1}{2},\frac{1}{2}\right){\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\left(q^2+{\delta }^2\right)}^{1/2-n}dq}$$

On making a further substitution of $q=\delta \tan \varphi$ , the integral becomes as follows

$$I={\delta }^{2-2n}B\left(n-\frac{1}{2},\frac{1}{2}\right)B\left(n-1,\frac{1}{2}\right)=\frac{\pi }{(n-1){\delta }^{2n-2}}$$

Interaction of atomic point with atomic ring

The interaction of a point with a ring can be categorized into two cases which are as follows: (1) the point is interacting with the ring from the side and (2) the point is interacting with the ring from the top or bottom. For the first case, the point $\mathbf{P}$ is assumed to be located at $(\delta ,0)$ . Furthermore, the center of the ring $Q(q,\theta )$ of radius q is assumed to be located at the origin where its coordinates are $Q=(q\cos \theta ,q\sin \theta )$ , as depicted in Figure 2(c). With the line element $\mathrm{qd}\theta$ , equation (6) becomes as follows

$$I={\displaystyle \underset{-\pi }{\overset{\pi }{\int }}\frac{q}{{\left[{(q-\delta )}^2+4q\delta {\sin }^2(\theta /2)\right]}^n}d\theta }$$

On making the substitution $t={\sin }^2(\theta /2)$ yields the following

$$I=\frac{2q}{{(q-\delta )}^n}{\displaystyle \underset{0}{\overset{1}{\int }}{t}^{-1/2}{(1-t)}^{-1/2}{(1-\mu t)}^{-n}dt}$$

where $\mu =-4q\delta /(q-\delta {)}^2$ . This integral can be written in a standard hypergeometric form as follows

$$\begin{array}{l}I=\frac{2q}{{(q-\delta )}^{2n}}\frac{\mathrm{\Gamma }(1/2)\mathrm{\Gamma }(1/2)}{\mathrm{\Gamma }(1)}F\left(n,\frac{1}{2};1;\mu \right)\\ =\frac{2\pi q}{{(q-\delta )}^{2n}}F\left(n,\frac{1}{2};1;\mu \right)\end{array}$$

The Pfaff transformation is utilized, ⁹⁴ $F(a,b;c,z)=(1-z{)}^{-b}F(c-a,b;c;z/(z-1))$ to produce a terminating hypergeometric series, thus

$$I=\frac{2\pi q}{{(q-\delta )}^{2n-1}(q+\delta )}F\left(1-n,\frac{1}{2};1;\frac{4q\delta }{{(q+\delta )}^2}\right)$$

In the case of an atomic point $\mathbf{P}$ with coordinates $P=(x,y,z)$ , and assumed to be located either at the top or the bottom of the ring $Q(q,\theta )$ , which is assumed to be located at the origin of the xy-plane with coordinates $(q\cos \theta ,q\sin \theta ,0)$ , so that

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }^2=(x-q\cos \theta {)}^2+(y-q\sin \theta {)}^2+z^2}\\ \multicolumn{1}{c}{=\beta -\alpha q\cos (\theta -{\theta }_0)}\end{array}$$

where $\beta =x^2+y^2+z^2+q^2$ , $\alpha =\sqrt{x^2+y^2}$ , and ${\theta }_0=\arctan (y/x)$ . Following the work by Tran-Duc et al., ⁶⁷ I becomes as follows

$$\begin{array}{c}\multicolumn{1}{c}{I=\underset{-\pi }{\overset{\pi }{\int }}\frac{q}{{[\beta -\alpha q\cos (\theta -{\theta }_0)]}^n}d\theta }\\ \multicolumn{1}{c}{=\frac{2\pi q}{{(\beta -\alpha q)}^n}F(n,\frac{1}{2};1;\frac{2\alpha q}{(\alpha q-\beta )})}\end{array}$$

(12)

