Enhancing the Control of a Magnetically Capped Molecular

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Enhancing the Control of a Magnetically Capped Molecular Nanocontainer: Monte Carlo Studies Tomasz Panczyk,*,† Tomasz P. Warzocha,‡ and Philip J. Camp§ †

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 § School of Chemistry, The University of Edinburgh, West Mains Road, Edinburgh EH9 3JJ, United Kingdom ‡

ABSTRACT: In a recent paper [Panczyk, T.; Warzocha, T. P.; Camp, P. J. J. Phys. Chem. C 2010, 114, 21299] we discussed some of the factors controlling the properties of a magnetically controlled molecular nanocontainer composed of a carbon nanotube and magnetic nanoparticles. This paper complements that study by (1) analyzing the role of the configuration of anchoring points on the carbon nanotube tips to which the magnetic nanoparticles are bound; (2) discussing possible effects of the presence of electrostatic stabilizing layers on the surfaces of the magnetic nanoparticles; (3) studying the effects of the carbon nanotube length; and (4) analyzing how the rate of increase of the magnetic field affects the uncapping processes. It is found that in some cases the nanocontainer might be locked in the double uncapped state. This effect can be minimized or eliminated by using either nanotubes longer than the sum of the magnetic nanoparticle radii or by utilizing electrostatic repulsion between charged nanoparticles. The magnetically triggered uncapping is better effected by using an instantaneous increase of the magnetic field strength. A slow increase of the field leads to alignment of nanocontainers with the field direction, and this effect reduces the effectiveness of magnetically triggered uncapping.

1. INTRODUCTION Composite systems comprising carbon nanotubes (CNTs) and magnetic nanoparticles (MNPs) represent an interesting option in production of new multifunctional nanomaterials. Magnetic nanoparticles have been either deposited on the sidewalls of carbon nanotubes1
3 or incorporated into the internal space of the nanotubes.4
6 In both cases the aim of fabricating these composite materials was to be able to control nanotube alignment by applying an external magnetic field, or to produce superparamagnetic nanoneedles.4 Normally, carbon nanotubes exist in randomly aligned bundles, and they exhibit substantially lower electrical and thermal conductivities than expected.7,8 To overcome this limitation, assemblies of aligned nanotubes have been proposed to exploit fully the properties of individual nanotubes in a bulk material.9
11 Magnetic nanoneedles, due to their unique properties, can be used as magnetic stirrers in microfluidic devices or magnetic valves in nanofluidic devices.12 Korneva et.al.4 have demonstrated that magnetic nanoneedles might also be used as nanoposts in fluidic chips for DNA separation. A theory of magnetostatic interactions between carbon nanotubes filled with magnetic nanoparticles has been proposed by Kornev et.al.5 and by Escrig et.al.,13 and can be applied to asses numerous potential applications of such materials. Carbon nanotubes can be used as supports for heterogeneous catalysts, and due to their superior electronic conductivity, high r 2011 American Chemical Society

surface area, and high mechanical and thermal stabilities, these materials are attracting increasing attention in the literature.14
16 In the case of CNT-based catalysts, metallic (and often magnetic) nanoparticles are normally bound to the sidewalls of carbon nanotubes. Terminally connected MNPs have also been engineered using plasma enhanced chemical vapor deposition.17 The presence of a hard magnetic nanoparticle (FePt L10 phase18) at the CNT tip is very promising for the realization of nanodevices, for example as tips for magnetic force microscopy, or as magnetically actuated nanoelectromechanical systems. CNT functionalization with FePt nanoparticles using organic linkers has also been reported. However, to the best of our knowledge, a selective binding at the CNT end has not been achieved yet.19 In recent publications20,21 we have proposed the concept of a magnetically controlled nanocontainer (NC) and demonstrated its potential features using Monte Carlo simulations. The crux of our concept is the design and optimization of a nanometer-sized capsule which might be switched from a capped to an open state by using some easy-to-apply, in vivo, external stimulus. The obvious area of application is nanomedicine and drug delivery technology, both rapidly developing branches of nanotechnology in terms of research and clinical trials.22
25 Received: January 5, 2011 Revised: March 17, 2011 Published: March 31, 2011 7928

