Magnetic Anisotropy Effects on the Behavior of a Carbon Nanotube

Nov 20, 2012 - Institute of Catalysis and Surface Chemistry, Polish Academy of ... Chair of Medical Biochemistry, Jagiellonian University Medical Coll...
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Magnetic Anisotropy Effects on the Behavior of a Carbon Nanotube Functionalized by Magnetic Nanoparticles Under External Magnetic Fields Tomasz Panczyk,*,† Mateusz Drach,‡ Pawel Szabelski,‡ and Anna Jagusiak§ †

Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30239 Cracow, Poland Department of Chemistry, Maria Curie-Sklodowska University, pl. M. Curie-Sklodowskiej 3, 20031 Lublin, Poland § Chair of Medical Biochemistry, Jagiellonian University Medical College, ul. Kopernika 7, 31034 Cracow, Poland ‡

ABSTRACT: The behavior of a multiwalled carbon nanotube functionalized by magnetic nanoparticles through triethylene glycol chains is studied using molecular dynamics simulations. Particular attention is paid to the effect of magnetic anisotropy of nanoparticles which significantly affects the behavior of the system under an external magnetic field. The magnetization reversal process is coupled with the standard atomistic molecular dynamics equations of motion by utilizing the Neel−Brown model and the overdamped Langevin dynamics for description of the inertless magnetization displacements. The key results obtained in this study concern: an energetic profile of the system accompanying transition of a magnetic nanoparticle from the vicinity of the nanotube tip to its sidewall, that is from the capped configuration to the uncapped one; range of the magnetic anisotropy constant in which the system performs structural rearrangements under the external magnetic fields; range of the magnetic field strengths necessary for triggering the structural rearrangements; and other effects like magnetic heating observed during the interaction of the system with the magnetic field. The determined properties of the studied system strongly suggest its application in the area of nanomedicine as a drug targeting and delivery nanovehicle.

1. INTRODUCTION Nanosized structures comprising carbon nanotubes (CNTs) and magnetic nanoparticles (MNPs) are currently of great interest due to their promising applications in the field of drug delivery.1−9 Valuable biomedical properties of CNTs are wellknown, and there exists a large amount literature devoted to CNT functionalization and their biomedical applications.10−16 The magnetic property, carried by MNPs, is commonly viewed as an extra factor facilitating manipulation of such objects (targeting) by means of an external magnetic field, providing visualization in magnetic resonance imaging or engineering hyperthermia of tumors under alternating magnetic fields.9,16 Attachment of MNPs to CNT surfaces is relatively easy to engineer and normally leads to composite objects where MNPs decorate surfaces of CNTs or locate in their inner cavities.1,2,17−23 Selective attachment of MNPs to CNT tips is less facile, but it has been engineered earlier.23 However, CNT tips functionalization with MNPs via covalent binding through an organic linker has not been reported so far. Perhaps this is due to experimental difficulties, or there is still a lack of interest in fabrication of such systems. Experimental difficulties related to creation of an organic linker between the CNT surface and MNP surfaces can be overcome as reported by Tsoufis et al.20 who were able to attach FePt nanoparticles to sidewalls of multiwalled CNTs via organic linkers in a few synthetic steps. A similar approach can be used for the development of reactive linkers at CNT tips and further attachment of MNPs to the linker ends. CNT tips are more reactive than sidewalls, and © 2012 American Chemical Society

there are many feasible and controllable methods of their functionalization, as reported by Prato et al.14 In a recent series of papers,24−27 we presented a careful analysis of the potential properties of single-walled carbon nanotubes functionalized by magnetic nanoparticles through alkane linkers attached to the CNT tips. By applying Monte Carlo simulations, we were able to demonstrate that such systems, called magnetically triggered nanocontainers (NCs), reveal very promising properties which can be beneficial in the area of drug targeting and delivery. Particularly, the NC can realize the scheme of “corking” and “uncorking” drug molecules in the inner cavity of carbon nanotubes launched by Hillebrenner et al.28 and by Hilder and Hill.29 The most critical stage in that scheme is the uncorking of the CNT at the target site to initiate drug release. This difficult stage can be easily implemented in the case of the NC since the application of an external magnetic field may lead to detachment of MNPs from the CNT tips and produce the uncorked state of the CNT. Moreover, magnetic field is commonly viewed as a safe and easy to apply external stimulus in vivo. Results of theoretical studies might be viewed as less convincing than a strict experimental proof of concept. However, a careful analysis of physics lying behind a given problem may provide fast and very useful information Received: October 14, 2012 Revised: November 19, 2012 Published: November 20, 2012 26091

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concerning future experimental works on the problem including their justification, choice of conditions, and working parameters. According to the above concept, we have determined the critical ranges of parameters which a functional NC must reveal. In particular, we found that one of the conditions is, in practice, rather difficult to satisfy.26 The determined critical linker lengths, i.e., not more than one or two CH2 segments, cannot be easily engineered, and this might be viewed as a serious obstacle in synthesis of the NCs. The current contribution, which is based on a more advanced model, brings new information about the nature of the NC and shows that the problem of very short linkers was, in fact, apparent. This conclusion comes from the application of a more realistic architecture of the NC, that is, multiwalled CNTs, functionalized tips, and triethylene glycol chains as linkers. Another key problem analyzed in this study is related to magnetic anisotropy of the MNPs. This property of MNPs was ignored in our previous studies in which we assumed that the magnetizations are fixed within the MNP body frames. As found in the current work, this assumption can be justified only in the case of extremely hard magnetic materials based on rare earth elements. For magnetic materials normally used in nanomedicine, the magnetization switching is a common phenomenon, and it should properly be accounted for in computer modeling. To that purpose, we developed a simple methodology for coupling standard atomistic molecular dynamics simulations with inertless magnetization switching processes within the magnetic nanoparticles. That methodology is based on the Neel−Brown model, and it is appropriate for the coherent rotation mechanism as discussed in the Methods section.