Interaction of atomic point with atomic spherical surface

The atomic point with Cartesian coordinates $\mathbf{P}=(0,0,\delta )$ is considered, which is at a distance $\delta$ from the center of an atomic spherical surface of radius a, parameterized using the spherical coordinates $\mathcal{S}(\theta ,\varphi )=(a,\theta ,\varphi )$ , as indicated in Figure 2(d). In terms of these coordinates, the integral required to evaluate equation (6) is given as follows

$$I=a^2\underset{-\pi }{\overset{\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\frac{\sin \theta }{{[a^2{\sin }^2\theta +{(a\cos \theta -\delta )}^2]}^n}d\theta d\varphi$$

Since the integrand in this case is independent of $\varphi$ , the integration involving $\varphi$ can be effected immediately and then by re-organizing the denominator to deduce

$$I=2\pi a^2\underset{0}{\overset{\pi }{\int }}\frac{\sin \theta }{{[{\delta }^2+a^2-2\delta a\cos \theta ]}^n}d\theta$$

(13)

which on making the substitution $t={\delta }^2+a^2-2\delta a\cos \theta$ becomes as follows

$$\begin{array}{c}\multicolumn{1}{c}{I=\frac{\pi a}{\delta }{\int }_{{(\delta -a)}^2}^{{(\delta +a)}^2}\frac{\mathrm{dt}}{t^n}=\frac{\pi a}{\delta (1-n)}{\left[\frac{1}{t^{n-1}}\right]}_{{(\delta -a)}^2}^{{(\delta +a)}^2}}\\ \multicolumn{1}{c}{=\frac{\pi a}{\delta (n-1)}[\frac{1}{{(\delta -a)}^{2(n-1)}}-\frac{1}{{(\delta +a)}^{2(n-1)}}]}\end{array}$$

Interaction of atomic point with infinite atomic cylindrical surface

Here, the interaction of an arbitrary atomic point $\mathbf{P}$ with an atomic cylindrical surface $\mathcal{C}$ of radius b and assumed to be infinite in length is determined. The cylinder is represented parametrically by the coordinates $\mathcal{C}(\theta ,z)=(b,\theta ,z)$ , where $-\pi <\theta \le \pi$ and $-\infty <z<\infty$ . Due to the rotational and translational symmetry of the problem, the point $\mathbf{P}$ in Cartesian coordinates is given by $(\delta ,0,0)$ , where $0\le \delta <b$ , as indicated in Figure 2(e). Accordingly, the distance from $\mathbf{P}$ to a typical surface element on $\mathcal{C}$ is given as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }^2=(b\cos \theta -\delta {)}^2+b^2{\sin }^2\theta +z^2}\\ \multicolumn{1}{c}{={\delta }^2+b^2-2\delta b\cos \theta +z^2}\\ \multicolumn{1}{c}{=(b-\delta {)}^2+4\delta b{\sin }^2(\theta /2)+z^2}\end{array}$$

so that, the following integral must be evaluated

$$I=b\underset{-\infty }{\overset{\infty }{\int }}\underset{-\pi }{\overset{\pi }{\int }}\frac{1}{{[{(b-\delta )}^2+4\delta b{\sin }^2(\theta /2)+z^2]}^n}d\theta \mathrm{dz}$$

By defining ${\lambda }^2=(b-\delta {)}^2+4\delta b{\sin }^2(\theta /2)$ and making the substitution $z=\lambda \tan \psi$ gives the following

$$\begin{array}{c}\multicolumn{1}{c}{I=b\underset{-\pi /2}{\overset{\pi /2}{\int }}{\cos }^{2n-2}\psi d\psi \underset{-\pi }{\overset{\pi }{\int }}\frac{1}{{\lambda }^{2n-1}}d\theta }\\ \multicolumn{1}{c}{=\mathrm{bB}(n-\frac{1}{2},\frac{1}{2})\underset{-\pi }{\overset{\pi }{\int }}\frac{1}{{\lambda }^{2n-1}}d\theta }\end{array}$$