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The Journal of Physical Chemistry C Though the detailed design of the NC has been described in our recent publications,20,21 it is worthwhile to recall some general aspects of its structure and the resulting properties. The NC is composed of a single-walled carbon nanotube and two magnetic nanoparticles which are attached to the CNT tips through single
CH2
linkers. In the absence of an external magnetic field (EMF) the most energetically favorable configuration of the NC is when both MNPs plug the terminal rings of the CNT; that is, the NC adopts the double capped state. Transfer to a semi or double uncapped state is accompanied by significant activation barriers (greater than 100 kJ/mol and depending on system parameters); thus, without an external stimulus, the NC cannot spontaneously reach one of these uncapped configurations. As shown in our recent paper,21 a transition to the uncapped state occurs only under strong external magnetic fields when the interaction of the MNPs with the EMF provides enough energy to overcome the activation barriers. Once uncapped, the NC remains in this state for a long (macroscopic) time due to another activation barrier (ca. 70
80 kJ/mol) for the capping process. These barriers are mainly dependent on the linker parameters (
CH2
); other system parameters do not affect the capping probability significantly. As has already been pointed out in our recent papers,20,21 the proposed nanocontainer can operate as a drug delivery system and offer all of the advantages of carbon nanotubes and magnetic nanoparticles. A significant conclusion is that such nanocontainers represent a feasible and easily engineered candidate for effecting drug release in vivo. Carbon nanotubes and magnetic nanoparticles are extensively studied in terms of their application as drug vectors.26
36 Their numerous advantages aside, the main problem arises while triggering drug release at the target site. Biodegradation, which may be caused by some chemical or external stimulus, changes in pH,32 and the application of near-infrared radiation37 are potentially useful for in vivo treatments; however, changing the properties of solutions, though proven to work successfully in vitro,24,38 does not appear feasible in living organisms. The proposed nanocontainer can release guest molecules absorbed in its interior upon applying a magnetic field. As the magnetic field is not normally harmful, its use as the external trigger for drug delivery systems is very promising and of course feasible. The conditions necessary for magnetically triggered uncapping have been inferred in our recent publication.21 We studied the roles of the CNT diameter, dispersion interaction strength, alignments of the magnetic moments, and external magnetic field strength. Only certain sets of parameters lead to a functional nanocontainer, reflecting a subtle balance between the molecular interactions arising from each variable. The conclusions drawn from that study are, however, valid only for one specific configuration of the anchoring points for linkers at the CNT tips, that is for the trans configuration with anchoring points set diametrically opposite one another and at opposite ends of the CNT. Other configurations were not analyzed, so the current study addresses this point since, as it turns out, the role of the configuration of the anchoring points is important and strongly affects the properties of the NC. We also present a more detailed analysis of the uncapping process of the NC by distinguishing semiuncapped and double-uncapped states and their occurrences under the EMF. Finally, the case of charge-stabilized MNPs is considered as this might be a convenient way of preventing agglomeration of the NCs in solution.

2. METHODS Detailed descriptions of the molecular model and the Monte Carlo simulation approach are provided in ref 21. The current

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Figure 1. Definition of the most important parameters of the model nanocontainer. The MNPs of radius rMNP are bound to the terminal rings of the CNT via single
CH2
groups, as shown in the rectangular area. μ is the magnetic moment associated with the MNP, and B illustrates the applied magnetic field. The vector connecting the anchoring point on the MNP surface and its center creates the angle j with the μ vector. The angle β defines the configuration of the anchoring points on the CNT tips, and s is the slit size between the surface of the MNP and the edge of the CNT (s defines the degree of capping).

