Introduction

Cancer treatment has been a target in medicine. Significant advances have been made in improving chemotherapy by employing various nanotubes for targeted and controlled delivery of anticancer agents [1,2,3]. So let’s give a brief explanation of the basic elements of drug delivery .First, we express the carbon nanotubes. Carbon nanotubes are one of the effective tools in the treatment of cancer [4]. In general, carbon nanotubes are classified into two categories: multi-walled carbon nanotubes in 1991 and single-walled carbon nanotubes discovered by Iijima et al. CNTs can be like a rolled graphite plate, thought to be composed of carbon atoms arranged in a hexagonal fashion [5,6,7,8,9,10]. Typically the diameters of these compounds are in nanometer range and can be metallic, or semiconductor, and have very interesting properties such as regular structure with high radius ratio, very low weight, high thermal conductivity and high surface area, thermal resistance [11,12,13,14,15,16,17,18]. Carbon nanotube and non carbon based nanotubes are used to study the adsorption. Carbon nanotube is used to study the adsorption of methane, krypton, xenon, hydrogen and metal cations from cadmium, copper, mercury, nickel, zinc, lead, trace elements and radioactive nucleic acids and volatile compounds such as dioxide, Non-aromatic are used in aquatic environment [19, 20] but non carbon based nanotubes is also been used in the adsorption of hydrogen please cite [21, 22]. Then, we are talking about the Gemcitabine anticancer drug. The main use of the gemcitabine anticancer drug is the chemotherapy of various cancers such as lung cancer, pancreatic tumor, bladder and breast cancer [23]. Occasionally, esophageal and lymphoma can also be treated its function is to be an anti metabolite and pyrimidine analogue. In the cell division cycle, it functions specifically in the S phase and inhibits DNA synthesis by activating the tissues. When DNA is rewritten, the drug is mistakenly incorporated into the DNA structure instead of cytosine nucleoside because it does not mate with other nucleoside, and it can interfere with DNA Encapsulation of the drug into the nanotube is one of the important methods of transporting the drug to the target tissue that can be performed by the nanotubes [24,25,26]. CNTs as effective drug delivery systems show good potential for cancer treatment, and they also have the ability to bind to specific cellular receptors and intracellular target molecules for targeted delivery of therapeutic agents [27,28,29,30,31]. Consequently our aim in this research is to obtain the thermodynamic functions of the Gemcitabine with SWCNT at different dielectric constant by computational method. The computational method used is the M06/6-311G method [32].

Method

M06 method

Mo6 is a part of highly parameterized approximate exchange–correlation energy functional in density functional theory (DFT). This functional is based on the meta-GGA approximation, i.e. it includes terms that depend on the kinetic energy density, and is based on complicated functional forms parameterized on high-quality benchmark databases. The M06 functional introduce 35 empirically optimized parameters, into the exchange–correlation functional [33].

SCRF method

For the SCRF model in quantum chemistry theory, it is necessary to determine the shape and volume of solute molecules for each set of compounds. Thus we have studied the solvation effects by using self-consistent reaction field (SCRF) approach, in particular using the polarizable continuum model (PCM). Polarized continuum model was used for the consideration of implicit effects of the solvent. In the PCM method, the molecular cavity is made up of the union of interlocking atomic spheres. The molecular free energy is calculated as the sum over three components:

$$ G_{\text{sol}} \, = \,G_{\text{es}} \, + \,G_{\text{dr}} \, + \,G_{\text{cav}} . $$

That Ges, Gdr, and Gcav, respectively, are electrostatic free energy, dispersion-repulsion free energy and cavitation free energy. Herein, we investigated the effects of various dielectric constants on the thermodynamic functions of the interaction of the Gemcitabine anticancer drugs with SWCNT through M06/6-311G method using Gaussian 09 software that in Scrf = PCM Scale factor 1.21 for all atoms except hydrogen and 70 areas are on each sphere [34,35,36].

The thermodynamic functions are obtained through the following equations [37]:

  1. 1.

