Role of Intermolecular Interactions in Assemblies of Nanocontainers

Article pubs.acs.org/JPCC

Role of Intermolecular Interactions in Assemblies of Nanocontainers Composed of Carbon Nanotubes and Magnetic Nanoparticles: A Molecular Dynamics Study Tomasz Panczyk,*,† Tatiana Da Ros,‡ Giorgia Pastorin,§ Anna Jagusiak,∥ and Jolanta Narkiewicz-Michalek⊥ †

Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences ul. Niezapominajek 8, 30239 Cracow, Poland Department of Chemical and Pharmaceutical Sciences, University of Trieste, Piazzale Europa 1, Trieste, 34127, Italy § Department of Pharmacy, National University of Singapore S4 Science Drive 4, 117543, Singapore ∥ Chair of Medical Biochemistry, Jagiellonian University Medical College, ul. Kopernika 7, 31034 Cracow, Poland ⊥ Department of Chemistry, Maria Curie-Sklodowska University pl. M. Curie-Sklodowskiej 3, 20031 Lublin, Poland ‡

ABSTRACT: In this work we analyze the effects of the interactions between nanocontainers composed of carbon nanotubes and magnetic nanoparticles, studying the stability of the capped forms of the nanocontainers in intermolecular collisions, in water solution. The stability depends on the collision line between two nanocontainers and increases when the considered objects are filled with cisplatin molecules. It is found that at room temperature the capped forms are always stable and the intermolecular collisions do not lead to leakage of drug molecules. However, the estimated range of intermolecular interactions suggests that the current architecture of the nanocontainers is unstable in terms of colloidal stability. Further functionalization of nanotubes sidewalls is necessary in order to reduce strong hydrophobic interactions between nanocontainers. Studies of the interaction with external magnetic field revealed that assemblies of the nanocontainers undergo magnetically triggered uncapping. The release of cisplatin follows a simple one-dimensional diffusion mechanism at long times; at initial times and for high initial concentration of drug molecules the effect of dragging is observed, that is, the diffusion of cisplatin becomes apparently a nonactivated process.

1. INTRODUCTION Carbon nanotubes (CNT) and magnetic nanoparticles (MNP) are widely studied as materials of great importance in medical diagnosis or as drug carriers. However, the studies are normally focused on their individual applications rather than on their conjugates. However, CNT-MNP conjugates provide even more interesting properties, though their fabrication might be very challenging. The magnetic properties of the MNPs are useful mainly in Magnetic Resonance Imaging (MRI); however, there are many other medical applications based on these properties like in vitro bioseparation, magnetically addressed drug delivery or hyperthermia.1 Similarly, CNTs are currently recognized as very promising drug carriers acting either as molecular cargoes or as a part of drug-CNT conjugates enhancing pharmacological activity of drugs.2 Coupling of magnetic nanoparticles with carbon nanotubes allows for creation of magnetically active nanotubes3 that, in turn, may act as multifunctional scaffolds for magnetization of cancer cells.4 Recently we proposed and carefully investigated another concept of coupling useful properties of CNTs and MNPs. Namely, by covalently linking MNPs to CNT edges (tips) using some linear molecules (e.g., polyethylene glycol or alkane © 2013 American Chemical Society

chains) a functional nanocontainer (NC), possessing extraordinary properties, can be created.5−7 We found that the NC may undergo reversible cycles of capping and uncapping the nanotube tips by applying an external magnetic field. This NC property might be extremely useful in processes of controlled drug delivery and release; moreover the triggering factor (external magnetic field) is safe and easy to apply in in vivo treatment. However, fabrication of the functional NCs might be a challenge since precise control of their molecular topology is a key factor. Further, the required property of the NCs depends on many parameters which need to be tuned in order to produce that specific activity of the system. The role of those parameters was carefully studied and a general conclusion is that the required ranges of parameters are within the limits found for materials commonly used in the fabrication of magnetic nanoparticles’ stable suspensions.7−9 However, our previous studies were focused on one single and isolated NC, as necessary starting point for more advanced studies. Received: October 31, 2013 Revised: December 13, 2013 Published: December 30, 2013 1353

dx.doi.org/10.1021/jp410736s | J. Phys. Chem. C 2014, 118, 1353−1363

The Journal of Physical Chemistry C

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the force field associated with the NC. Their detailed description (as well as method of coupling the processes of magnetization reversal with the standard atomistic molecular dynamics scheme) is provided in ref 9. The key assumptions are that the MNPs are single domain particles revealing the uniaxial magnetic anisotropy and the magnetization reversal proceeds according to the coherent rotation mechanism.15 Both assumptions are justified according to the results of experimental studies of cobalt and a few other nanoparticles.16 The force field component associated with the interactions of cisplatin was setup according to the methodology outlined in our recent paper7 and both molecular topology and interaction parameters for cisplatin were taken from the work by Lopes et al.17 That force field assumes a standard topology of CP molecule, that is, nonaquated and uncharged cis-(NH3)2PtCl2 structure. It should be noted that CP undergoes various hydration reactions in aqueous media producing among the others a positively charged [(NH3)2PtCl(H2O)+] complex, CPW+. That hydration reaction is characterized by pKa values not smaller than 2 and its rate decreases with the increasing concentration of Cl− ions.18 Thus, normally, the concentration of CPW+ in physiological fluid is small (∼1%). Further, the concentration of CPW+ in the inner cavity of a CNT is difficult to predict since the potential field coming from the nanotube walls might alter the occurrence of that hydration reaction. Therefore, we neglected the possible presence of small number of CPW+ ions in our model. The electrostatic interactions between partial charges are explicitly accounted for using Coulomb potential, however, due to the implicit solvent model assumed,14 the electrostatic interactions are screened according to the Debye screening model, that is, by applying the exponential decay of forces with the separation distance. For the considered conditions the Debye screening length is set to 8 Å. It corresponds to properties of physiological fluid, that is, 0.145 mol L−1 NaCl solution. The cutoff distance for both electrostatic and Lennard-Jones interaction is 30 Å. For the dispersion and electrostatic interactions between MNPs and cisplatin, the cutoff distance is 125 Å. All calculations were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (lammps) code19 with several extra classes written from scratch and working on magnetic torques. Normally, calculations were performed in NVT ensemble using 2.0 fs time step. The number of atoms creating a single NC was 47370, while the number of cisplatin molecules confined in a single NC was 170. The latter was determined according to two rationales: (i) we need possibly large number of CP molecules in order to reach possibly high resolution of release patterns and (ii) we cannot use very high loadings because we work on the level of implicit solvent model (ISM)14 and we need some free space available inside the NC in order to keep the assumptions of the ISM reliable. Therefore, we use 170 CP molecules per single NC, this gives molar concentration of CP inside the NC 3.2 mol L−1 and volume fraction of space occupied by CP 0.33. That concentration is much higher than the saturated solution of CP in bulk water at 25 °C, that is, 0.008 mol L−1. However, due to potential field created by the CNT walls such a concentration is reliable and does not lead to nucleation of CP into clusters. The calculations were based on the Langevin dynamics20 which, according to the fluctuation−dissipation theorem, links the dynamics of a system with friction forces what, in turn, allows for a proper description of the real time. The friction