study uses the same definitions, force field, and, unless mentioned specifically, the same sets of interaction parameters. Thus, to avoid repetition of already published material we provide here only a brief description of the methods applied. Figure 1 illustrates the concept of the NC and highlights the most important parameters which we have to deal with. An MNP with diameter dMNP = 2rMNP is bound to a CNT tip via the linker highlighted by the rectangular box. The properties of the linker were discussed in detail in ref 21. Chemically, it is a single
CH2
group contributing to the total energy of the system via bond stretching and bond bending potentials.39 The bondstretching potential is Ustretch ¼

Kr ½ðr12
re Þ2 þ ðr23
re Þ2  2

ð1Þ

where Kr = 9.65  106 K nm
2 and re = 0.154 nm. The bondbending potential is Ubend ¼

Kθ ½ðθ1
θπ Þ2 þ ðθ2
θe Þ2 þ ðθ3
θπ Þ2  2

ð2Þ

where Kθ = 6.25  104 K rad
2 and θe = 114°, i.e., ∼2 rad. The values of the potential parameters are representative of nalkanes39 and should also be correct in the present case. The equilibrium angles for θ1 and θ3 are assumed to be θπ = 180.0°. In this work we will use only one combination of the MNP and CNT diameters, that is, dMNP = 90 Å and a CNT diameter of 35.2 Å. The assumed diameter of the MNP ensures a high enough value of the magnetic moment, μ, necessary to induce magnetically triggered uncapping under reasonable magnetic field strengths, B. As before,21 we assume that μ is fixed in the MNP body frame, which is appropriate for a ferromagnetic nanoparticle. We study CNTs of the same chirality (45, 0) and with lengths lCNT = 84 and 126 Å. Because the properties of the NC depend on the mutual alignment of the magnetic moments associated with both MNPs, we define, for systematic study, j as the angle between the 7929

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magnetic moment vector μ and the vector connecting the center of the MNP to the anchoring point on its surface. The angles jL and jR therefore describe the alignment of μ with respect to the anchoring points on the left-hand side (L) and right-hand side (R), respectively. The degree of capping is characterized by the distance s between the MNP and the terminal ring of the CNT. It is defined as the maximum distance between the surface of the MNP and the surface of any one
CH group located on the terminal ring of the CNT; this is given by the center
center distance between the MNP and the
CH group minus (dMNP þ σCH)/2, where σCH = 3.8 Å is the Lennard-Jones range parameter for the
CH group within the united-atom representation.40 A new parameter appearing in this study is the angle β which defines the configuration of the anchoring points on the CNT tips. Its meaning is straightforward if we look at the inset of Figure 1, which shows the head-on view of a CNT. β = 180° means the trans configuration, as analyzed previously,21 whereas β = 0° corresponds to the cis configuration. The total interaction between the MNPs contains two contributions, the dispersion interaction and the magnetic dipole interaction. The former is described by the Hamaker potential with a given value of the Hamaker constant Acc.41,42 If the MNPs are spherical objects of radius a (in the present case a = rMNP) then the interaction energy between them is given by " # Acc 2a2 a2 r 2
4a2 þ 2 2 þ ln Ucc ¼
6 r 2
4a2 r r2 " # Acc σ 6 r 2
14ar þ 54a2 r 2 þ 14ar þ 54a2 2r 2
60a2 þ þ
37 800 r r7 ðr
2aÞ7 ðr þ 2aÞ7

ð3Þ where r is the separation distance, σ is the LJ diameter of the constituent atoms, and Acc is the Hamaker constant. The interaction between an MNP (a colloidal particle, denoted by c) and a single atom (denoted by s), e.g., a carbon atom within the CNT, is given by " # 2a3 σ 3 Acs ð5a6 þ 45a4 r 2 þ 63a2 r 4 þ 15r 6 Þσ 6 1
Ucs ¼ 9ða2
r 2 Þ3 15ða
rÞ6 ða þ rÞ6 ð4Þ Equation 4 is derived in the same way as eq 3 but with one of the diameters of the nanoparticles set equal to zero. Acs in eq 4 is the Hamaker constant calculated using the mixing rule Acs = (AccAss)1/2 where Ass = 144εLJ and εLJ is the energy parameter in the LJ potential for the single atom. In the current study we use only one value of the Hamaker constant Acc = 8  10
20 J which is in the middle of the values studied previously.21 The magnetic dipole interaction is computed using the Dormann
Bessais
Fiorani pair potential43
45 Udip ¼