Figure 1. Architecture of the nanocontainer considered in this study. The magnetic NPs are bound to the nanotube tips by triethylene glycol chains. The external ring of the multiwalled nanotube is saturated by amide groups, while the internal rings are formally saturated by hydrogens. The magnetization vector of the NP is denoted by m⃗ , and it can rotate according to the potential (4). The angle θ is defined by the position of the magnetization m⃗ and the easy axis e⃗ which is fixed within the body frame of the NP. The easy axis is assumed to coincide with the vector connecting the MNP center and the anchoring point A. A given atom, i.e., carbon, nitrogen, oxygen, or hydrogen, is denoted by appropriate color, as shown in the bottom part of the figure.

2. METHODS All calculations were performed using the large-scale atomic/ molecular massively parallel simulator (lammps) code30 with several extra classes written from scratch and working on magnetic torques. The force field consisted of several interaction types which are shortly discussed in subsequent paragraphs. 2.1. Magnetic Interactions. Magnetization reversal within a magnetic particle occurs due to two processes: the Brownian rotation which is based on the reorientation of the particle as a whole and the Neel rotation which stands for the magnetization reorientation within the particle body frame. If the particle does not interact with an external field or other magnetic particles, the former process follows classical rotational diffusion mechanism. However, in our case the nanoparticle is rigidly bound to the linker at the anchoring point A (Figure 1), thus there appears torque τ⃗l acting on the center of the nanoparticle τl⃗ = rNP ⃗

× fl ⃗

where m⃗ ij are the magnetization vectors of particles ij and rd⃗ d is the dipole−dipole separation vector. The magnitude of particle magnetization is a product of its saturation magnetization Ms and volume V, |m⃗ | = MsV. Because we considered only the simplest case of uniaxial magnetic anisotropy, then, according to the Stoner−Wohlfarth model (SW), the anisotropy energy coming from the displacement of magnetization from its equilibrium orientation, Ea, is given by

Ea = K aV sin 2 θ

(1)

where Ka is the anisotropy constant and θ is the angle between magnetization m⃗ and the easy axis e.⃗ Thus, any displacement of magnetization orientation induces the anisotropy torque τ⃗a acting on the NP center of mass

where fl⃗ is an instantaneous force acting on the anchoring point. The presence of other magnetic NPs and external magnetic field, B⃗ , induces magnetic torques due to dipole−dipole interactions τ⃗d and dipole−field interactions τ⃗B 1 3 τd⃗ ∼ − 3 (m⃗ i × m⃗ j) + 5 (m⃗ j · rdd⃗ )(m⃗ i · rdd⃗ ) rdd rdd

(2)

τB⃗ = m⃗ i × B⃗

(3)

(4)

τa⃗ = m⃗ i × ∇⃗Ea

(5)

The same torque but of opposite direction acts on the magnetization vector m⃗ i. The Brownian motion of the nanoparticle can thus be determined by solving the classical equation for angular velocity of the nanoparticle ω⃗ NP 26092

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where τ⃗m = τ⃗d + τ⃗B − τ⃗a is the net magnetic torque acting on the moment m⃗ ; Δt is the integration time step; and R⃗ is uniformly distributed random numbers with zero mean and unit standard deviation. The Brown and Neel rotations are, as seen in eq 11, coupled by adding the angular velocity of the NP in the rhs of eq 11. The instantaneous orientation of magnetization is thus given by

(6)

where I is the NP inertia. Accordingly, the NP instantaneous orientation is given by de ⃗ = ω⃗ NP × e ⃗ dt

(7)

dm⃗ = ωm⃗ × m⃗ dt

Magnetization reversal according to the Neel process is generally a complex phenomenon. Exchange interactions between individual spins, magnetic anisotropy, external field, and thermal agitation lead to a few qualitatively different mechanisms of the net magnetization switching even in the case of single-domain nanoparticles.31 Such mechanisms like nucleation, curling, or coherent rotation have been found experimentally32 as well as in various theoretical studies.33,34 Particularly, the dynamics of magnetization reversal is fully described by the stochastic Landau−Lifshitz−Gilbert equation33,34 which predicts quantitatively all mechanisms mentioned above. In the case of the system under investigation, the inertless Neel process is much faster than the Brown rotation; thus, a full description of the dynamics of magnetization reversal according to the Landau−Lifshitz−Gilbert equation is not necessary, and we may rely only on its stationary solution. That solution has been found by analyzing the Fokker−Planck equation which governs time evolution of the nonequilibrium probability distribution of magnetic moment orientations associated with the stochastic Landau−Lifshitz−Gilbert equation.33 Thus, on introducing the appropriate Neel time t0, which is the characteristic time of free diffusion in the absence of potential, the stationary solution of the Fokker−Planck equation has the Boltzmann distribution when