Therefore, the aim of the current study is an analysis of the effects coming from the interaction between multiple NCs on their overall performance as magnetically controlled nanocapsules. Particularly, we need to confirm the stability of drug containing NCs under typical working conditions and when collisions between two NCs occur in solution. It is also important to estimate the range of dispersion interactions between two NCs, range that, in turn, controls the nucleation of many NCs into bigger agglomerates. Finally, the process of magnetically triggered uncapping, occurring in clusters of two NCs, needs dedicated studies since this process might be affected by intermolecular interactions.

2. METHODS The topology and the force field controlling properties of the NCs are actually the same as those utilized in our recent work9 devoted to magnetic anisotropy effects in processes of magnetically assisted uncapping of NCs. However, due to the presence of two NCs in the simulation box, the sizes of the NCs have been slightly reduced in order to keep the computational costs on a comparable level. A single NC is composed of a three-walled carbon nanotube with the inner and outer diameters 23.5 and 37.7 Å, respectively. The corresponding chiral indices are (30,0), (39,0), and (48,0). The CNT length was 200 Å, the hemispherical tips were removed and the extremities of the CNT itself were saturated by amide groups as possible stoppers. The choice of amide groups for functionalization of the CNT tips is justified because amidation reaction is a standard step in experimental procedures of attachments of various functions to terminal rings of carbon nanotubes. 10 We assumed a uniform distribution of amide groups on the outermost nanotube ring. The terminal carbons on the inner CNTs tips were left without saturation by hydrogens because this has negligible effect in systems’ energies11 and this would additionally slow down the computations due to high vibration frequencies of hydrogen atoms. The MNPs were bound to CNT tips by triethylene glycol chains. The MNPs were composed of 80 Å in diameter magnetic cores which are covered by 10 Å thick silica shells, so the total diameters of the MNPs are 100 Å. The magnetic cores were assumed to be cobalt nanoparticles while the silica shells act as protective and stabilizing agents. The presence of silica reduces the range of dispersion forces originating from metallic nanoparticles, provides electrostatic stabilization in solution and provides functional silanol groups for attachment of polyethylene glycol linkers.9 The force field associated with the NC consists of several types of interactions. The interatomic interactions are set up according to the General Amber Force Field (GAFF).12 The structure of the nanotube remains rigid since the CNT deformation due to interactions with, for example, magnetic NPs is, at the considered conditions, unlikely.11 A very important component of the force field is the dispersion interaction between MNPs and other species. It can be adequately described within the Hamaker theory of dispersion interactions.13 Additionally, due to large size of the simulation box (500 × 500 × 500 Å), it is impossible to treat solvent (water) molecules explicitly. Therefore, the recently developed14 implicit solvent model, based on Hamaker theory, is utilized here. The magnetic interactions due to magnetic anisotropy of the MNPs, external field and dipole−dipole interactions complete 1354



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forces acting on each atom were determined from the EinsteinStokes equation and were suitably rescaled in order to match the diffusivity of cisplatin in water. That diffusivity was determined in our recent paper involving explicit water molecules.7