    Internal energy:

Internal energy is the sum of all the kinetics and potential energies of the constituent particles of a system that in statistical thermodynamics is as follows:

$$ E = NK_{\text{B}} T^{2} \left( {\frac{\partial \ln q}{\partial T}} \right)_{V} . $$
(1)
  1. 2.

    Entropy is:

$$ S = R\ln \left( {q\left( {V,Y} \right)e} \right) + RT\left( {\frac{{{\text{d}}\ln q}}{{{\text{d}}T}}} \right)_{p} . $$
(2)
  1. 3.

    Enthalpy:

The enthalpy represents the amount that represents the total thermal energy of a system. The enthalpy is the sum of the internal energy of the system and the product of the pressure in its volume that in statistical thermodynamics is as follows:

$$ H = RT\left[ {T\left( {\frac{\partial \ln q}{\partial T}} \right)_{N,V} + 1} \right]. $$
(3)
  1. 4.

    Finally, the following equations are used for Gibbs free energy. The energy available to generate the work from a reaction is called Gibbs free energy.

$$ \begin{aligned} G = - RT\ln \left( {\frac{q}{N}} \right)_{p} \hfill \\ q = q_{\text{tr}} q_{\text{rot}} q_{\text{vib}} q_{\text{el}} q_{\text{nucl}} \hfill \\ q_{\text{tr}} {\text{ = (2mkT )}}^{{\frac{3}{2}}} \frac{V}{{h^{3} }} \hfill \\ q_{rot} = \sum\limits_{J = 0}^{\infty } {\sum\limits_{M = - J}^{J} {e^{{ - \frac{{E_{j}^{\text{rot}} }}{KT}}} } } \hfill \\ q_{\text{vib}} { = }\sum\limits_{n = 0}^{\infty } e^{{ - \frac{nh\upsilon }{KT}}} \hfill \\ q_{\text{el}} { = }\sum\limits_{i} {e^{{ - \frac{{E_{i}^{\text{el}} }}{KT}}} } \hfill \\ q_{\text{nucl}} { = }\prod\limits_{j} {(2S_{j} } + 1) \hfill \\ \end{aligned} $$
(4)
$$ \Delta G(T) = \Delta H(T) - T\Delta S(T). $$
(5)

Gibbs free energy, enthalpy, entropy and internal energy are the most important thermodynamic relationships that these are illustrated in equations. The partition function \( \left( q \right) \) is divided into 5 parts that these include transition \( \left( {q_{\text{tr}} } \right) \), rotation \( \left( {q_{\text{rot}} } \right) \), vibration \( \left( {q_{\text{vib}} } \right) \), electron \( \left( {q_{\text{el}} } \right) \) and nuclear \( \left( {q_{\text{nucl}} } \right) \) that in formula 4 they were all expressed [37]. Ab Initio calculations show that the complexation of the Gemcitabine anticancer drug with the carbon nanotube in this interaction induces an effective change that leads to a less bonding energy for this interaction in solvent environment [38,39,40] here we want to describe the obtained results.

Results

Investigation of DFT calculations is extremely relevant to diagnose the potential applications of carbon nanotubes as efficient Carrier for targeted drug delivery including Gemcitabine anticancer drug [38,39,40]. In this project, the interaction between Gemcitabine anticancer drug and SWCNT is investigated that the SWCNT has a length of 10 Å and the bond length of each carbon–carbon is 1.41 Å. In the first step, the Gemcitabine anticancer drug and the SWCNT were plotted separately and optimized by M06/6-311G (Figs. 1, 2).

Fig. 1
figure 1

Optimized structure of Gemcitabine anticancer drug

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Fig. 2
figure 2

Optimized structure of SWCNT

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We first calculated the Charge per atom by M06 method. Then, we approach the drug to the SWCNT in different directions. Because we do not know in advance how the drug interacts with the nanotube. Drug sites with more negative charge are more likely to interact. We bring the drug closer to the carbon nanotube from directions that have a more negative charge (Fig. 3). So, the drug is approaching the nanotube in four different directions which is shown in Figs. 4, 5. According to the results in Table 1, configuration 1 is the most stable configuration. It is predicted that the Gemcitabine anticancer drug has approached SWCNT because it has the lowest energy.