μ0 μ2 ^
3ð^ ½^ μ μ μ1 3 ^r Þð^ μ2 3 ^r Þ 4π r 3 1 3 2

ð5Þ

where μ = |μ1| = |μ2|, μ0 is the magnetic permeability of free space, r is the length of the separation vector between the MNPs, and the circumflex denotes a unit vector. The interaction of the MNP with the external field is a simple dot product of μ and B. All calculations are carried out at T = 300 K.

As the MNPs in solution are very likely to carry an electric charge the role of the electrostatic interactions will be qualitatively discussed in section 3.2. A colloidal suspension normally consists of macroions (in this case, charged MNPs), counterions, salt ions, and solvent molecules. This multicomponent mixture can be modeled on an effective level within the DLVO theory.47
51 It involves only the (negatively charged) macroions, whose Coulomb repulsion is screened exponentially by the surrounding counterions and salt ions. The total interaction between two charged MNPs is therefore the sum of the dispersion interactions (Ucc) and the magnetic interactions (Udip) defined previously, and the screened Coulomb interaction described by the Yukawa term47 UYuk ðrÞ ¼

ðZe0 Þ2 expðkdMNP Þ expð
krÞ r 4πεε0 ð1 þ kdMNP =2Þ2

ð6Þ

where Z is the number of elementary charges (e0) the macroion gains in the solution, ε0 is the dielectric permittivity of the vacuum, and ε is the relative dielectric constant of the solvent. The inverse Debye screening length κ involves contributions from the counterions and the salt ions. The salt contribution is measured by the ionic strength I (in mol L
1).47 Assuming charge neutrality between macroions and counterions, the inverse Debye length can be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1000e20 ðZF þ 2INA Þ ð7Þ k¼ ε0 εkB T where F is the macroion concentration (in mol L
1) and NA is Avogadro’s number.

3. RESULTS AND DISCUSSION 3.1. Activation Barriers and Locking Phenomenon. The most important factor controlling the properties of the NC is the total energy profile while making the transition from the capped to the uncapped state. We can distinguish two types of transition by analyzing the states of the left-hand side and right-hand side of the carbon nanotube: capped
capped to capped
uncapped (denoted as the cc f cu transition) and capped
uncapped to uncapped
uncapped (denoted as the cu f uu transition). Any transition is accompanied by a transient increase in the total energy of the system (Ut) due to deformations of the
CH2
linker and the loss of dispersion attractions. The Ut profiles, or more precisely the minimum total energy paths for the transitions, have been determined using umbrella sampling and without the EMF. The Hamiltonian of the system was modified by adding the term 1/2 τ[(sL
s0L)2 þ (sR
s0R)2] where s0L,R is the target value of sL,R. Smooth sampling of the configurations with (sL, sR) close to (s0L, s0R) has been obtained by choosing τ = 5 kJ Å
2. The minimum energy path for the transition from one state to another can thus be determined by scanning ranges of (s0L, s0R) values and recording the smallest total energy of the system for a given set of (sL, sR). Typical minimum-energy paths obtained in this way [from 2  106 MC steps for each set of (s0L, s0R)] are shown in Figure 2. Left panels show the ccfcu transitions, i.e., s0L = 1.25 Å, corresponding to the position of the Ut minimum in the capped configuration, while s0R changes from 1.25 to 35.5 Å, which is the position of the second minimum in Ut corresponding to the uncapped configuration. Right panels show the continuation of the uncapping process, i.e., s0R is kept at 35.5 Å 7930