Now on making the further substitution $t={\sin }^2(\theta /2)$ yields the following

$$\begin{array}{c}\multicolumn{1}{c}{I=\frac{2b}{{(b-\delta )}^{2n-1}}B(n-\frac{1}{2},\frac{1}{2})}\\ \multicolumn{1}{c}{\times \underset{0}{\overset{1}{\int }}{t}^{-1/2}{(1-t)}^{-1/2}{(1-\mu t)}^{1/2-n}\mathrm{dt}}\end{array}$$

(14)

where $\mu =-4b\delta /(b-\delta {)}^2$ . This integral is now in the Euler form as follows

$$\begin{array}{c}\multicolumn{1}{c}{I=\frac{2b}{{(b-\delta )}^{2n-1}}B(n-\frac{1}{2},\frac{1}{2})\frac{\mathrm{\Gamma }(1/2)\mathrm{\Gamma }(1/2)}{\mathrm{\Gamma }(1)}}\\ \multicolumn{1}{c}{\times F(n-\frac{1}{2},\frac{1}{2};1;\mu )}\\ \multicolumn{1}{c}{=\frac{2\pi b}{{(b-\delta )}^{2n-1}}B(n-\frac{1}{2},\frac{1}{2})F(n-\frac{1}{2},\frac{1}{2};1;\mu )}\end{array}$$

Note that in terms of the usual parameters of the hypergeometric function where $c=2b$ , and by employing a quadratic transformation (see Erdélyi et al. ⁹² equation (24) on page 64), the integral I yields the following

$$\begin{array}{c}\multicolumn{1}{c}{I=\frac{2\pi b}{{(b-\delta )}^{2n-1}}B(n-\frac{1}{2},\frac{1}{2}){\left(\frac{b-\delta }{b}\right)}^{2n-1}}\\ \multicolumn{1}{c}{\times F(n-\frac{1}{2},n-\frac{1}{2};1;\frac{{\delta }^2}{b^2})}\\ \multicolumn{1}{c}{=\frac{2\pi }{b^{2n-2}}B(n-\frac{1}{2},\frac{1}{2})F(n-\frac{1}{2},n-\frac{1}{2};1;\frac{{\delta }^2}{b^2})}\end{array}$$

Next, the total interaction of an atomic point $\mathbf{P}$ , which is offset from the axis by a distance $\delta$ , and a cylindrical surface $\mathcal{C}$ of radius b are considered where $\delta >b$ . In this case, the calculation follows along similar lines to the above except that the terms are rearranged so as to pick up a different solution of the hypergeometric equation, which is a solution with the argument inverse to that given in the previous section.

The same cylinder defined in cylindrical coordinates by $\mathcal{C}(\theta ,z)=(b,\theta ,z)$ , where $-\pi <\theta \le \pi$ , and $-\infty <z<\infty$ is determined. The atomic point with Cartesian coordinates $\mathbf{P}=(\delta ,0,0)$ is defined, but in this case, $\delta >b$ . Following the above steps, an expression for the distance $\rho$ , from the point $\mathbf{P}$ to an arbitrary area element on the surface of the cylinder $\mathcal{C}$ is as follows

$${\rho }^2=(\delta -b{)}^2+4\delta b{\sin }^2(\theta /2)+z^2$$

In a similar manner to that described above, the integral I is of the following form

$$I=\frac{2\pi b}{{(\delta -b)}^{2n-1}}B(n-\frac{1}{2},\frac{1}{2})F(n-\frac{1}{2},\frac{1}{2};1;-\frac{4\delta b}{{(\delta -b)}^2})$$

whereupon on again employing the quadratic transformation

$$I=\frac{2\pi b}{{\delta }^{2n-1}}B(n-\frac{1}{2},\frac{1}{2})F(n-\frac{1}{2},n-\frac{1}{2};1;\frac{b^2}{{\delta }^2})$$

Some important mathematical formulae are derived which may be exploited to calculate the interaction energy between two nanostructures. Analytical expressions for an atomic point (i.e. a single atom) with various shaped molecules have been determined. In more complicated atomic configurations involving two or more molecules, another surface integral of the atomic point must be evaluated to determine the total interaction energy of the system. In the following sections, a number of nanotechnology applications are surveyed which have exploited these formulae to determine the properties of the systems.