Fig. 3
figure 3

The charge of atoms in the Gemcitabine anticancer drug

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Fig. 4
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a Configuration 1. b Configuration 2. c Configuration 3. d Configuration 4

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Fig. 5
figure 5

The angles and lengths of the Bond in Configuration 1

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Table 1 Total energy of different configurations with M06/6-311G Method
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Changes in transplant length and angles, although very small, indicate an interaction between the drug and SWCNT that shown in Fig. 3 and Tables 2 and 3.

Table 2 Bond length in Gemcitabine anticancer drug and Configuration 1 by M06/6-311G method
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Table 3 Angles in Gemcitabine anticancer drug and Configuration 1 by M06/6-311G method
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The negative amount of bond energy indicates that the Gemcitabine anticancer drug is adsorbed on SWCNT that this amount of bond energy is obtained from the relation (6) [38,39,40], which is a strong interaction due to its large amount.

$$ E_{\text{bin}} = E_{\text{Config1}}^{\text{tot}} - \left( {E_{\text{CNT}}^{\text{tot}} + E_{\text{drug}}^{\text{tot}} } \right). $$
(6)

According to Table 4, the top percent of the gap band energy shows the percentage of changes of Homo–Lumo band energy that \( \frac{{\Delta E_{\text{g}} }}{{E_{\text{g}} }} \) Obtained from the relation (7):

Table 4 The values obtained for configuration1 by M06/6-311G
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$$ \frac{{\Delta E_{\text{g}} }}{{E_{\text{g}} }} = \frac{{E_{\text{g(config1)}} - E_{\text{g(CNT)}} }}{{E_{\text{g(CNT)}} }} $$
(7)
$$ E_{\text{g}} = E_{\text{Lumo}} - E_{\text{Homo}} . $$
(8)

Due to its high percentage (\( \frac{{\Delta E_{\text{g}} }}{{E_{\text{g}} }} \)), It can be said that the electrical properties of CNTs will change a lot after the interaction, which confirms the effect of Gemcitabine anti-cancer adsorption on SWCNT. The difference between the highest molecular orbitals having an electron and the lowest molecular orbital without an electron is called Eg that is obtained from the relation (8) and shown in Table 4.

Table 5 indicates which in NBO method which about interactions in both filled and virtual spaces which could raise the analysis of intermolecular interactions. The results of NBO show which the \( {\text{LPN}}_{ 1 7} \to \sigma^{ * } {\text{C}}_{61} \) donor–acceptor interactions are most significant ones. The NBO calculations are applied for quantitative analysis of the electron delocalization from the lone pair electrons of the nitrogen atoms of Gemcitabine to the sigma antibonding orbitals of C of SWCNT. In complex of Gemcitabine-SWCNT, the lone pairs of Nitrogen atom take part as donors and the σ*-orbitals of Carbon of carbon nanotube act as acceptors. As can be seen, the stabilization energy of \( {\text{LPN}}_{17} \to \sigma^{ * } {\text{C}}_{61} \) interaction reduces on crossing from the gas phase to the solvent phase. In addition, the Results show which the occupation number alterations of donor (OND) orbitals for complex of Gemcitabine-SWCNT are sensitive to the solvent environment.The occupation number values of acceptor (ONA) orbitals in gas, ethanol, methanol and water phase are different. Thus, it can be deduced that the gas phase has a relatively low effect on N17….C61 bond of complex of Gemcitabine-SWCNT.