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The Journal of Physical Chemistry C while s0L changes from 1.25 to 35.5 Å; thus, finally, the system ends up in the double uncapped uu configuration. Figure 2 shows two cases of the arrangements of the anchoring points on the CNT tips: the case with β = 0° giving the cis configuration, and β = 180° leading to the trans configuration

Figure 2. Minimum-energy paths for the transitions cc T cu T uu determined for the (45, 0) CNT with dMNP = 90 Å, Acc = 8  10
20 J, |μ| = 4.4  104 μB (Bohr magneton), and lCNT = 84 Å. The top panel stands for β = 0° (the cis configuration of the anchoring points on the CNT tips) whereas the bottom one corresponds to β = 180° (the trans conformation). In both cases the alignment of the magnetic moments is the same, (jL, jR) = (0, 0). All of these calculations were carried out at T = 300 K and with B = 0.

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which was, in fact, the case analyzed previously.21 Snapshots of the NC configuration in various situations underline the importance of the angle β as a parameter controlling its properties. In particular, the energies of the uu states differ significantly, and obviously this will lead to distinct differences in the behavior of the NCs for different angles β. However, as follows from Figure 2, the transitions cc T cu for both β values reveal the same energy profiles meaning that the activation barriers E(cc f cu) = U6¼ t (cu)
Ut(cc) and E(cu f cc) 6¼ = U6¼ t (cu)
Ut(cu), where Ut denotes the energy at the transition state (TS), are almost the same for any value of β. The next stage of uncapping, that is, the cu f uu transition, leads to qualitatively new features. Namely, for β = 0 the total energy minimum associated with the uu state becomes sharp and significantly deeper than that for β = 180°. It means that the activation barriers E(uu fcu) = U6¼ t (uu)
Ut(uu) will differ significantly for different arrangements of the anchoring points, i. e., β values. It seems, however, that the barriers E(cu f uu) = U6¼ t (uu)
Ut(cu) will not be affected strongly by β, similar to the case of the cc T cu transition. Looking at the insets of Figure 2, we can easily recognize that the sharp and deep minimum of Ut for the uu state is due to close contact of the MNPs in this case. The CNT length is smaller than the sum of the diameters of the MNPs, and thus, the surface-to-surface distance might be as small as the balance of dispersion and magnetic interactions allows for. The roles of the angle β and the alignment of the magnetic moments (jL, jR) are important not only in the β = 0 case. Figures 3 and 4 show a complete analysis of the activation barriers for a number of combinations of these parameters. Figures 3 and 4 show large amounts of data displayed in a compact form, and therefore merit detailed discussion. The dashed grid lines divide the graph into vertical segments corresponding to some value of β as denoted on the x axis. The y axes correspond to the activation energies as denoted by the axis titles, and they are separated by solid grid lines. Within each panel defined by the solid and dashed grid lines, a given point represents the energy value for one combination of jL and jR angles. So, within each panel we can

Figure 3. Activation barriers (in kJ/mol) determined for the (45, 0) CNT with dMNP = 90 Å, Acc = 8  10
20 J, and lCNT = 84 Å. Each panel on the graph shows how a given activation barrier changes with the alignment of magnetic moments (jL, jR) for a given arrangement of anchoring points on the CNT tips β. 7931

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Figure 4. Activation barriers (in kJ/mol) determined for the (45, 0) CNT with dMNP = 90 Å, Acc = 8  10
20 J, and lCNT = 126 Å, i.e., longer by 42 Å than that in Figure 3. Each panel on the graph shows how a given activation barrier changes with the alignment of magnetic moments (jL, jR) for a given arrangement of anchoring points on the CNT tips β.