t0 =

1 |m⃗ | λ 2γkT

The above set of equations allows us to model the dynamics of the magnetization reversal of NPs as a function of the anisotropy barrier and applied field. Though eqs 8−12 are based on the simplest approach to the treatment of the Neel rotation, they preserve the most important properties of the coherent rotation model. Additionally, the outlined model can be easily coupled with the standard Newton’s dynamics which needs to be applied to other nonmagnetic atoms. 2.2. Interatomic Interactions. The nonmagnetic component of the force field was set up according to the General Amber Force Field (GAFF).36 For the terminal amide groups, located on the outer ring of the CNT (Figure 1), an acetophenone molecule has been used as a template for the determination of bond lengths, angles, force constants, and other parameters related to torsion and dihedral coefficients. These amide groups were linked to every second carbon atom on the terminal rings of the outer CNT. The inner CNTs were assumed to be formally terminated by hydrogens though we did not account for that explicitly in calculations. The linkers between CNT and NPs were built by assuming that one of the terminal amide groups undergoes functionalization, leading to a triethylene glycol chain. In these chains the third oxygen is linked to the anchoring atom on the NP surface. The anchoring atoms are formally assumed to be silicon atoms being a part of the silica shell covering magnetic cores of NPs. All interaction parameters and topology associated with the terminal amide groups and linkers were generated using the automatic atom and bond-type perception scheme implemented in AmberTools 12.37 The multiwalled carbon nanotube (MWCNT) was generated as a sequence of five concentric zigzac (n,0) single-walled nanotubes differing by nine units in chiral indices. This gives a mean interlayer separation distance of 3.46 Å.38 The MWCNT as a whole was treated as a rigid body which is justified in the case of medium interactions between NPs and CNTs, as found recently.38 The length of the studied MWCNT was 200 Å, while the inner and outer diameters were 39.15 and 67.34 Å, respectively. 2.3. Dispersion Interactions. Because the simulation box was large (500 × 500 × 500 Å), it was impossible to treat solvent (water) molecules explicitly. Instead, we utilized the recently developed implicit solvent model based on the Hamaker theory of dispersion interactions.39 According to that model, the energy of interaction between a single atom and a large solid sphere across solvent, Ucs, is given by

(8)

where γ is a gyromagnetic ratio and λ is the dimensionless damping coefficient that measures the magnitude of the relaxation (damping) term relative to the gyromagnetic term in the dynamical equation.33 We can further assume that the magnetization reversal proceeds according to the coherent rotation mechanism; that is, the magnitude of magnetization does not change during rotation of the moment, and thermal agitation helps cross the energy barrier associated with magnetic anisotropy (Neel−Brown model). That mechanism has been experimentally confirmed for the case of cobalt and a few other nanoparticles.35 Accordingly, in our case the magnetization reversal is a simple free rotational diffusion process with an activation barrier. Thus, a mean time spent at a given orientation tN obeys the exponential law t N = t0 exp(E /kT )

(9)

where E is the net activation barrier E = K aV sin 2 θ − m⃗ ·B⃗

(10)

Ucs = Acs

The dynamics of inertless magnetization reversal of the magnetic moments is therefore described by the overdamped Langevin dynamics. According to that scheme, the angular velocity of magnetization ω⃗ m reads λγ ωm⃗ = τm⃗ + ω⃗ NP + M sV

2λγkT R⃗ MsV Δt

(12)

2a3σ 3 9(a 2 − r 2)3

⎡ (5a6 + 45a 4r 2 + 63a 2r 4 + 15r 6)σ 6 ⎤ ⎢1 − ⎥ ⎣ ⎦ 15(a 2 − r 2)6

(13)

where a is the radius of the nanoparticle and r is the center− center distance between the atom and the nanoparticle. The

(11) 26093

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the crucial conditions for the NC to be “operational” is the thermodynamic preference of the capped configuration over the uncapped one at normal conditions. Both configurations should be separated by a significant energy barrier, preventing spontaneous transition from the capped to uncapped configuration. We have recently found that the above condition cannot be easily satisfied for NCs composed of single-walled carbon nanotubes and NPs linked to the CNT by alkane chains. Only very short linkers, i.e., one or two CH2 segments, led to the mentioned energetic balance, while longer linkers produced an inverse relationship.26 In the current study, we deal with definitely a more realistic model inspired by the known chemistry of carbon nanotubes. It accounts for the presence of functional groups at the tips which normally develop upon chemical treatment of pristine nanotubes, i.e., oxidation, amidation, and addition reactions.10−15 Accordingly, the linkers are assumed to be composed of polyethylene segments instead of previously studied alkane chains. Finally, the studied diameters of nanotubes imply application of multiwalled nanotubes since the diameters of single-walled ones do not normally exceed 10 Å. Thus, a key question is whether this new model satisfies the general condition concerning the energy balance between the capped and uncapped configurations. 3.1. Total Energy of the NC at Various Configurations. Thermodynamic preference of a given configuration can be recognized by studying its free energy and tracking how it changes from one configuration to another. A systematic approach to that problem needs introduction of an order parameter which strictly defines configurations of the system on a given transition path. For the particular case considered here, the most interesting transition path is the transfer of a MNP from the configuration in which it caps the nanotube tip to the configuration when it locates on the nanotube sidewall, i.e., leaving the nanotube tip uncapped. The most convenient order parameter is, thus, an angle σ defined by coordinates of the three following points: MNP center, carbon atom on the outermost nanotube tip ring to which the linker is attached, and the center of mass of the innermost nanotube tip ring. The angle σ grows as the MNP detaches from the tip and goes toward the sidewall. Thus, drawing the free energy profile as a function of σ gives insight into available stable configurations of the NC on the route from the capped to the uncapped state. Because we are dealing with the implicit solvent model and we consider only a single NC in the simulation box (periodic images are beyond the cutoff), the analysis of the free energy (or potential of mean force) is equivalent to the analysis of the total energy of the NC. To couple the structure of the NC with the order parameter σ, an umbrella bias potential kσ(σ − σ0)2 was utilized. The parameter σ0 defines the target configuration of the NC, whereas kσ is a constant which determines how fast the NC configuration converges to the target one. In the first step, we determined the equilibrium value of σ corresponding to the fully capped state of the NC. The calculations started from a configuration which was close to the capped one. After an equilibration period, the system spontaneously reached the capped state, and the σ0,eq value was determined using unbiased calculations. It was found that σ0,eq = 55.6 ± 0.1°, and this value was used in the umbrella sampling to constrain one end of the NC in the equilibrium capped state. The other end of the NC was used for scanning of various values of σ0 (i.e., configurations of the NC) in biased calculations. In that way, the total energy profile as a function of