Mechanics of nanostructures

Nanostructures such as carbon nanotubes, nanopeapods, nanocones, and carbon onions exhibit outstanding physical and mechanical properties such as their high strength, high flexibility, and low weight, and they provide a basis for the creation of many novel nano-devices. One particular application which has attracted much attention is the nano-oscillator,^12,37,95,96 which is able to generate frequencies in the gigahertz range, ¹² and which may form the basis of a number of ultrahigh-frequency devices in the computer industry. Since the discovery of ultra low friction by Cumings and Zettl, ⁹⁵ double-walled carbon nanotube oscillators have been widely studied using both molecular dynamics simulations and experiments.^{12,13,96–98} In addition, carbon nanotubes have received much attention for medical applications, especially their use as nanocontainers for drug and gene delivery. In particular, a well-known self-assembled hybrid carbon nanostructure, so-called nanopeapods, may be regarded as a model for possible drug carriers, where the carbon nanotube can be thought of as the nanocontainer, and the ${\mathrm{C}}_{60}$ molecular chain can be considered as the drug molecule. ⁹⁹ Nanocones have received less attention in the literature, primarily because only a small amount are produced in the production process. ¹⁰⁰ However, the narrow vertex of the cone makes an ideal candidate as a nanoprobe in scanning tunneling microscopes. ¹⁰¹

The Lennard-Jones potential together with the continuous approximation has been successfully employed in a number of studies to determine the van der Waals interaction energy and the force between two interacting non-bonded nanostructures. In particular, several authors determine the molecular interaction between a fullerene and carbon nanotubes.^8,36–46 Girifalco ³⁶ determines the interaction energy between two ${\mathrm{C}}_{60}$ fullerenes and extends the study in Girifalco et al. ⁶ to find the energy between two identical parallel carbon nanotubes of infinite length and between a carbon nanotube and a ${\mathrm{C}}_{60}$ fullerene. Girifalco et al. ⁶ also provide the value of the interaction constants in the Lennard-Jones potential for carbon atoms in graphene–graphene, C₆₀–C₆₀, and C₆₀–graphene. Furthermore, Hodak and Girifalco ⁸ propose an energy formula for universal graphitic systems including the interaction of an ellipsoid inside a single-walled carbon nanotube. In general, it is possible to combine both the continuous and discrete approaches to model the interaction between two nanostructures. As shown in both Hilder and Hill ⁹ and Verberck and Michel, ¹⁰² the single-walled carbon nanotube is modeled continuously, while the fullerene is modeled as a discrete atomic structure.

Cox et al.^38,39 have proposed the important notions of suction and acceptance energies for the encapsulation behavior of an atom and a ${\mathrm{C}}_{60}$ fullerene when sucked inside a carbon nanotube. The suction energy is defined as the total work performed by the van der Waals interactions on an atom or molecule entering the carbon nanotube. The acceptance energy is the total work performed by van der Waals interactions on the atom or molecule entering the nanotube, up to the point that the van der Waals force once again becomes attractive. ³⁸ The forces acting on the atom or a ${\mathrm{C}}_{60}$ fullerene interacting with carbon nanotube of finite length can be approximated by two equal and opposite Dirac delta functions operating at the extremities of the tubes, as shown in Figure 3. Once the atom or molecule is encapsulated inside the tube, these forces tend to keep them oscillating inside, and this is the physical basis of the nano-oscillator.

Figure 3.

Plot of forces for (a) atom oscillating inside (6, 6) carbon nanotube and (b) ${\mathrm{C}}_{60}$ fullerene oscillating inside (10, 10) carbon nanotube.