Table 5 NBO analysis of CNT-Gemcitabine complex for configuration 1, occupation numbers of donor (OND) and acceptor (ONA) orbitals and their energies (in kcal/mol) in some important orbitals at M06/6-311 g level of theory for configuration1
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AIM analyses

A topological analysis is performed to compute the electron density \( (\rho ) \) and its second derivative \( (\nabla^{2} \rho ) \) applying AIM method.AIM method Suggests which for the covalent bond interactions, the \( \rho \) is large and value of the \( \nabla^{2} \rho \) is negative. Table 6 displays the computed topological parameters of complex of (Gemcitabine–SWCNT) in the gas, Ethanol, Methanol, Water phase. The \( \rho {\text{N}}_{17} \ldots C_{61} \) values show which the complex of (Gemcitabine – SWCNT) has the least electron density in the solvent phase and the most electron density in gas phase. The determinate interactions display the positive \( \nabla^{2} \rho \).The positive values of \( \nabla^{2} \rho \) of N17….C61 \( (\nabla^{2} \rho {\text{N}}_{17} \ldots C_{61} ) \) bond shows a reduction in the electrical charge during the transplantation.

Table 6 Topological parameters of CNT-Gemcitabine complex in different solvents at M06/6-311 g level of theory for configuration1
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Finally, we investigate the effects of dielectric constant on this interaction with M06/6-311G level using Gaussian 09. Therefore in this project, the cavity radius was calculated and applied for Gemcitabine anticancer drug and SWCNT and complex of (Gemcitabine–SWCNT) that the calculated cavity radius for Gemcitabine anticancer drug and SWCNT and complex of (Gemcitabine–SWCNT) are 5.08 and 5.75 and 6.45 Å, respectively. Tables 7, 8, 9 show the Gibbs free energy, enthalpy and entropy values of the Gemcitabine anticancer drug and the SWCNT and complex of (Gemcitabine–SWCNT) in different dielectric constant at temperatures of 298,300,305,310 K respectively.

Table 7 Calculated thermodynamic values of G, H (kcal/mol), S (kcal/mol K) at different dielectric constants at different temperatures for the gemcitabine anticancer drug in the M06/6-311G level
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Table 8 Calculated thermodynamic values of G, H (kcal/mol), S (kcal/mol.K) at different dielectric constants at different temperatures for the carbon nanotube in the M06/6-311G level
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Table 9 Calculated thermodynamic values of G, H (kcal/mol), S (kcal/mol K) at different dielectric constants at different temperatures for the compound of (Gemcitabine -SWCNT) in the M06/6-311G level
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Figures 6, 7, 8 show that the Gibbs free energy, enthalpy and entropy value are dependent on the dielectric constant. It is also observed that by increasing the temperature from 298 to 310 K, the Gibbs free energy of the Gemcitabine anticancer drug and the SWCNT and complex of (Gemcitabine–SWCNT) will decrease in the gas, ethanol, methanol, and water phase. So that the lowest Gibbs free energy is at 310 K. At temperature range of 298–310 K, the Gibbs free energy decreases with increasing dielectric constant. The Gibbs free energy is the most negative value when the Gemcitabine anticancer drug and SWCNT and the complex of (Gemcitabine–SWCNT) are placed in water solvent with dielectric constant of 78.39. It is also observed that by increasing the dielectric constant of the solvent in the temperature range of 298–310 K, the calculated enthalpy values shift to negative values that the most negative amount of enthalpy in water occurs and the entropy values are positive, and the most positive value of entropy happens when the solvent is water (Tables 7, 8 and 9).

Fig. 6
figure 6

Thermodynamic values of a G (kCal/mol), b S (kcal/mol K) for Gemcitabine anticancer drug

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Fig. 7
figure 7

Thermodynamic values of a G (kCal/mol), b S (kcal/mol K) for SWCNT

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Fig. 8
figure 8

Thermodynamic values of G (kCal/mol), b S (kcal/mol K) for complex of (Gemcitabine-SWCNT)

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Now consider the following reaction:

$$ {\text{Gemcitabine}} + {\text{SWCNT}} \to {\text{Gemcitabine}} - {\text{SWCNT}} . $$
(9)