Hamaker constant Acs can be computed using the following mixing rule Acs = ( Acc +

A ww (1 + σw /a)3 − 2 A ww )

( 144ε − v A ww )

(14)

where σw is the diameter of a water molecule; ε and σ are the usual Lennard-Jones parameters; and Acc and Aww are the Hamaker constants for NP material and water, respectively. The factor v is the ratio of the number of water molecules replaced by the nanotube to the number of atoms within the nanotube.39 We further assumed that the magnetic NP is composed of 80 Å in diameter magnetic core and a 10 Å thick silica shell. Thus, as we found recently,38 its interaction with other atoms is dominated by the nature of the shell material. We used in calculations Acc = 6.5 × 10−20 J, i.e., representative of silica, and Aww = 3.7 × 10−20 J, i.e., the value for water. Because the NPs are covered by silica, we also assumed that they carry some negative charge. According to the literature data, the charge density of silica NPs at 0.145 mol L−1 ionic strength of NaCl electrolyte is ca. 0.1 C m−2.40 The presence of charge was accounted for in calculations by applying Yukawa potential.26,27 Given a large separation distance between NPs, the electrostatic repulsion between NPs was, in fact, negligible. However, this electrostatic repulsion should be considered as a factor preventing agglomeration of many NCs into one cluster at high concentrations. As shown in ref 38, the assumed charge density of silica shells produces repulsive forces between silicacovered MNPs strong enough to prevent them from collapsing into one cluster due to magnetic and dispersion attraction. According to GAFF, the atoms creating linkers and terminal amide groups carry some partial charges as well. Due to the application of the implicit solvent model, these electrostatic interactions were screened according to the Debye screening model, i.e., by applying the exponential decay of forces with separation distance. For the considered conditions, the Debye screening length was set to 8 Å. 2.4. Computational Details. Calculations were performed in NVT ensemble using a 1.0 fs time step. The number of atoms creating the nanocontainer was 65 756, and the temperature was controlled by applying the Langevin thermostat with a damping factor δ = 6.7 fs. This value mimics the presence of implicit water molecules at similar conditions, i.e., density, viscosity, and temperature. A correct choice of that parameter, which is actually a mean time upon which an atom velocity is altered according to Langevin dynamics, allows for a fairly good description of the dynamics without involving explicit solvent molecules. A frictional force Ff originating from solvent viscosity η is proportional to atom velocity v, so that Ff = −(m/(3πηd))ν and hence δ = (3πηd)/m, where m is mass and d diameter of a given atom. Thus, δ = 6.7 fs corresponds to a carbon atom in water at temperature 300 K. The damping factors for other atoms were suitably rescaled according to their sizes and masses. The cutoff for the colloid−single atom interactions, i.e., when using eq 13, was 100 Å, whereas for the LJ and Coulomb interactions we used 12 Å as the cutoff distance.

3. RESULTS AND DISCUSSION Previous studies, devoted to a similar class of systems, led to several conclusions concerning the construction of the magnetically controlled nanocontainer.24−27 Recall that one of 26094

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σ was determined. Results shown in Figure 2 represent the averaged values of energy and σ from three independent scans.

blocked, we may state that the dominating form of the NC at ambient conditions will be the capped one. In the previous study,26 we found that the application of longer linkers (more than 2 CH2 segments in alkane chains) leads to inversion of the relation between energies in the capped and uncapped states. It makes such structures useless in terms of their application as drug delivery vehicles. In the current model, the linker lengths are much longer than two CH2 segments; nevertheless, we observe the requested energy profile. The question arises: which property is responsible for keeping the preferred energy balance while going from the previously studied model26 to the current one? The key differences between these two models are: application of multiwalled CNT and triethylene glycol chains (combined with amide grups) as linkers instead of alkane chains. The role of those modifications is seen when splitting the total potential energy of the NC into contributions coming from bonds, angles, dihedrals, and pairs energies. As seen in Figure 3, the

Figure 2. Total energy profile corresponding to the transfer of one MNP from the capped to the uncapped configuration.

As seen in Figure 2, the capped configuration corresponds to the absolute minimum in the total energy. The uncapped state corresponds to the second minimum, and both states are separated by the energy barrier. Thus, the shape of the total energy profile satisfies the general condition which a functional NC must obey. The energy well is narrower in the capped state, and the uncapping transition need to cross a steep and high (∼ 85 kJ mol−1) energy barrier. It means that the MNP is tightly trapped in the capped configuration, and its escape needs significant energy input. Given a large mass of MNPs (1.8 × 106 g mol−1) and the steep and high energy barrier, the mean residence time in the capped state will be long. That time cannot, however, be determined from simulations, as it definitely reaches macroscopic values. In a recent paper,26 we did a simple estimation of the residence times using the concept of activated hopping probability, valid for usual light atoms. However, the previously assumed attempts' frequency of ∼1010 s−1 was probably overestimated because it was identified with the frequency of conformational changes in n-alkanes, that is, molecules composed of light atoms. It is obvious that if one of the terminal atoms in a chain has a mass of the order of 106 g mol−1 then its oscillation frequency is significantly reduced. According to the harmonic oscillator model, that frequency would be reduced by 2 orders of magnitude, as it is proportional to the square root of mass. Obviously, the harmonic oscillator model is not applicable here, but the general idea of estimating the attempt frequency seems reasonable. Thus, we may assume that the mean residence time is a product of a more likely value ∼108 s−1 times the Boltzmann factor related to the height of the activation barrier. This gives 5.8 × 106 s in the capped state and 119 s in the uncapped state. Definitely, neither process (transition from capped to uncapped or vice versa) can be probed using unbiased molecular dynamics simulations. The estimated times, however, still support the application of the current architecture of the NC as a drug delivery system. The difference in energies for capped and uncapped states, i.e., ∼27 kJ mol−1, might serve as an estimation of the relative equilibrium concentrations of both forms of the NC. Thus, when looking at the uncapping process as a kind of chemical reaction, the ratio of capped to uncapped forms of the NC at equilibrium is proportional to exp(27/2.5) = 4.9 × 104 at 300 K. Given that the uncapping process is strongly kinetically