see how the activation energy changes when jL and jR run from 0° to 180° in steps of 45°. Figure 3 leads to the conclusion that the activation energies E(cc f cu) as well as E(cu f cc) are insensitive to the position of the anchoring points on the CNT tips. The observed oscillations are due to changes in alignments of the magnetic moments, and their variations are about 3 kJ/mol, which is comparable to kBT. Put simply, at the corresponding distances the magnetic dipole interaction energy is small. The barrier E(cc f cu) is about 178 kJ/mol which means that the cc and cu states are very well separated and a spontaneous cc f cu transition cannot occur. The barriers for the reverse process (cu f cc) are about 85 kJ/mol; thus, as we estimated previously,21 a spontaneous cu f cc transition can occur though it may take quite a long time. The second stage of the uncapping, i.e., the cu f uu transition, is accompanied by barriers similar to those in the first stage (about 178 kJ/mol), and similarly E(cufuu) does not depend on β, though a visible periodicity as a function of (jL, jR) appears for β = 0°. However, the reverse process, that is uu f cu, is strongly affected by the configuration of the anchoring points. For β < 45° and for the uu states the distance between the MNPs becomes short which enhances the dispersion and the magnetic dipole interactions. Therefore, we observe large E(uu f cu) values reaching 150 kJ/mol and a nice periodicity as a function of (jL, jR). The highest barrier appears for (jL, jR) = (90°, 90°) when the magnetic dipoles form a nose-to-tail (ff) alignment in the uu state. This is a very stable configuration, and spontaneous capping (the uu f cu transition) is very unlikely to occur. In other words, if the NC attains such a state then it would be locked in this state and so would not be functional. Figure 4 shows analogous results as Figure 3, but in this case the CNT length has been enlarged by 50% to 126 Å. Comparing both figures we can draw a very simple but important conclusion. Namely, the effects coming from the close contact of the MNPs for β < 45° have been eliminated for the longer nanotube. The barrier E(uu f cu) remains at the same level as E(cu f cc)

(about 85 kJ/mol) for any value of β. It means that the transition uu f cu is now thermodynamically accessible in a reasonable time scale. Thus, the final conclusion concerning the design of a functional nanocontainer is that it should be composed of a nanotube significantly longer than the sum of the radii of the MNPs. This ensures that the locked uu states of the NC will never occur 3.2. Effects of Charge Stabilization of the MNPs on the Occurrence of the Locking Phenomenon. The above considerations need some revision if we account for the presence of a stabilizing layer on the MNP surface. Bare magnetic nanoparticles readily agglomerate due to magnetic and dispersion interactions unless their surfaces are covered by some steric or electrostatic stabilizing layers. Depending on the concentration they might form chainlike clusters, rings, or 3D networks.44,45 Thus, in order to ensure the stability of the NCs in solution, the MNP surfaces must also be covered by a stabilizing layer. So far, we have not specified the chemical nature or other parameters of such a layer. We implicitly assumed that a bare magnetic core is covered by a thin stabilizing shell containing a carbon atom to which
CH2
can be bound. As the NC is intended to work in an aqueous solution the most obvious means of stabilization is by electrostatic repulsion between charged colloidal MNPs. We can formally assume that the bare magnetic cores are covered by thin silica or other oxide shells34,46 which normally become negatively charged in electrolyte solutions. The
CH2
linkers can be formally bound to surface oxygen atoms, so the simulation model does not change, but another contribution to the system energy appears, i.e., the electrostatic repulsion between macroions. The problem of the stability of charged colloidal particles in suspensions is complex, and we are not going to address it in detail now.47
51 Instead, we discuss how the electrostatic repulsion between charged MNPs may contribute to the potential energy minima in the uu states of the NC and try to understand whether the risk of locking the NC in the uu states exists also in the presence of the electrostatic stabilizer. 7932

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3.3. Effects of the Angle β on the Magnetically Triggered Uncapping. The uncapping processes (the cc f cu and cu f uu

Figure 5. Total interaction energy between two charged MNPs (with dMNP = 90 Å, Acc = 8  10
20 J, |μ| = 4.4  104 μB) at concentration F = 10
4 mol L
1 in a solution with ionic strength I = 0.145 mol L
1, T = 300 K, and B = 0. The alignment of the magnetic moments is ff, which corresponds to the strongest possible magnetic attraction.