The values of \( \Delta G_{\text{reaction}} \) and \( \Delta H_{\text{reaction}} \) and \( \Delta S_{\text{reaction}} \) can be obtained from the following relation:

$$ \Delta Q = Q_{\text{Gemcitabine - SWCNT}} - (Q_{\text{Gemcitabine}} + Q_{\text{SWCNT}} ) $$
(10)
$$ \Delta Q^{\text{BSSE}} = \, Q_{{{\text{Gemcitabine - SWCN}}T}} - (Q_{\text{Gemcitabine - SWCNTghost}} + Q_{\text{SWCNT - Gemcitabineghost}} ). $$
(11)

The calculated interaction energies have corrected for basis set superposition errors (BSSE) [38,39,40]. ∆QBSSE of Gemcitabine drug molecule on SWCNT and SWCNT surface is perturbed. In order to correct the Reciprocating state for eliminating the error of BSSE for ∆QBSSE is obtained by applying Eq. 11. Figure 9 (Table 10) shows that by the temperature increases, the \( \Delta G^{\text{BSSE}}_{\text{reaction}} \) values increase because it has the lowest value at 298 K, so that reaction 9 at 298 K is most stable. It also shows that with increasing temperature, the \( \Delta H^{\text{BSSE}}_{\text{reaction}} \) values decrease, so that the minimum value is 310 K. It also shows that with increasing temperature, the \( \Delta S^{\text{BSSE}}_{\text{reaction}} \) values decrease that the most positive value is observed at 298 K, and the reaction is more stable. So in general water solvent is the best environment for reaction 9. The calculated Mulliken charge values per atom are shown in Table 11.

Fig. 9
figure 9

Thermodynamic changes of a ∆GBSSEreaction , b ∆HBSSEreaction (kcal/mol), c ∆SBSSEreaction (kcal/mol K), d Ebin BSSE (kcal/mol) for reaction 9

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Table 10 The values of thermodynamic changes of G, H (kcal/mol), S (kcal/mol.K), Ebin (kcal/mol) and ∆QBSSE calculated at different dielectric constants at different temperatures for reaction between Gemcitabine anticancer drug and carbon nanotube in M06/6-311G level for configuration1
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Table 11 Average Mulliken charge values per atom for complex of (Gemcitabine-SWCNT) in three different solvents and gas phase
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Nuclear magnetic resonance parameters

NMR spectroscopy is a method that uses the magnetic properties of specific atomic nuclei to determine the physical and chemical properties of the atoms or molecules in which they are present. This is based on the phenomenon of nuclear magnetic radiation and it can provide information on the structure, reaction state, and chemical environment of the molecules. We use here Ab initio calculations to calculate nuclear magnetic shielding through M06/6-311G method using Gaussian 09. The calculated magnetic shielding anisotropy (∆σ, ppm), shielding asymmetry (η) and the chemical shift tensor (δ) calculated for C61, N17, C1, C2, N3, N5 and C6 nuclei of Gemcitabine drug with carbon atoms of the open end of a SWCNT system and the graphs of calculated isotropic magnetic shielding constants σiso (ppm), anisotropic magnetic shielding tensors σaniso (ppm), in gas phase, water, methanol and ethanol are presented in Fig. 10 (Table 12). As was envisaged, the NMR shielding tensors of C61, N17, C1, C2, N3, N5 and C6 nuclei are drastically affected by the atom to that they are bonded and by the kind of the bond to the neighboring atom. The results in Table 1 display which intermolecular interactions play a fundamental role in determining C61, N17, C1, C2, N3, N5 and C6. It is clear which one atom in Gemcitabine–SWCNT system has maximum σiso value in ratio the other atoms of complex of (Gemcitabine–SWCNT). This value is related to the N17 atom. The maximum ∆σ has value for C2 atom is in complex of (Gemcitabine – SWCNT). C2 has the largest value (δ) in the complex of (Gemcitabine–SWCNT) system. C61 has the largest intermolecular effect.