Cox et al.^40,43,44 also study the mechanics of spherical and spheroidal fullerenes entering carbon nanotubes. Particularly, Figure 4 shows the energy profiles for spheroidal ${\mathrm{C}}_{70}$ and ${\mathrm{C}}_{80}$ fullerenes interacting with carbon nanotubes for various offset distances $\epsilon$ and tilt angles $\psi$ , and two distinct and approximately equal local minima are observed. Baowan et al. ⁴² determine the encapsulation mechanics of the ${\mathrm{C}}_{60}$ into a carbon nanotube where the ${\mathrm{C}}_{60}$ is initiated outside the tube in the absence of any applied external forces. ⁴² Once a number of ${\mathrm{C}}_{60}$ fullerenes are encapsulated inside the tube, two patterns emerge which are termed zigzag and spiral, ⁴¹ and the composite nanostructures are referred to as nanopeapods. Moreover, the spiral motion of carbon atoms and ${\mathrm{C}}_{60}$ fullerenes inside single-walled carbon nanotubes is investigated by Chan et al. ⁴⁵ and Chan and Hill. ⁴⁶

Figure 4.

Contour plot of interaction energy for (a) ${\mathrm{C}}_{70}$ fullerene for a 8.0-Å radius nanotube and (b) ${\mathrm{C}}_{80}$ fullerene for a 8.3-Å radius nanotube, showing two distinct and approximately equal local minima.

For two concentric cylindrical carbon nanotubes, Zheng and Jiang ¹² determine the van der Waals restoring force between the inner and outer shells of a multi-walled carbon nanotube and subsequently predict a gigahertz frequency of the oscillatory motion. Baowan and colleagues^14,15 determine analytical expressions for the suction energy and offset configurations of double-walled carbon nanotubes and also predict the gigahertz frequency for the nanotube oscillators. A similar approach has been adopted by Cox ¹⁶ to model the behavior of forced double-walled nanotube oscillators. Ansari and colleagues^18–20 consider the effects of geometrical parameters on the force distributions for the oscillatory behavior of double-walled carbon nanotubes. The effect of capped ends of double-walled carbon nanotubes is also studied by Baowan, ¹⁷ and the effect of tube radii is investigated by Tiangtrong and Baowan. ²¹

Ruoff and Hickman ¹⁰³ consider the interaction between a spherical fullerene and a graphite sheet. For spherical carbon onions $C_{N_1}@C_{N_2}(N_2>N_1)$ , Iglesias-Groth et al. ¹⁰⁴ also adopt the Lennard-Jones potential and the continuous approximation to determine the interlayer interaction. Using the formula of Iglesias-Groth et al., ¹⁰⁴ Guérin ¹⁰⁵ obtains the interaction energy between the interlayer of carbon onions which is in excellent agreement to that obtained from a discrete atom–atom summation model given in Lu and Yang. ¹⁰⁶ Furthermore, Baowan et al. ¹⁰ predict the interlayer spacing for each shell of the carbon onions. Moreover, they observe that the equilibrium spacing decreases as the shell is further away from the inner core, and this is due to the decreasing curvature for larger spheroids. Thamwattana et al. ¹¹ also exploit the Lennard-Jones potential and the continuous approximation to focus on various interactions involving a fullerene and other carbon nanostructures, and analytical expressions are obtained. The study by Thamwattana et al. ¹¹ confirms that molecules are likely to be at a certain distance apart in order to minimize the total interaction energy.

Henrard et al. ¹⁰⁷ use a similar technique to that proposed by Girifalco ³⁶ and obtain the potential for bundles of single-walled carbon nanotubes. Cox and colleagues^22–24 study extensively the mechanics of carbon atoms and nanotubes oscillating in carbon nanotube bundles and again utilizing the Lennard-Jones potential together with the continuous approximation, and the results obtained can be used to predict the oscillator bundle configuration which optimizes the suction energy and therefore leads to the maximum frequency oscillator.