Using the DLVO model, briefly outlined in the Methods section, we can qualitatively predict how the activation barriers may change when the NC is in aqueous solution. Figure 5 shows how the effective interaction energy between two charged MNPs changes for various charges, Z, while the other parameters are the same as in Figures 2
4. We assumed that the ionic strength I = 0.145 mol L
1 (the physiological value52) and that the MNP concentration F = 10
4 mol L
1 (corresponding to an MNP volume fraction of 0.0230). Figure 5 clearly shows that without a charge the MNPs would agglomerate due to very deep energy minimum at close distances. Of course, more reliable conclusions concerning the stability of an ensemble of NCs in solution would require dedicated calculations as the entropic factors play very important roles in such cases. However, it is clear that, without stabilizing layers, the ensemble of NCs in solution would tend to create large clusters and would be useless for targeted molecular delivery. The problem of the stability of the NCs in solution needs dedicated studies involving expensive calculations, and this will be a focus of future work. We can see from Figure 5 that in the range Z = 100
175 the potential energy reveals two distinct minima with depths depending on Z. They are separated by repulsive barriers at a distance of about κ
1, and so the results shown in Figures 2 and 3 will be significantly altered in the case of charged MNPs and with values of β allowing close contact of the MNPs. The uu states are very unlikely to occur spontaneously; they can, however, appear upon interaction with the EMF. Then, the energy barriers at κ
1 might be easily surmounted, and when β < 45°, the MNPs might end up at the closest-contact energy minimum. For some values of Z (ca. 125) that minimum is relatively deep and the locked uu states might appear also in the case of charged MNPs. Although the E(uu f cu) barrier would be significantly smaller than those shown in Figure 3, a spontaneous transition uu f cu would still be much slower (if not completely blocked) than that for a longer nanotube (Figure 4) or with β > 45°. Thus, even if we account for the presence of the electrostatic stabilization of the MNPs, an NC composed of a nanotube longer than the sum of the radii of the MNPs represents a safer choice since we completely avoid locking the NC in the uu state.

transitions) cannot occur spontaneously due to high activation barriers. They can, however, be triggered by the interaction of the NC with an EMF, as shown previously.21 A good way of visualizing the propensity for magnetically triggered uncapping over wide ranges of system parameters is a map of flash uncapping defined in ref 21. It shows whether a given alignment (jL, jR) leads to an NC which rapidly transforms from the initial fully capped state (cc) to any of the uncapped ones (cu and uu) just after switching on the EMF of a given strength, B. Currently we use a slightly modified definition of the flash uncapping and use two approaches for studying its occurrence. We start from the cc state and perform a standard unbiased MC run for 2  105 steps with B = 0. Next, that equilibrated configuration is used for studying two ways of applying the EMF. The first approach aims at understanding how a randomly oriented impulse field affects the state of the NC. Thus, the simulation is continued for 105 steps with B switched on, and its orientation is random. Afterward, the field is switched off, and the simulation is continued for the next 105 steps. During these 2  105 MC steps the state of the NC is monitored, and if the NC achieves any of the uncapped states in less than the first 105 consecutive steps and remains in this state until the end of the run, it is assumed that flash uncapping occurred. The second approach consists of two stages of the interaction with the EMF. The equilibrated cc state is subjected to the interaction with a randomly oriented and weaker magnetic field B = 0.5 T for 105 steps. This field cannot induce uncapping, but it does lead to alignment of the whole NC with the field direction via a rigid-body rotation. Afterward the field strength is increased to the target value without changing its direction, and the simulation is continued for 105 steps. Finally, the field is switched off, and calculations continued for the last 105 steps. As in the previous approach the state of the NC is monitored, and flash uncapping is detected in the same way using the last 2  105 steps. So, this approach mimics a ramped increase of the magnetic field strength. Figure 6 shows maps of flash uncapping for a series of NCs differing only by the configuration of the anchoring points β. Other parameters of the NCs are the same as in Figure 3; i.e., this is the case of the short CNT. The maps were determined according to the first approach, which demonstrates the effect of exposure to a short, randomly oriented, impulse field. Each point represents the threshold for uncapping, determined from a sequence of individual simulations at different applied magnetic field strengths. No averaging has been attempted, but in spot checks, three independently generated maps were found to give the same overall picture. The pattern obtained for β = 180° (the trans configuration) is essentially the same as those determined in the previous study.21 It exhibits a symmetry axis jR = 180
jL separating two equivalent areas of (jL, jR) combinations leading to magnetically triggered uncapping. When β decreases, going toward the cis configuration, the patterns change as well which is obvious as the (jL, jR) combination creating the antiparallel alignment (most sensitive to B) depends on the configuration of the anchoring points on the CNT tips β. The most interesting observation is that each pattern consists of a solid area (or areas) of activity and the remaining area where, no matter what is the field strength, a considerable amount of isolated spots of activity appear. To understand this phenomenon, we analyze Figure 7 which shows maps of flash uncapping for the same systems but determined 7933