Figure 3. Contributions to the total potential energy of the NC coming from bonds, angles, dihedrals, and pairs energies as a function of the NC state.

deformations of linkers induced by the transfer of the MNP from the capped to the uncapped state affect weakly the energy associated with bonds, angles, and dihedrals. Long linkers are very flexible, and due to many available degrees of freedom, they always can attain a conformation which minimizes strains induced by changes of the angle σ. The total energy profile in Figure 2 is, thus, determined by the energy of pairs interactions which, in turn, are dominated by the dispersion interactions. It may be concluded that the key factor responsible for the desired energy profile is the appearance of the multiwalled carbon nanotube in the current model of the NC. It has recently been shown that colloid nanoparticles bind to carbon nanotube tips stronger than to sidewalls provided that the nanotube is more than 3-walled.38 This is a result of enhanced dispersion interactions occurring due to a larger number of carbon atoms located at the tips. The interaction with sidewalls 26095

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container under a magnetic field. Exposition of a magnetic nanoparticle to an external magnetic field leads to its magnetization reorientation according to the Neel and Brown processes. Under strong fields, i.e., when m⃗ ·B⃗ ≫ KaV, the magnetization might overcome the anisotropy barrier, according to the fast Neel process, and align with the direction of the field. Let us estimate the magnetic field strength under which the anisotropy barrier of 80 Å in diameter cobalt NPs is overcome. The magnetization saturation of Co NPs is Ms = 1070 kA m−1, as reported by Massart et al.,46 thus the anisotropy energy for Ka = 106 J m−3 equals the magnetic energy |m⃗ ||B⃗ | at |B⃗ | = 0.93 T. Accordingly, a Ka larger by 1 order of magnitude corresponds to a ten times larger field strength, i.e., 9.3 T. Hence, the reported anisotropy constants for Co NPs imply that field strengths in the range 1−10 T are enough to induce magnetization reversal according to the Neel mechanism. The same range of magnetic field strength was found in our previous studies as necessary to trigger the uncapping process which occurs mainly according to the Brown mechanism.24−27 Thus, there is a question whether the Brown process is still strong enough to move MNPs away from the CNT tips after the magnetization flip according to the Neel process. Figure 4 shows what happens during exposition of the NC to a strong 12 T magnetic field. The initial structure of the NC was prepared starting from a configuration close to the capped one and allowing it to relax for 2 ns. Then, we got the equilibrium configuration in which both MNPs stuck to the CNT tips, and the magnetizations were aligned with the easy axes directions. That configuration was next used to study the influence of the external magnetic field on the structure of the NC. As a measure of the instantaneous configuration of the NC, we defined another order parameter which is more illustrative than the previously defined angle σ. Namely, it is the maximum distance between the MNP surface and any of the carbon atoms located on the innermost nanotube ring. Such a parameter has a strict physical meaning: it is the size of the slit between the MNP surface and the entrance to the free space inside the nanotube. Thus, it measures the degree of capping/uncapping which, in turn, informs how big a guest molecule can formally enter (or escape) the CNT interior. Results shown in Figure 4 correspond to three values of Ka; 108 J m−3 is definitely too high to be identified with Co NPs; however, this example shows the behavior of the NC in the limit of fixed magnetization directions within MNP body frames because the applied field is too weak to induce the magnetization reversal. The other Ka’s are within the limit of the reported values for cobalt NPs, and for both cases the 12 T field strength is enough to overcome the anisotropy barriers. This is clearly seen in the snapshots; the directions of magnetization (red spots on the MNP surfaces) are turned from the easy axes directions to the field direction. We found that process (Neel) to be very fast, taking no more than a few picoseconds. Afterward, as seen, the Brown process initiated structural rearrangements in some cases; in other cases, nothing happens after completion of the Neel process. As seen in the top part of Figure 4, the very high Ka suppresses the Neel rotation, and magnetizations are always aligned with the easy axes. The total torque acting on the MNPs is thus dominated by the magnetic torque, eq 3. The anisotropy torque, eq 5, is small because the magnetizations remain in their anisotropy potential wells due to extremely high energies necessary to make any displacement from those