Fig. 10
figure 10

NMR parameters calculated for complex of (Gemcitabine-SWCNT) in a gas phase, b water, c ethanol and d methanol

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Table 12 Computed NMR parameters for complex of (Gemcitabine – SWCNT) in three different solvents and gas phase
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Effect of carbon nanotube length and diameter

Another factor, we examined in the interaction of the Gemcitabine anticancer drug with the carbon nanotube was the diameter of the carbon nanotube. The Gemcitabine drug connected to the carbon nanotube from the wall. The binding energy between the Gemcitabine anticancer drug and the carbon nanotube is obtained from the following relationship:

$$ BE = n\ln \left| d \right|. $$
(12)

The diameter of the carbon nanotube is d. It is observed that the Gemcitabine-SWCNT bond becomes longer as the carbon nanotube diameter increases. So we conclude that with increasing SWCNT diameter we have an exponential increase during Bond Length and a decrease in bonding energy (see Fig. 11).

Fig. 11
figure 11

a The binding energy between the Gemcitabine radicals with SWCNT as a function of the SWCNT diameter. b Length of Gemcitabine -SWCNT bond as a function of CNT diameter

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Another factor to consider in Gemcitabine-SWCNT interaction is the different lengths of SWCNT. If the SWCNT length is increased, the bonding energy between the Gemcitabine-SWCNT will increase. The Gemcitabine-SWCNT bond length is a function of the SWCNT length. Therefore, as the carbon nanotube length increases, the Gemcitabine-SWCNT Bond Length decreases that this is a linear process. And because here the length 10 Å and diameter 5 Å had the same bond energy, this nanotube was selected for calculations (see Fig. 12).

Fig. 12
figure 12

a The binding energy of the Gemcitabine radicals with the SWCNT as a function of SWCNT length. b Length of Gemcitabine -SWCNT bond as a function of SWCNT length

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Solution phase

In order to comprehend the role of solvent on the interaction of Gemcitabine anticancer drug with the SWCNT, the stability and solubility of studied models in solvent phase are studied and the achieved results are collected in (Table 13). Table 13 electrophilicity of complex of (Gemcitabine-SWCNT) increased, that suggests an increase in charge transfer from Gemcitabine anticancer drug to SWCNT. The absolute hardness and softness are main properties for measuring a molecular stability and reactivity. Table 13 shows the absolute hardness of complex of (Gemcitabine-SWCNT) decrease which reflects the high reactivity of Gemcitabine anticancer drug with the SWCNT. The complex of (Gemcitabine-SWCNT) Ebin was calculated to be − 35.463 kCal/mol in gas phase and − 36.203 kCal/mol in water phase which is a decrease in Ebin from gas to liquid.

Table 13 Quantum chemical descriptors for the complex of (Gemcitabine-SWCNT)
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Conclusions

In this project, it appears that the results of the SCRF method in particular using the polarizable continuum model (PCM) are sensitive to the solvent polarity of the surroundings. The interaction among molecules of solvent and solute will enhance by increasing the molecule dipole moment. As a result, interaction of solute molecules will reduce, that efficacies Gibbs free energy and entropy.

Thus Gibbs free energy and enthalpy Values of Gemcitabine anticancer drug and SWCNT and the complex of (Gemcitabine-SWCNT) decreased when the dielectric constant of the solvent increases but the entropy increases. Increasing the temperature, \( \Delta G^{\text{BSSE}}_{\text{reaction}} \) increases so that it is the lowest value for 298 K. The most positive value of \( \Delta S^{\text{BSSE}}_{\text{reaction}} \) is for 298 K so the reaction 9 for 298 K is spontaneous. The best solvent for this reaction is water. The results obtained by M06/6-311G level indicate adsorption of the drug into nanotubes is physical. Thermodynamically, Formation of drug-CNT complex in water is more favorable with respect to other solvent because it has the most negative Gibbs free energy. Moreover, the AIM and NBO analyses are used to elucidate the nature of N17…C61 bond interactions in complex of (Gemcitabine-SWCNT). In this complex, the created interactions display the positive \( \nabla^{2} \rho \).The positive values of \( \nabla^{2} \rho \) at complex of (Gemcitabine-SWCNT) of N17–C61 bond demonstrate reduction of electronic charge along the bond path. The point is that the attained data at the AIM method are entirely conforming to the found data in the NBO framework.