The equilibrium configurations of carbon atoms and ${\mathrm{C}}_{60}$ fullerenes inside carbon nanotori have been determined by Hilder and Hill,^25–27 Chan and colleagues,^28,29 and Sumetpipat et al. ³⁰ Even though complicated analytical expressions are derived, the energy profiles are easily obtained utilizing algebraic packages such as Maple. Furthermore, the interaction energy between two nanocones has been investigated by Baowan and Hill^31–33 and Ansari et al., ³⁴ where the spacing between the two cone surfaces is determined to be 3 Å. The equilibrium arrangement between two carbon nanostacked cups, which are truncated cones that are found as the hollow cores of carbon nanofibers,^108–112 is determined by Baowan et al. ³⁵ again using the Lennard-Jones potential and the continuous approach.

MOFs and gas storage

MOFs comprise metal atoms or clusters that are linked periodically by organic molecules to establish an array such that each atom forms part of an internal surface. MOFs have delivered the highest surface areas and hydrogen storage capacities for any physisorbent and are shown to be the most practically promising material for gas storage. ¹¹³ Exposed metal sites^114,115 pore sizes, ¹¹⁶ and ligand chemistries^117,118 have been found to be the most effective routes for increasing the hydrogen enthalpy of adsorption within MOFs. The MOF adsorbent that presently holds the record for gravimetric hydrogen storage capacity at room temperature is the first structurally characterized beryllium-based framework, Be-BTB (benzene tribenzoate). Be-BTB has a Brunauer–Emmett–Teller (BET) ¹¹⁹ surface area of 4400 m²g⁻¹ and can adsorb 2.3 wt% hydrogen at 298 K and 100 bar. ¹²⁰ We refer the reader to Furukawa et al. ¹²¹ for a comprehensive review of the chemistry and the applications of MOFs.

The so-called Topologically Integrated Mathematical Thermodynamic Adsorption Model (TIMTAM), as proposed by Thornton and colleagues,^81,82 assumes the ideal building blocks described in the section on analytical expressions for idealized molecular building blocks to represent the cavity of the structure, and then, these expressions are exploited to calculate the potential energy interactions between the gas and the adsorbate. The major advantage of the TIMTAM approach is that it provides analytical formulae that are computationally instantaneous, and therefore, many distinct scenarios can be rapidly investigated which evidently serves to accelerate material design. ⁸¹ A schematic representation for MgC₆₀@MOF for a MOF cavity impregnated with magnesium-decorated C₆₀ is shown in Figure 5, where the TIMTAM model is utilized to determine the energy level in the cavity for the magnesium atom. ⁸¹ Moreover, the same approach has also been proved as a useful technique to investigate the effect of pore size in MOFs.^{80–82,86,87}

Figure 5.

Model for ${\mathrm{C}}_{60}@\mathrm{MOF}$ where Mg atom locates within cavity surface at radius $r_1$ , $\rho$ is distance between gas molecule and center of cavity, and b denotes radius of ${\mathrm{C}}_{60}$ . The color bar indicates the energy value of MgC₆₀@MOF.

Furthermore, Chan and Hill ⁸⁴ investigate the storage of hydrogen molecules inside graphene-oxide frameworks comprising two parallel graphenes rigidly separated by perpendicular ligands. These authors find 6.33 wt% for GOF-28 at a temperature of 77 K and a pressure of 1 bar which is consistent with several experimental and other computational results.^122–124 Based on the assumption of no steric hindrance and a small electronic barrier, Chan and Hill ⁸³ model the interaction of a rigidly suspended benzene molecule within a MOF, which is then used as a building block in more complex MOFs.