is little affected because the successive walls are more and more far away from the surface of the colloid NP. The “functional” energy profile, observed in previous studies25−27 based on single-walled CNTs, was obviously due to strains in linkers conformations. As mentioned, only extremely short linkers composed of a single or two CH2 segments provided the required stronger binding of NPs at the capped states. Simply, the uncapped states were associated with strong deformations of linkers which resulted in an increase of the total potential energy of the whole NC. In the current model the mechanism is different: strains are reduced due to high flexibility of the linkers in both conformations of the NC so that the energy profile is determined by the dispersion interactions between the MNP and the carbon nanotube. 3.2. Uncapping Processes under Strong Magnetic Fields. Magnetic anisotropy is an intrinsic property of MNPs, and it is a complex function of a building material, its crystallographic structure, shape, and many other factors. Because the magnetic anisotropy constant Ka defines a barrier preventing magnetization flips within the MNPs according to the Neel rotation, it must be viewed as a crucial factor affecting properties of the NC. In the previous studies,25−27 we assumed ferromagnetic ordering within the volumes of the MNPs and formally set an infinite value of the anisotropy constant (magnetization was fixed within the body frame of the MNP). As we are aiming at a more realistic representation of the NC, a discussion concerning finite values of Ka is necessary. The reported values of Ka for bulk metallic cobalt are 4.5 × 105 and 2.5 × 105 J m−3 for hcp and fcc structures, respectively.41 It is well-known, however, that Ka is much larger for metal nanoparticles than for the bulk. For bulk samples, Ka is primarily due to magnetocrystalline anisotropy, whereas for nanoparticles the dominant contributions to magnetic anisotropy arise from stresses, surface effects, and the shape of the granules.42,43 Thus, for very small NPs consisting of about 30 Co atoms, the low-temperature value of Ka was found to be 3 × 107 J m−3, i.e., 2 orders of magnitude larger than for bulk metal.43 In other studies involving 40 Å in diameter cobalt NPs and temperatures about 600 K, Ka = 6 × 106 J m−3 was found.44 On the other hand, Respaud et al.41 found Ka in the range 0.83−1.0 × 106 J m−3 for ultrafine (15 Å in diameter) Co NPs. As seen, the Ka value seems to be dependent on the NP sizes and probably also shapes distributions, and other factors affect the values of Ka determined experimentally in different laboratories. Thus, it is difficult to predict a strict Ka value for a given case, but knowing the above experimental values we can get a notion about the physically reasonable range of its values. The assumed diameter of magnetic cores in our model, 80 Å, is significantly larger than that mentioned above, thus we might expect that actual Ka values for our cobalt NPs should be rather closer to 106 J m−3 than to 107 J m−3. Obviously, cobalt is neither the only nor the best material for the magnetic caps of the NC. Materials based on rare earth elements often exhibit enormously high magnetization saturation and magnetic anisotropy. For example, SmCo5 exhibits a Ka value as high as 1.3 × 107 J m−3 for bulk material.45 It is likely that SmCo5 nanoparticles would reveal a Ka value about 1 order of magnitude larger. Therefore, such materials represent a reasonable alternative to cobalt if Co NPs revealed too low for a proper function of the NC magnetic anisotropy. Let us consider how the presence of finite magnetic anisotropy affects the behavior and structure of the nano26096

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final configuration. The effect of field direction obviously depends on the alignment of the easy axes. Here, for clarity, we study only one case; i.e., both axes are defined by the anchoring points on the MNP surfaces and their centers. However, any combination of the easy axes directions would lead to the observed magnetically triggered uncapping though for other field directions. The only exception is a case when both axes and the field are parallel at the moment of switching on the field. Then, the uncapping cannot occur because the system is already at the minimum energy state. Here, we observe that the NC may end up either in double uncapped (y direction) or single uncapped (x and z directions) states. In each case, the transition needs some time; the slowest corresponds to the z direction, and it seems that 10 ns was not enough to bring the system to the final state. A more realistic anisotropy constant for cobalt NPs, i.e., 107 J −3 m , leads to some reduction of its susceptibility to the magnetically triggered uncapping. The most facile is the case when the field is applied along the y direction, whereas the x direction becomes inactive. Application of the field along the z direction leads to double uncapped states, but the time necessary to complete this process is long. For Ka = 107 J m−3, the magnetic energy is larger than the anisotropy energy; therefore, the magnetizations quickly align with the field direction, and the magnetic torques, eq 3, vanish. The Brown rotation is thus driven by the anisotropy torques according to eq 5. It means that the rates of uncapping processes as well as their occurrences directly and almost exclusively depend on the height of the anisotropy barrier. Clearly, application of field strengths larger than the anisotropy barrier (in terms of the associated energies) is not justified. This is because the magnitude of the Brown processes is determined only by the height of the anisotropy barrier and the angle between magnetization and the associated easy axis. The results shown in the bottom part of Figure 4 indicate that further reduction of Ka by 1 order of magnitude deactivates the NC totally. However, the studied time window is very narrow when compared to a macroscopic scale. It is possible that the uncapping will occur later on, but we cannot confirm that directly. In this case, the magnetic anisotropy energy barrier is KaV = 161 kJ mol−1, thus it is almost twice the energy barrier for the uncapping transition (Figure 2). However, taking into account the angular dependence of the anisotropy potential (∼sin2 θ) the further fate of the NC is difficult to predict. It is likely that in the y case the uncapping will occur because the stationary state angles between magnetizations and easy axes were found to be ca. 110 and 65° (for the L and R end of the NC). This means that the effective anisotropy energy is in the range 130−140 kJ mol−1, and there is still some surplus of energy enabling the transition into the uncapped state. The other directions of field lead to these angles in the range 30−50°, so the anisotropy torques are significantly reduced in these cases, and there is little chance to overcome the uncapping barrier in short times. It may be concluded that Ka = 106 J m−3 is the lower limit of activity in the flash uncapping mechanism. A slow uncapping seems to be possible for the present architecture of the NC even for somewhat lower anisotropy constants and combined with an extra thermal agitation due to temperature fluctuations. However, such a process cannot be directly probed using unbiased molecular dynamics simulations. 3.3. Triggering the Uncapping Processes under Moderate Magnetic Fields. As concluded in the previous

Figure 4. Temporal evolution of the NC structure after exposition to 12 T external magnetic field. The initial configuration was the fully capped structure brought to equilibrium without the external magnetic field applied. The magnetizations of the MNPs were aligned with the directions of the easy axes. The symbols x, y, and z mean the direction of the applied field; i.e., the direction z coincides with the direction of the CNT axis at the first time step, while the x and y are orthogonal to each other. Codes L (left) and R (right) are used for formally distinguishing both ends of the NC in the calculations. The slit size determines how the surface of the MNP has grown away from the innermost nanotube ring; for slit sizes above 39.15 Å the innermost nanotube is fully uncovered, while 53.24 Å is reached when the MNP moves behind the outermost nanotube. Thus, the shaded areas represent the range of slit sizes corresponding to the CNT wall width. The snapshots show the final configurations of the NC after 10 ns together with the indication of the magnetic field directions (arrows). The red spots on the surfaces of the MNPs show projections of their magnetization directions.