3. RESULTS AND DISCUSSION 3.1. Systems Topology and Organization of the Studies. Interactions between many NCs in solution become important at higher concentrations of NCs, that is, when the distance between individual NCs becomes comparable to the interaction length. Using molecular dynamics simulations, we can directly observe how these interactions affect the behavior of the entire system. However, due to very large sizes of individual NCs we are able to analyze only two NCs in the simulation box simultaneously. This limitation comes from enormous computational costs related to solution of motion equations with a mixture of short and very long cutoffs and small and very large atomic masses existing in the system. Nevertheless, analysis of the interaction between only two NCs allows us to draw some important conclusions concerning many-body effects in assemblies of the NCs in solution. Because the NCs are not isotropic in terms of shapes and force field topology, we will focus on four initial arrangements that seem to be particularly interesting while studying interparticle collisions in solution. These arrangements are shown in Figure 1. In any of the configurations shown in Figure 1, the nanotubes interiors were filled with cisplatin molecules and the systems were allowed to relax for 4 ns. From analysis of the time dependence of systems’ total energy it was found that 200 ps of simulation time were enough for the equilibration. During that equilibration period the nanotubes were fixed in their initial positions, while MNPs, linkers, and cisplatin molecules were allowed to move according to classical MD trajectories. In any case, the initial distances between the points of impact were chosen in such a way that the interaction energy between the NCs, UBB, became negligible, so that the equilibration process was not affected by the body−body interactions. From the last 2 ns of the equilibration period some molecular properties of the systems were determined. Figure 2 shows the radial distribution functions, rdfs, for the distances between Pt− Pt and N−Cl atoms in CP molecules and also for the distances between MNP and Pt. Analysis of these rdfs leads to the conclusion that cisplatin (CP) molecules are oriented in such a way that Cl atoms point toward NH3 groups. This is due to the charge distribution within the CP molecule. Positive partial charges are located on hydrogen atoms creating the NH3 groups while the negative charges are located on Cl atoms. Therefore, the minimum distances between Pt−Pt are larger than those between N−Cl. The rdfs for the MNP-Pt distances indicate that CP molecules adsorb on the MNPs surfaces. There is large density excess at the vicinity of the MNPs surfaces. This is due to quite strong interaction between CP and MNPs. The effective energy of interaction comes from either dispersion or electrostatic interactions. The latter is responsible for spatial orientation of CP molecules in reference to MNPs surfaces, that is, the NH3 groups point toward the MNP surfaces since they carry positive partial charges (see inset in Figure 2). As already mentioned, the MNPs are negatively charged. The rdfs for the MNP−Pt distances reveal three layers of CP adsorbed on the surfaces of MNPs. Their densities decrease

Figure 1. Side views of the studied initial arrangements of two NCs before collision and the subsequent magnetically triggered uncapping. The small insets show the corresponding top views. In parts C and D, one of the MNPs has been removed for presentation purposes. The arrows indicate the points of impact analyzed according to the controlled collisions tests.

Figure 2. Radial distribution functions (rdfs) for the distances between Pt−Pt, N−Cl, and MNP−Pt for cisplatin (CP) molecules confined in the internal space of the NC.

with the distance from the MNPs and beyond about 80−100 Å from the MNP centers no extra peaks can be observed. It 1355



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example, a random distribution of amide groups over the threewalled CNT tip we could anticipate a slight increase of the interaction energy because of closer contact of some amide groups with the MNPs surfaces. The uncapped state, that is, when the MNPs locate on the CNT sidewall, can be viewed as the least stable since the depth of the potential well drops to −56 kJ mol−1. This agrees with the required energy profile of the NC22 because the uncapped states should spontaneously convert into the capped ones. The apparent instability of the uncapped states is, however, a matter of the observation time scale. Due to large masses of the NCs, the transition into the capped states can reach macroscopic times. Thus, once uncapped the NC may stay in that state for hours or even weeks,7,8 as estimated in our previous works. In the uncapped state of the initially filled NC, the dispersion interaction component is almost the same as in the unfilled state. There are, however, large differences in the total energies meaning that the electrostatic component is important in that case. Indeed, the electrostatic interactions are long-ranged thus, even in the uncapped state, the MNPs interact with the CP molecules residing in the CNT interior. Another phenomenon enhancing the range of the electrostatic interaction is the presence of the released CP molecules in the surroundings of the MNPs. They appear because the uncapped state of the NC allows for migration of the CP molecules from the CNT interior to the CNT sidewalls7 in the first stage of the release process. This leads to significant energy increase and enhances the energy fluctuations. The electrostatic component would be slightly enhanced in case of a presence of hydrated CPW+ complexes. It would lead to slightly stronger stabilization of the capped forms of the NC since the concentration of CPW+ is supposed to be small. 3.2. Collision Tests. Stability of drug-containing NCs in solution is a crucial requirement for their function as drug delivery vehicle. Intermolecular collisions (especially the NC− NC collisions) may potentially lead to transient detachments of MNPs from the CNT tips what, in turn, may lead to uncontrolled leakage of drug molecules. The depths of the potential wells which keep the MNPs stuck to the CNT tips are deep, as shown in the previous subsection. However, the NCs are very heavy when compared to masses of typical molecules (mass of a single NC is ∼3.9 × 106 g mol−1) and hence they carry very large momentum. Moreover, the mass is not distributed uniformly within the NC. The CNT rods are much lighter than the MNPs which, in turn, are connected to the CNT by very light linkers. Therefore, collisions between multiple NCs may lead to destruction of the subtle energy balance existing within the capped state of the NC and initiate the uncontrolled leakage of drug molecules. Analysis of direct collisions between two NCs provides, thus, very important conclusions concerning the stability of the NCs. However, due to friction forces acting during the motion of the NCs in solution, it is difficult to define strict parameters of the collisions. Particularly, the total energy of the system is not conserved as the system interacts with the heat bath. The most useful parameter of collision would be an initial kinetic energy (impact temperature) of the colliding objects.23 However, that energy is not conserved either because the friction forces continuously tend to restore the target temperature defined by the heat bath. Thus, the impact temperature would depend on the initial distance between colliding objects. A collision between two objects consists essentially of two stages; the first is the stage of approaching and subsequent total

implies that at the considered conditions the CP does not form clusters/aggregates inside the CNT. This is in agreement with previous findings where density profiles along the nanotube axis were directly determined.7 Experimental studies suggest, however, that CP forms aggregates visible in TEM.21 This qualitative difference is due to two factors: (i) We use a much narrower nanotube with an internal diameter 23.5 Å, while the nanotubes used in experimental studies had an average diameter 100 Å. This means that in our model the potential energy field coming from nanotube walls is stronger and prevents creation of clusters. (ii) The concentration of CP in our model system is smaller than in ref 21 and this obviously reduces the likelihood of clusters formation. Analysis of the potential energy between MNPs and other components of the NCs leads to interesting conclusions. These energies are listed in Table 1 and are split into dispersion and Table 1. Mean Interaction Energies between MNPs and Other Components of the System for Various States of the NCs, That Is, Capped and Filled with CP Molecules, Capped and Without CP Molecules in the System, and Uncapped and Without CP Molecules in the Systema state