For the specific gas molecule, benzene, Tran-Duc and colleagues^65–69 extensively investigate the equilibrium configuration of benzene dimer adsorbed on graphene sheet, ${\mathrm{C}}_{60}$ fullerene, and carbon nanotubes. They obtain an analytical expression as a function of the distance between the gas molecule and the material surfaces and the rotational configuration of the benzene itself. This analysis might be exploited to improve the design of the gas storage system. ⁶⁹ For methane, Adisa et al.^1,2,70–73 investigate the encapsulation and packing of methane in various carbon nanostructures such as spherical fullerenes, nanotubes, and nanobundles. In terms of clean energy and the effect on the environment, the theoretical study ⁷³ indicates a promising future using natural gas storage in molecular structures.

Molecular selective and separation

Water molecule has a simple chemical structure and is often a basic unit in many biomolecules. The determination of water separation can be envisaged as the first step in a study of the selective separation of more complicated molecules. Hilder and Hill ⁵⁹ determine the maximum velocity for a single water molecule entering a carbon nanotube, and their model predicts that the radius of the carbon nanotube must be at least 3.464 Å for acceptance of a water molecule, and that a radius of 3.95 Å provides the maximum uptake or suction energy. Chan and Hill ⁶¹ utilize the same mathematical technique to investigate the transport of water through carbon nanotubes and suggest that their results rapidly reduce the computational time for the full numerical calculation. As an alternative for molecular selectivity, Garalleh et al.^62–64 determine the interaction energy between water and various other biomolecules.

Ions are atoms or molecules in which the total number of electrons does not equal the total number of protons, giving the atom a net positive or negative electrical charge. On using the applied mathematical approach, Chan and Hill^75,76 investigate the interaction energy between various types of atoms and ions, namely, Mn²⁺, Au, Pt, Na¹⁺, and Li¹⁺ on graphene sheet. These authors determine the equilibrium position for the atom/ion on the surface of the graphene sheet and the minimum intermolecular spacing between two graphene sheets. Furthermore, carbon nanotubes have been used to facilitate the transport and separation of atoms and ions.^77,78 Similarly, Rahmat et al. ⁷⁹ determine the suction and the acceptance behavior of ${\mathrm{C}}_{60}$ molecule, Li⁺, Na⁺, Rb⁺, and Cl⁻ ions and ion–water clusters into peptide nanotubes. A critical tube radius of 8.5 Å is determined such that all the ions are accepted into the peptide nanotubes, whereas the ${\mathrm{C}}_{60}$ is rejected. This work has many potential applications involving ion separation, including drug delivery systems and high-performance alkali batteries using nanomaterials as components.

The selective separation of biomolecules is a critical process in food, biomedical, and pharmaceutical industries. Baowan and Thamwattana ⁵⁰ utilize the Lennard-Jones potential function and the continuous approximation to separate trypsin and lysozyme using mesoporous silica. These authors predict that the silica pores with radii lying in the range 17.23 and 21.24 Å will only allow lysozyme to be encapsulated. Using the same approach, Thamwattana et al. ⁵¹ investigate three model configurations for bovine serum albumin to be encapsulated inside carbon nanotubes, as indicated in Figure 6. They conclude that a critical radius of pore or tube is crucial for the design to facilitate maximum loading of proteins and drug molecules.

Figure 6.

Three possible configurations for bovine serum albumin encapsulated in a carbon nanotube and modeled: (a) as three-connected spheres, (b) as a prolate-ellipsoid, and (c) as a right-cylinder.

The far more complicated biomolecule DNA has been examined by Alshehri et al.,^52–55 who determine equilibrium positions of a DNA strand absorbed onto a graphene sheet or encapsulated inside a carbon nanotube. These authors find that a space of approximately 20 Å is required for the absorption of DNA onto the graphene sheet. ⁵³ Moreover, they observe that double-stranded DNA is encapsulated inside a single-walled carbon nanotube of radius larger than 12.30 Å, and they show that the optimal radius of the single-walled carbon nanotube to enclose a double-stranded DNA is 12.8 Å.^52,54,55 Furthermore, since lipid bilayers, lipid nanotubes, and liposomes are potential candidates for use in molecular separation, Baowan et al. ⁵⁶ determine the penetration and encapsulation of ${\mathrm{C}}_{60}$ fullerene through/in lipid bilayer family. Furthermore, the penetration and encapsulation of silica nanoparticles are examined in Baowan et al. ⁵⁷ and silver and gold nanoparticles in Baowan and Thamwattana. ⁵⁸ Although electrostatic interaction energy may arise from hydrophobic layers of the lipid, these authors show that the dominant energy contribution originates from the van der Waals interactions.