positions. Therefore, the magnetic energy fully converts into Brown rotation, and the NC readily responds to the magnetic field which induces a fast transition to the uncapped state. However, the final configuration of the NC strongly depends on the direction of the applied field. It implies that the uncapping transition is faster than the rotation of the entire NC. This is due to huge components of the NC inertia tensor: two very large masses connected by a relatively light but long rod require a long time for reorientation. As the uncapping transition takes a few nanoseconds, the initial mutual orientation of the NC and the external field determine the 26097

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section, application of very strong magnetic fields is not justified because the Brown rotation is driven mainly by magnetic anisotropy torques. That mechanism occurs when the energy of the magnetic dipole in the field is larger than the anisotropy barrier. Then the magnetization direction aligns with the field direction, and the magnetic torque vanishes. Therefore, the field strength |B⃗ | = KaV/|m⃗ | is enough for overcoming the anisotropy barriers because the process is additionally thermally agitated. Moreover, application of such fields, i.e., comparable to the anisotropy barriers, maximizes displacements of magnetizations from the easy axes. As a result, the magnetizations locate close to the top of the barriers, and the anisotropy torques become the strongest. Figure 5 shows how the application of the field strengths, adjusted to the magnitude of the anisotropy barriers, i.e., |B⃗ | =

transition. This observation indicates that for small anisotropy barriers the uncapping process is thermally assisted. The NC waits for a bigger temperature fluctuation, and after receiving a sufficient thermal energy input it slightly escapes from the bottom of the potential energy well. Then, the anisotropy torques become strong enough to drive the uncapping process further. That mechanism may lead to the uncapped states of the NCs with Ka’s even smaller than those shown in Figure 5. As mentioned, the considered time window is very narrow because of the usual limitations in molecular dynamics simulations. Thus, the lack of activity for Ka = 106 J m−3 shown in Figure 4, and also found for conditions assumed in Figure 5, might be due to the limited observation time. However, given that Ka = 2 × 106 J m−3 is enough to produce the activity of the NC in the flash uncapping transition, we can conclude that cobalt NPs (suitably prepared though) might be used for the construction of functional NCs. Very important conclusions might be drawn from observations of the NC’s temperature during the magnetically triggered uncapping processes. A selection of such data is shown in Figure 6. The temperature profiles correspond to various Ka’s

Figure 6. Instantaneous temperature of the NC during exposition to the magnetic field along the y direction. All parameters are the same as in Figure 5.

with the associated field strengths, as analyzed in Figure 5. We can find out that during the uncapping process the temperature of the system for Ka = 107 J m−3 rapidly grows up to 380 K; for lower Ka the temperature jump is smaller; while for 2 × 106 J m−3 it is almost invisible. Similar phenomena are commonly observed in experimental studies of interaction of magnetic nanoparticles, suspended in fluids, with alternating magnetic fields. That phenomenon was found useful in novel, though still experimental, approaches to tumor treatments using hyperthermia.6,16 The temperature increase in suspension of spherical MNPs is mainly due to the exchange of kinetic energy between rotationally excited MNPs and fluid molecules. Taking into account the architecture of the NC, we may expect the same or even better transformation of the magnetic energy into thermal energy. This is because the magnetic field induces both rotational and translational motions of MNPs during the uncapping process. The kinetic energy exchange between translational degrees of freedom is more effective than between rotational and translational ones. Therefore, the drug delivery and controlled release aside, the NC may also act as an efficient hyperthermia agent. The magnetic heating can also be viewed as a side effect which, however, can be very useful in enhancing the rate of drug release from the NC interior.

Figure 5. Temporal evolution of the NC structure after exposition to the external magnetic field |B⃗ | = KaV/|m⃗ |. Meanings of all symbols and their description are the same as in Figure.4.

KaV/|m⃗ |, affects the uncapping process. By comparing the cases for Ka = 107 J m−3 in Figures 4 and 5, we can find out that the application of weaker fields leads to a more facile response of the NC to the field. In Figure 4 the x direction was inactive, whereas in Figure 5 the uncapping occurs quickly for the same field direction. However, in general, the weaker fields mean the longer time of the uncapping transition. In the middle and the bottom parts of Figure 5, we can observe a kind of induction time of uncapping; i.e., there is some initial time period during which the NC stays capped before commencing the uncapping 26098

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3.4. Vanishing, Reversed, and Rotating Fields. Figure 7 shows the results of further studies of the NC. The top part of

The above scenario allows us to draw some conclusions concerning the application of alternating magnetic fields for triggering the NC uncapping. Assuming that the field strength changes sinusoidally in time, or there is a pulse width modulation (PWM) applied in time, we should get the same picture as in Figure 7. That is, initially the NC will undergo the flash uncapping according to the temporal evolution profiles shown in Figures 4 and 5. Afterward, when the field strength is zero, the NC will stay in the uncapped state, and magnetizations will be pushed toward the easy axes directions. Next, depending on the field timing, there will be full magnetization reversal (sinusoidal field) or displacement of magnetizations from the easy axes directions to the old field direction (PWM). Generally, we should get the same picture as in the case of static fields with the only difference being the periodic magnetization displacements according to the Neel process. Thus, the magnetic heating effect will, probably, be smaller or at most the same as in the case of ordinary spherical MNPs because the main component of the energy dissipation mechanism is the Neel process in this case. The above predictions are, of course, valid for relatively large Ka’s, i.e., when the NC is able to make the uncapping in times shorter than the field period. When the uncapping needs more time than the field period, the situation becomes more complex, and any predictions about the final NC state would be unreliable. Another way of the magnetic field application is a rotating field approach. Figure 8 shows temporal evolution of the NC

Figure 7. Behavior of the uncapped NC, obtained at conditions corresponding to the top part of Figure 5, i.e., Ka = 107 J m−3, after switching off the external magnetic field (top part) and, afterward, applying the field in the opposite direction (bottom part). In the top part, the symbols x, y, and z are used only for distinguishing the three configurations of the NC obtained in the previous stage of calculations. In the bottom part the arrows show the direction opposite to the direction of the currently applied field.