⟨Utot⟩, kJ/mol

capped, filled with CP capped, unfilled uncapped, unfilled uncapped, filled with CP

−182 ± 8 −93 ± 4 −56 ± 2 −114 ± 18

⟨Udisp⟩, kJ/mol −82 −80 −54 −51

± ± ± ±

7 3 2 13

⟨Uel⟩, kJ/mol −100 ± 8 −13 ± 4 −2 ± 2 −63 ± 16

⟨Utot⟩ is the mean total potential energy per single MNP, ⟨Udisp⟩ is its component coming from dispersion interactions, while ⟨Uel⟩ is the component coming from electrostatic interactions. a

electrostatic components. Because the interaction energy of a single MNP informs about the stability of a given state of the NC, we can conclude that the capped states of the NCs are additionally stabilized by the presence of CP molecules. The stabilization is mainly due to electrostatic interactions between CP and MNPs since the dispersion interaction component is almost identical in both filled and unfilled states of the NC. The very deep potential energy well −182 kJ mol−1 found in the capped and filled state of the NC implies that any spontaneous uncapping (e.g., due to thermal fluctuations) is highly unlikely. Further, to initiate the CP release from that state, large amount of energy must be transferred to the system. This is very useful conclusion since the presence of the encapsulated drug molecules does not worsen but improves the overall energy balance between the capped and uncapped states of the NC. Thus, in case of high loadings of CP molecules, as normally found in experiments,21 the capped states of the NC will be even more stable. In ref 21, the mass ratio of CP to MWCNT was found to be 0.621, and this is much higher than in the present model system, that is, 0.09. The lack of CP molecules inside the NC decreases the depth of the potential well to −93 kJ mol−1. However, the capped state is still relatively stable in this case. The electrostatic component (−13 kJ mol−1) is much smaller than in the case of the filled state but it is still non-negligible as it comes from interaction with amide groups located on the terminal rings of the outermost nanotube. Thus, the presence of amide groups provides an extra energy stabilizing the capped states. The amount of that energy will, to some extent, depend on the distribution of those groups on the CNT tips. Considering, for 1356



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potential energy between the NCs. In the case of the arrangement A such a point does not exists because of electrostatic repulsion between MNPs. Thus, the NCs were placed at the point of minimal dispersion interaction energy between the MNPs. As shown in Figure 3, the collisions lead to transient temperature increases and detachments of MNPs from the nanotube tips. The ranges of these phenomena depend on the initial collision velocities. For example, the collision with the velocity 0.25 Å ps−1 corresponds to the temperature ∼625−650 K. The initial kinetic energy carried by both NCs, which must be transferred to the heat bath, reaches 4875 kJ mol−1. As seen, the heat bath is able to restore the target temperature in about 100 ps. However, the collision destroys the initially capped states of the NCs leading to slit sizes >7 Å. This is definitely an unstable state of the NC and there occurs leakage of CP molecules from the interior of the nanotube, as observed in simulations. On the other hand, such a highly energetic collision is very unlikely at conditions under which the NCs are designed to work. Local temperature fluctuations at about 310 K cannot reach the range of 600−650 K, thus the discussed situation is only illustrative. The above results imply rather that the capped state of the NC would be unstable at 600 K. Table 2 shows the critical parameters determined from the collision tests for all the collision trajectories shown in Figure 1: These are the initial temperature reached by the system as a result of setting a given velocity along the collision trajectory, maximum slit size recorded upon collision, and the occurrence of CP molecules leakage. Analysis of these parameters leads to a few interesting conclusions. Namely, the stability of the capped states of NCs depends on the collision trajectory. As already mentioned, the mass is not distributed uniformly within the volume of the NC and this is the reason of various sensitivities of the NCs to the collision induced uncapping. In the case of the arrangements C and D, the impact points are located on the nanotubes, thus, at the moment of collision the MNPs take over the nanotubes due to very large inertia. As a result the detachment of the MNPs occurs easily as seen in the slit sizes in Table 2. The collision trajectory A seems to be the most resistant since even at the temperature 520 K no detachment occurs. This is understandable because in that arrangement the direct collision between two MNPs is realized. There occurs reflection upon collision but the nanotube, as it is much lighter than MNPs, easily follows the MNPs trajectories without a large retardation. It makes such collision trajectories strongly resistant to the uncapping. We can generally state that the temperature 310 K, at which the collision tests were performed, is safe in terms of a risk of collision induced uncapping and subsequent leakage of CP molecules. Collisions performed with the velocity 0.1 Å ps−1 generated the instantaneous temperatures of 355−375 K and no uncapping occurred at these conditions. There was some weakening of the capped states, as seen in Figure 3, but the systems quickly returned to the ground states without the leakage of CP molecules. For illustrative purposes, we also performed the same collision tests but without the CP molecules encapsulated in the NCs interiors. The maximum slit sizes obtained in those cases are collected in Table 2. The lack of CP molecules makes the NCs less stable in collision trajectories B - D and the general conclusion is that the lack of CP molecules reduces the stability of the capped forms in intermolecular collisions. This is consistent with the previous finding concerning the energies of