Targeted drug delivery

The prospect that nanocapsules may realize the “magic bullet” concept, as first proposed at the beginning of the 20th century by the Nobel Prize winner Paul Ehrlich (1854–1915), generated immense interest in their development. The ideal drug carrier, or “magic bullet,” is envisaged as a transporter of drugs or other molecular cargo to a specific site in the body which then unloads the cargo in a controlled manner. Although this notion may sound like science fiction, the advent of nanotechnology means that it is rapidly becoming scientific fact. Despite the prominence of carbon nanotubes in the broader area of nanotechnology, the field of nanotube biotechnology is in its infancy, and there is still much work that needs to be accomplished before specific products can be produced. Drug delivery is one of the most promising biomedical applications of nanotechnology, and as stated by Hillebrenner et al. ¹²⁵ in a review of template-synthesized nanotubes for biomedical delivery applications, “The future challenges for nanotubes as drug delivery vehicles are substantial but not insurmountable.”

Again following the Lennard-Jones potential function and the continuous approximation approach, Hilder and Hill determine the energy behavior and suction characteristics in the encapsulation of various drugs into nanotubes including cisplatin,^126,127 paclitaxel, and doxorubicin ¹²⁸ and into other nanotubes such as boron nitride, silicon, and boron carbide. ¹²⁹ Their results predict an appropriate tube size that gives rise to the optimum encapsulation mechanics for a prescibed drug molecule. Moreover, Hilder and Hill ¹³⁰ also examine nanosyringes comprising double-walled carbon nanotubes to inject DNA or anticancer drugs directly into the cell. The reader is referred to Hilder and Hill ¹³¹ for a comprehensive review on the various models for drug release using a nanotube carrier, as indicated in Figure 7. Ansari et al. ¹³² employ both a hybrid discrete–continuum model and molecular dynamics simulation to study the offset of cisplatin in a single-walled carbon nanotube and conclude that the methods give comparable results.

Figure 7.

Outline of proposed drug delivery process: (a) nanotube surface is functionalized with a chemical receptor and drug molecules are encapsulated, (b) open end is capped, (c) nanocapsule is ingested and locates to target site due to functionalized surface, (d) cell internalizes capsule, for example, by receptor-mediated endocytosis, (e) cap is removed or biodegrades inside cell, and (f) drug molecules are released.

Summary

This review has focused on the use of applied mathematical modeling to determine the mechanical energy behavior of nanostructures. We have concentrated on those applications for which the 6-12 Lennard-Jones potential energy function and the continuous approximation apply. The continuous approximation assumes that the total molecular interaction energy of the system involving a discrete atomic configuration can be approximated by a uniform distribution of atoms, either throughout a region or over a bounding surface. First, analytical expressions for various molecular building blocks are evaluated to provide the major contribution to the interaction energy of the actual complicated atomic configuration. The principal mechanical properties of the nanostructure can then be determined from the energy distribution, which gives rise to the force distribution, equilibrium configurations, and oscillatory behavior. Moreover, this applied mathematical approach has been illustrated to determine the gas storage characteristics of MOFs. The same approach has also been successfully applied to both organic and inorganic molecules, including systems of biomolecules involving protein and enzyme selective separation and targeted drug delivery. In summary, although this robust mathematical approach has already been successfully exploited in many applications, its most important future role will likely be either as a component or as the first iteration in large-scale computational calculations to determine the molecular interaction energy of complex atomic configurations, and such use will significantly increase computational efficiency.