Figure 7 shows what happens after switching off the external magnetic field when the NC is in the uncapped state obtained after 10 ns exposition to the field. The current initial states are thus the final states from Figure 5 for Ka = 107 J m−3. We may thus conclude that the vanishing field does not change much the NC configuration provided that it has already reached the uncapped state. Only the z case returns to the capped state because in this case the NC has been left without the field prior to passing the activation barrier for uncapping. As already discussed, the spontaneous recapping needs macroscopic times; therefore, its occurrence within the studied 10 ns is very unlikely. The bottom part of Figure 7 shows further evolution of the NC structure when the field was applied again but in the direction opposite to that previously used for the uncapping. Intuitively we would expect a magnetically assisted recapping; that is, the MNPs should quickly be driven to the vicinity of the CNT tips. However, the results are totally different. After field inversion, there occurs only a quick magnetization reversal according to the Neel process. Then, the magnetic moments find the second and equivalent energy minima around the easy axes, and the energetic status of the whole NC does not change significantly. Simply, the magnetic torques, induced by the field inversion, act only for a short time. It is too short to induce the Brown rotation pushing the MNPs toward the CNT tips. Therefore, the final configuration of the NC remains unchanged in terms of topology, but the magnetizations are flipped according to the new field direction.

Figure 8. Temporal evolution of the NC structure during exposition to the rotating magnetic field. The initial capped structure of the NC was subjected to the magnetic field which changes its direction every 2 ns according to the sequence x−y−z repeated 3 times.

structure during application of the magnetic field according to periodic changes of the field direction. The initial structure of the NC was identical to those used in Figures 4 and 5, that is, it was the fully capped equilibrium structure. The magnetic field was applied in such a way that its direction changed every 2 ns according to the sequence shown on the greyed area in Figure 8. The field strength was constant and corresponded to the magnitude of the anisotropy barrier. As seen in Figure 8 we will get a number of intense transitions from the capped to uncapped states (and vice versa) for one of the NC ends. This, of course, very particular picture is a result of the choice of the field timing and the initial alignment of the NC and the field direction. Generally, in the case of rotating fields, we should observe periodic capping and uncapping transitions. This is because the NC is very sensitive to the field direction as shown in Figures 4 and 5. The final state of the NC, upon application of the rotating field, is impossible to predict, but during the exposition for a finite time the NC will surely undergo a lot of structural changes. 26099

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Notes

Moreover, there will be a huge magnetic heating effect due to the intense migration of the MNPs in twisting magnetic field. It seems that this is the most promising method of engineering a magnetically triggered drug release from the interior of the NC at a target site.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Polish National Science Centre (NCN) Grant N N204 205240.

4. SUMMARY AND CONCLUSIONS New insights concerning the properties of the magnetically controlled nanocontainer, found in this study, can be summarized as follows: •The required energetic balance between the capped and uncapped configurations of the NC can be achieved by applying multiwalled carbon nanotubes. The lengths of linkers as well as their chemical identities are then not very significant provided that they are flexible and do not hinder the motion of MNPs from one position to another. This is because the dispersion interactions at the tips are significantly enhanced in the case of MWCNTs. Moreover, they govern the shape of the total interaction energy profile while going from the capped to the uncapped configuration. The difference in energy in these two states is not huge, ∼27 kJ mol−1; similarly, the activation barriers for capping (∼58 kJ mol−1) and uncapping (∼85 kJ mol−1) transitions are not high as well. However, due to very large masses of MNPs, the time scale for these transitions reach macroscopic values. Generally, the energy profile of the new architecture of the NC supports its application as a drug targeting nanovehicle. •Analysis of the magnetic anisotropy values provides useful information about the choice of magnetic materials for the construction of magnetic caps. Considering reasonable magnetic field strengths, i.e., not larger than 12 T, we found that magnetic anisotropy constants of the order of 108 J m−3 lead to a perfect ferromagnetic state of the MNPs; i.e., the magnetization reversal cannot be induced under such field strength. Such a high anisotropy constant can be found only in the case of the hardest magnetic materials based on rare earth elements. Lowering the anisotropy constant to values representative of cobalt NPs, i.e., 106−107 J m−3, leads to facile magnetization reversals within the MNPs. However, the magnetically triggered uncapping of the NC still occurs for the anisotropy constants slightly larger than 106 J m−3. •Application of very strong fields is less effective than moderate ones. This is due to a fast alignment of magnetizations with the field direction under strong fields. This, in turn, leads to vanishing of magnetic torques. Under weaker fields, e.g., comparable to the demagnetizing field of magnetic anisotropy, the magnetizations cannot totally align with the field direction, thus the magnetic torques continuously assist the Brown rotation and facilitate migration of the MNPs. We observed a pronounced magnetic heating effect during the magnetically assisted uncapping processes. Its intensity as well as susceptibility of the NC to uncapping strongly depend on the initial field direction. Once the NC is uncapped, the field reversal does not lead to the recapping transition. The NC remains uncapped though the magnetization flips according to the Neel mechanism. Generally, the alternating magnetic fields will probably lead to similar effects like the static ones. However, the rotating fields may lead to intense multiple uncapping and recapping transitions.

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