conversion of the kinetic energy into the potential energy. The second stage is the reflection accompanied by conversion of the potential energy into the kinetic energy. In case of inelastic collisions some energy dissipation may occur during the event.23 The largest net velocity between colliding objects is when the system crosses the minimum of the potential well between the colliding objects. Just after this point repulsion initiates and the collision begins. So, in the case of the NCs colliding in the dissipative environment, the best choice of the collision parameter is the value of the velocity component along the collision line taken at the minimum of the NC−NC potential energy. By setting various velocity values it is possible to monitor the effective temperatures reached by the system and the stability of the NCs at the considered conditions. The stability of the NC needs to be arbitrarily defined. The most obvious choice of the stability parameter is a slit size between the surface of the MNP and the nanotube tip. A critical slit size, below which the NC is still considered as stable, depends on the size of the encapsulated molecules. In the case of CP molecules the critical size can be identified with the van der Waals diameter of the largest atom (chlorine, 4.4 Å), which compares well with the minimal radius of CNT (4.8 Å), where CP molecules can penetrate.24 For smaller molecules, the critical slit size must be accordingly reduced but even for molecular hydrogen it is not smaller than 3.0 Å.25 Figure 3 shows the results of collision tests performed for the arrangements B and D. Analogous studies have been done for all the arrangements shown in Figure 1. The velocities of both NCs were altered in such a way that the velocity components along the collision lines increased by 0.1, 0.15, 0.2, and 0.25 Å ps−1. These velocities were imposed at the point of minimal

Figure 3. Results of collision tests for the arrangements B and D shown in Figure 1. Top panels show how the slit sizes between MNPs and CNT tips evolve upon collision with the initial velocities shown on the legends (in Å ps−1). For a given velocity only the strongest detachments are plotted. Bottom panels show the corresponding temperature profiles. 1357



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Table 2. Parameters Determined from the Collision Testsa velocity, Å ps−1

0.1

trajectory

0.15

0.2

0.25

0.1

0.15

temperature, K

0.2

0.25

0.1

0.15

max. slit, Å

A unfilled

375

420

520

625

B unfilled

375

420

505

625

C unfilled

375

420

500

610

D unfilled

355

420

500

600

3.27 2.02 2.93 3.48 3.52 5.83 3.32 3.34

3.16 2.22 4.42 6.52 5.52 5.77 5.92 6.82

4.22 3.38 5.93 7.72 5.42 9.77 7.33 9.42

0.2

0.25

leakage 7.13 7.53 7.02 8.79 6.63 8.57 8.52 10.39

no

no

no

yes

no

no

yes

yes

no

yes

yes

yes

no

yes

yes

yes

a The collisions were performed according to the trajectories shown in Figure 1 with the initial velocities along the collision lines 0.1, 0.15, 0.2, and 0.25 Å ps−1. The initial temperatures found during the collisions, maximum slit recorded and the occurrence of CP molecules leakage are arranged from the lowest to the highest velocity in each row. The numbers arranged in the rows titled “unfilled” show the maximum slit sizes obtained under the same conditions but for empty NCs.

the capped states shown in Table 1. However, the collision trajectory A indicates that in such an arrangement the empty NCs are more stable than the filled ones. This is due to the fact that the nanotube parts of empty NCs are even lighter than the filled ones. Thus, the nanotubes may easier follow the trajectories of the reflected MNPs without a damage of the (capped) structures. 3.3. Interaction Energy between NCs. Interaction energy between NCs in solution is a factor which controls stability of a suspension of the NCs at higher concentrations. If that energy is too high then an ensemble of the NCs will probably agglomerate in big clusters after some time. Thermal energy normally prevents agglomeration in case of light molecules but extended colloid particles usually need some extra treatment. Magnetic nanoparticles (e.g., iron oxide) are widely used in ferrofluid technology26 and as contrast agents in magnetic resonance imaging.1 A common method of their stabilization is the creation of a protective shell which either reduces the dispersion interaction energy by a suitable choice of shell material (refractive index) or produces electrostatic repulsion due to surface charge of that shell material. Carbon nanotubes are normally insoluble in water but suitable functionalizations lead to developing of various functions on their tips and/or sidewalls what, in turn, makes them soluble in water and allows for further attachments of, for example, bioactive moieties.27 Our model system partially accounts for the abovementioned approaches to stabilization of MNPs and CNTs. We assumed that magnetic cobalt cores are covered by thin negatively charged silica layers. Carbon nanotubes are, in turn, functionalized at the tips; their sidewalls are left without modification. The question is whether such a chemical state of the NCs prevents agglomeration of the NCs into a big cluster. Figure 4 shows how the total potential energy Ubb between two NCs changes upon collisions discussed in previous section. Depending on the initial alignment (collision line) Ubb takes various values which differ significantly between each other. Ubb also evolves in time (decreases) meaning that the system moves toward some equilibrium configuration. However, as seen in RMSD values the movement is slow (2−2.5 Å ns−1) because of very large masses of the NCs. The final state of each system would probably be similar to D, that is, a configuration that enhances the contact area between CNTs and is close to a global minimum of the potential energy between two NCs. However, the time necessary for such a transition will be different in each case.

Figure 4. Total potential energy between two NCs and root of mean squared displacement (RMSD) from the initial positions upon collisions occurring along the collision lines A−D.

Upon collision along the line A the energy Ubb is positive meaning that after 4.5 ns of the simulation time the NCs still experience repulsion being due to surface charges of the MNPs. After a sufficiently long time (inaccessible in MD simulations), this system probably ends up in a configuration similar to D as well. Thus, the effective interaction energy between two NCs at equilibrium might reach −700 kJ mol−1 no matter what was the initial collision trajectory. This is very high energy and an ensemble of the NCs will probably agglomerate into a cluster after sufficiently long times. We can thus conclude that the present architecture of the nanocontainer needs further treatment in order to produce the colloidal stability in solution. This could be done by developing hydrophilic functional groups on the nanotubes sidewalls.10,27 Moreover, such a treatment increases the biocompatibility and reduces cytotoxicity of nanotubes.28,29 1358



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On the other hand, the risk of agglomeration is important only at higher concentrations of the NCs in solution. The studied simulation box corresponds to the concentration of ∼100 g L−1, which is very high when compared to typical concentrations used in, for example, cell viability assays, which is of the order of a few tens of mg L−1.21 Assuming the concentration of the NCs is 0.1 g L−1, it can be easily estimated that the average distance between two NCs is ∼4000 Å, thus, it is much longer than the intermolecular interactions range. Therefore, suspensions of the NCs at concentrations about 0.1 g L−1 might be stable for quite long times; however, when delivered to target sites by means of, for example, EPR effect, the local concentration might increase and a fast agglomeration happen. 3.4. Magnetically Triggered Uncapping and Release of Cisplatin. In order to assess the occurrence of the magnetically triggered uncapping in clusters of the NCs we performed dedicated calculations. Thus, each configuration obtained in the relaxation studies (i.e., upon collisions) has been exposed to an external magnetic field. The conditions leading to a fast transition from the capped to the uncapped state of a single isolated NC are well-known from our previous studies.7,9 Here, we still assume that the MNPs reveal the saturation magnetization 1070 kA m−1, which is representative of cobalt nanoparticles.30 The magnetic anisotropy constant of the MNPs is assumed to be 107 J m−3, which is within the limits found for cobalt NPs, as discussed previously.9 The external field strength was set to 9.3 T and it acted along the principal axes of the simulation box. Under the above conditions, the doublets of the NCs behave similarly to single and isolated ones, though some important new phenomena can be observed. Figure 5 shows the time evolution of the systems A−D with the external magnetic field applied. The structures of the systems are described by the slit sizes, similarly like in Figure 3. Thus, changes of the topologies leading to total or transient uncappings are unambiguously depicted by the slits values. In Figure 5, t = 0 means that the systems are in configurations obtained after collision tests for small collision velocities, thus, in each case the silt sizes are close to zero meaning that the systems are in double capped configurations. As seen in Figure 5, the external field leads to vigorous changes in slit sizes. It should be recalled that mutual orientation of both MNPs belonging to a single NC or the CNT length has no meaning in terms of the occurrence of the uncapping.6,9 It depends solely on the initial angle between field direction and magnetic moment of each MNP. As discussed in ref 9, that angle induces torque trying to align the magnetic moment along the field direction and according to Neel rotation mechanism. This, in turn, induces the magnetic anisotropy torque which tries to rotate the whole MNP according to Brown rotation mechanism. That rotation may be blocked for some rotation axes by the bond linking the MNP with the CNT. As a result, that hindered rotation transforms into translation of the MNP which finally leads to uncapping. As seen, the uncapping mechanism is quite complex and depends on many factors like the mentioned initial angle between the field and magnetic moment, easy axis orientation, the position of the anchor on the MNP surface and the magnitude of magnetic anisotropy constant. Therefore, we observe in Figure 5 various sensitivities of the studied systems to the magnetic field; moreover, the uncapping profiles will obviously change significantly upon modification of the field direction or other relevant parameters.

Figure 5. Temporal evolution of the systems A−D defined as the time dependence of the slits between MNPs and carbon nanotube edges. Each color identifies a given system and every system is described by the time dependence of 4 slits denoted by the same color. The external magnetic field acts for 8 ns (from 0 to 8 ns), afterward the field is switched off and the systems freely evolve for the next 12 ns. Each panel on the graph shows how the direction of the applied field affects the uncapping processes, thus, Bx, By, and Bz mean that the field was applied along the x-, y-, and z-axis, respectively.

Generally, we can classify the behavior of the studied systems into four cases: (i) rapid uncapping, (ii) delayed uncapping, (iii) transient uncapping, and (iv) no uncapping. Further, the behavior of a given system strongly depends on the field direction. It seems that y-direction is the most effective one since almost all the systems underwent rapid uncapping in that case. The z-direction produces mostly transient uncappings and after field removal the systems returned to the capped states. The delayed uncapping occurs in the x-direction case, simply the system needs some time for adjusting its orientation to the field direction and after that time stronger magnetic torques appear which finally lead to the uncapping. The discussion concerning the effect of the field direction is only illustrative because in practice any control of the NCs orientation in space is impossible. However, from the results in Figure 5, we may conclude that for any field direction the 1359



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the interaction induced uncapping is unlikely to occur. Another approach is the creation of hydrophilic shells on the sidewalls of the nanotubes. Then, the range of intermolecular interactions could be reduced and the occurrence of that phenomenon minimized. Additionally, it would enhance the solubility of the NCs in water and prevent from agglomeration at higher concentrations. The release of drug molecules (cisplatin) from a single isolated NC was carefully studied in our previous paper.7 We found that the process consists of two stages: (i) migration from the nanotube interior to its sidewalls and (ii) desorption from the sidewalls into the bulk. Both stages of the cisplatin release are accompanied by activation barriers, the sum of which is about 25 kJ mol−1. In the current study, involving two interacting NCs, the molecular mechanism of cisplatin release is actually the same. The total activation barrier for a single molecule is currently slightly higher (∼31 kJ mol−1), but this is due to application of narrower nanotube in construction of the NCs. Figure 7 shows the release patterns for the considered systems, that is, the number of cisplatin molecules whose coordinates are within the internal space of the nanotubes versus time. Similarly to Figure 5, at t = 0, the systems are in double-capped configurations and the external field is switched on. The applied field lasts only for the first 8 ns. The shapes of the release patterns are similar for each system and for any direction of the external field. Some differences are obviously of the stochastic nature. Interestingly, switching the field off has no visible effect on the release curves. Because all the curves in Figure 7 come from statistically independent runs they can be averaged giving a smoother and more illustrative result. Such averaged release pattern is shown in Figure 8. As follows from the analysis of Figure 8, the release of cisplatin can be divided into two stages; the first one, which takes about 12 ns, corresponds to a fast decay of concentration; the second one, after 12 ns, is definitely slower. Thus, there are a few significant differences between the current findings and the release pattern found in the previous study7 concerning the release from a single isolated NC. Of course, there are some differences in conditions in these two cases; the most important are the initial concentration of cisplatin and the diameter of the nanotube. Previously7 we used a five-walled nanotube with the inner diameter 39.15 Å; currently the nanotubes have three walls and the inner diameters are 23.5 Å. The initial concentrations of cisplatin inside the nanotubes were 2.1 mol L−1 and 3.2 mol L−1 for the previous and the current case, respectively. As already mentioned, a narrower nanotube enhances the activation barrier for diffusion of cisplatin. Therefore, we observe a smaller mean release rate than previously, i.e. in the time window 0−10 ns the mean release rate is currently ∼2.8 ns−1 per nanotube. In the previous studies involving a wider nanotube, the rate was ∼11.7 ns−1, despite the lower initial concentration used. Previously, the maximum simulation time was 10 ns, currently we extended that time to 20 ns and it allowed us to detect the mentioned second (slow) stage of the release process. Further extension of the simulation time was technically questionable since our computer resources needed 5−6 weeks for generation of a single 20 ns run. The release pattern seems to be more complex than observed previously and its analysis needs more detailed discussion. Let us assume that the release of cisplatin is still controlled by onedimensional diffusion mechanism and the concentration gradient at the outlet of the nanotube is of a linear shape. Its

uncapping will occur; it might be rapid and total uncapping or just transient one existing only during exposition of the system to the field. In some cases the NCs might be transferred to the double uncapped configurations in the other ones only one side of the nanotube will be opened. Generally, the assemblies of the NCs, though strongly anisotropic in properties, should respond to any field direction giving finally significant fraction of the uncapped structures. An important new phenomenon observed in NCs assemblies is the uncapping occurring after switching off the applied field. This is seen for systems B and C after about 15 ns of the simulation time. The physical origin of that process is related to strong interaction between nanotubes sidewalls. As shown in Figure 4, the intermolecular interaction energy between two NCs might reach several hundreds of kJ mol−1, thus, it exceeds the energy of the MNPs interaction with the nanotubes tips, which is about 90−180 kJ mol−1 in the considered cases. Because the diameters of the MNPs are larger than the diameters of CNTs, the MNPs are pushed aside the nanotubes axes and the intermolecular-interaction-induced uncapping occurs. Figure 6 shows the final structure of the system B in

Figure 6. Final structure of the system B in which the intermolecularinteraction-induced uncapping occurred. The arrow indicates the MNP which has been detached from the nanotube tip due to strong interaction between nanotubes and the lack of enough space between the nanotube and the MNP.

which the discussed phenomenon has occurred. Obviously, that effect must occur no matter what was the initial orientation between the NCs. After sufficiently long time all configurations A−D or any other will end up in a configuration similar to that one shown in Figure 6. The arrow in Figure 6 indicates the MNP which has been detached from the nanotube tip due to strong interaction between nanotubes. The lack of enough space resulted in uncontrolled uncapping of the NC. Therefore, this phenomenon should be avoided as it leads to uncontrolled leakage of drug molecules before application of the triggering factor. The most obvious approach to getting rid of the interactioninduced uncapping is a suitable adjustment of diameters of both nanotubes and MNPs. If the diameter of a MNP is comparable or smaller than the diameter of the outermost nanotube, then 1360



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Figure 8. Averaged release patterns from Figure 7 (solid red lines) and fitting results using eqs 1, 2, and 3. Top panel shows the fitting results assuming that Da = 6.7 × 10−6 cm2 s−1 and the bottom panel shows the result for Da = 5.0 × 10−6 cm2 s−1. In both cases, Db = 10−8 cm2 s−1, and other parameters are the same as in simulations. The bottom panel shows that a slight reduction of Da gives a perfect fit of eq 2 to the short time branch of the release pattern. That reduction, however, worsens the fit obtained from the full eq 1

describes one-dimensional diffusional flux of molecules between two cylinders; the first is initially filled with N0 molecules, while the second is initially empty. Let us further assume that the distance at which the concentration drops, δ, is related to the nanotube length L and the number of molecules residing in the inner cavity of the nanotube N0 − N, that is, δ = L/(N0 − N); thus, it is simply an average fraction of the nanotube length belonging to a single molecule. Under the above assumptions eq 1 becomes an ordinary differential equation but its solution cannot be expressed by elementary functions. Instead, we can easily find its solutions for two important limiting cases, that is, for the initial times when N → 0 or for a long time limit, t → ∞. For N → 0, each term (N0 − N) ≈ N0 and the solution of the resulting differential equation with the boundary condition N(0) = 0 reads:

Figure 7. Magnetically triggered release of cisplatin from the systems A−D. N0 − N is the number of cisplatin molecules still residing in the nanotubes interiors. The initial structures of NCs are the same as described in Figure 5. The magnetic field is applied at t = 0 and lasts for 8 ns. Each panel on the graph shows the release patterns for various field directions.

slope, however, might be some function of time. Then, the rate at which the molecules are leaving the internal space of the nanotubes can be described by the following microkinetic equation: N −N 1 dN N = 0 − DS dt δSL δS 2Dt

(1)

where N is the number of molecules which have left the nanotube, S is the area of the nanotube tip, L is the nanotube length, D is the diffusivity, and δ is a variable parameter related to the slope of the concentration gradient. The meaning of both terms on the r.h.s. of eq 1 can be easily understood; namely, the first term is the concentration of cisplatin inside the nanotube and the second term is the concentration in a virtual cylinder of the same diameter like the nanotube and the length determined by the diffusion length of cisplatin, that is, (2Dt)1/2. The difference between these two concentrations divided by δ gives the concentration gradient which slope is determined by the value of δ. Thus, eq 1

⎛ N ⎞ N N0 − N = exp⎜ − 0 2Da t ⎟ − 1 + 0 2Da t ⎝ L ⎠ L N0 2Da t ≈ (2) L So, at initial times the concentration inside the nanotube changes as a square root of time. The second limit, that is, t → ∞ simplifies eq 1 in such a way that the second term on the r.h.s. vanishes. Thus, the solution of the resulting equation with a more general boundary condition N(t1) = N1 reads: 1361


The Journal of Physical Chemistry C N0 − N =

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overall release pattern was well described by the “normal” activated diffusion model, that is, eq 3. The above considerations allow us to draw some general conclusions concerning the release of drug molecules (particularly cisplatin) from the NCs. Namely, the rate of release depends on the nanotube diameter, the narrower nanotube the slower release because the activation barrier for diffusion increases with the decreasing nanotube size. Further, the long time part of the release pattern should obey eq 3 (at least approximately since eq 1 is an approximation as well). Moreover, eq 3 should correlate release data from any nanotubes, not necessarily NCs, and the release should reveal typical features of an activated process, for example, the temperature dependence. At very high initial loadings and during the magnetically triggered uncapping, the effect of dragging of drug molecules by the detaching MNPs may be visible. Then, the release rate might be very high and apparently nonactivated, however, that effect should disappear within nanoseconds regime. It is worth noting, that possible presence of charged CPW+ complexes would make the effect of dragging even more visible.

L2N1 + D bN0(N0 − N1)(t − t1) 2

L + D b(N0 − N1)(t − t1)

(3)

Note that eq 3 is identical to the equation derived in ref 7 and which was found to be a very good formula for correlation either simulation or experimental data. The individual symbols denoting the diffusivities in eqs 2 and 3, that is, Da and Db mean that in these two limiting cases the diffusivity of cisplatin might be different. This conclusion comes from fitting of eq 1 to the simulation data using numerical computations. The result of that fit is shown in Figure 8 as the solid line labeled as “model”. In order to reproduce the simulation data using eq 1 we had to adjust only the diffusivity because other parameters are known. The rapid change of the release rate at 12 ns needs, however, an assumption that at this point the diffusivity D in eq 1 rapidly drops down in a stepwise fashion. The estimated values of diffusivities at the initial time Da = 6.7 × 10−6 cm2 s−1 and at longer times (after 12 ns) Db ≤ 10−8 cm2 s−1 provide interesting clues concerning the release mechanism. The value of Da is identical to the nonactivated diffusivity of cisplatin inside the nanotube determined in ref.7 The Db, in turn, is comparable to the activated diffusivity of cisplatin for the migration out of the nanotube interior. This means that normally the process should proceed with the diffusivity Db and should be described by the limiting eq 3, as found in ref 7, in analysis either simulation or experimental data. In this study we, however, observe an “anomaly” related to the existence of the branch of the release pattern governed by the nonactivated diffusion Da and the limiting solution (2). This anomalous part obeys the t0.5 time dependence in contrast to previous analysis7 where such dependence was definitely rejected. The question arises what is a physical justification of such a rapid change of the diffusivity and why the release initially proceeds like a free diffusion without any activation barriers. Another question is why the nonactivated part of the release was not detected in the previous study concerning the similar system.7 The answers come from analysis of the differences between the current system and the previous one7 involving a wider nanotube. The most visible difference is that we currently use two interacting NCs, however, it is not clear that this fact affects the release rate at initial times when the molecules in both nanotubes are beyond the interaction range. Another significant difference is the initial concentration of cisplatin; currently it is 1.52 times higher than the highest concentration studied in ref.7 Further, the diameter of the innermost nanotube is only 0.6 of the diameter used previously. All that means that the molecules are much more densely packed in the internal space of the nanotubes and larger fraction of cisplatin molecules are within the interaction range with the MNPs. Because the external magnetic field leads to rapid migration of magnetic nanoparticles out of the nanotube tips, some amount of cisplatin molecules receive an extra acceleration toward the nanotube outlets due to intermolecular interaction with the MNPs. Physically, that effect is equivalent to a local heating which, in turn, helps crossing local activation barriers. Thus, this is the reason why the first stage of the release proceeds as the nonactivated process and why it finishes suddenly (simply, the number of those excited molecules is limited). In the previous study such mechanism was detected either though, due to smaller density of cisplatin molecules the nonactivated section of the release was much shorter (less than 1 ns). Therefore, the

4. CONCLUSIONS This work provides important physical insights into possible phenomena occurring in assemblies of the nanocontainers. Particularly, we found the following: − The presence of drug molecules inside the NCs enhances their stability, that is, interaction energy between MNPs and the nanotubes tips is higher when the NCs are filled with cisplatin molecules. − Collisions between NCs in solution and at room temperature cannot induce their uncapping, however, at higher temperatures the collision induced uncapping might occur and its likelihood depends on spatial orientation of the colliding NCs. − Interaction energy between NCs turns out to be very high; it means that additional chemical treatment is necessary in order to prevent their agglomeration at higher concentrations. The electrostatic repulsion between charged MNPs is not enough in the considered cases and extra functionalization of the nanotubes sidewalls is needed. − Strong interaction between NCs may lead to uncontrolled uncapping when the diameters of the MNPs are larger than the diameter of the nanotube. − The magnetically triggered uncapping is sensitive to the field direction but for any direction a significant fraction of the uncapped states (total, transient, or delayed) can be observed. − The release rate of cisplatin depends on the nanotube diameter because that factor affects the activation barrier for diffusion. − At high initial loadings, the release pattern might reveal an apparent nonactivated section; this is due to the effect of dragging the drug molecules by the detaching MNPs. − The long time part of the release from the NCs or any nanotubes should obey simple one-dimensional activated diffusion mechanism.

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AUTHOR INFORMATION

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Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Polish National Science Centre (NCN) Grant No. N204 205240. REFERENCES

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