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Figure 6. Maps of the alignments of magnetic moments (jL, jR) for which the flash uncapping occurs for different configurations of the anchoring points on the CNT tips, β, denoted on the axis titles. The uncapping is triggered by a short impulse of the magnetic field B. A given color represents the B value under which flash uncapping occurs and the NC takes the cu or uu state. The lack of a point means that for that (jL, jR), uncapping does not occur. Calculations were performed for (45, 0) CNT, dMNP = 90 Å, Acc = 8  10
20 J, T = 300 K, and lCNT = 84 Å.

according to the second approach; i.e., the magnetic field strength is ramped up in two steps. Figure 7 leads to the striking conclusion that for β e 90° the NCs are almost inactive. Traces of activity remain only at the corners for β e 30°, and some randomly distributed, isolated spots are observed. Only the configurations close to trans seem to be little affected by the two-step method of applying the magnetic field. Thus, it is clear that exposing the NC to a weak field and allowing it to relax via a rigid-body rotation, prior to the application of the strong field, drastically reduces its ability to undergo magnetically triggered uncapping. By comparing Figures 6 and 7 it is clear that all configurations are affected by initial alignment of the whole NC with the field direction. It may lead either to some reduction in the probability of uncapping or to total inactivity.

The strong sensitivity of the NC observed in Figure 6, no matter the angle β or (jL, jR) combination, is due to random mutual alignment of the magnetic moments and the field direction. It turns out that the probability of the uncapping is greater than the probability of rigid-body rotation for strong enough fields. Therefore, even if (jL, jR) creates the alignment which is little sensitive to the field (close to parallel), one or both MNPs might be strongly kicked by the field and detach from the CNT tip faster than the torque would rotate the whole NC minimizing the magnetic energy. These observations suggest that a practical realization of magnetically triggered uncapping should involve a pulse-wave modulation of the EMF strength. It is likely that after application of some number of pulses statistically all NCs would get uncapped no matter what are the values of jL, jR, and β. This 7934

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Figure 7. Maps of the alignments of magnetic moments (jL, jR) for which the flash uncapping occurs for different configurations of the anchoring points on the CNT tips, β, denoted on the axis titles. The results correspond to the ramped increase of the field strength; i.e., prior to the application of a strong field the NC is allowed to relax under a weak magnetic field of 0.5 T. A given color represents the B value under which flash uncapping occurs and the NC takes the cu or uu state. The lack of a point means that for that (jL, jR), uncapping does not occur. Calculations were performed for (45, 0) CNT, dMNP = 90 Å, Acc = 8  10
20 J, T = 300 K, and lCNT = 84 Å.

is because any orientational ordering of NCs induced by the EMF impulse (not necessarily leading to uncapping) will quickly be destroyed by thermal motion during the time between pulses. Thus, a new set of randomly distributed angles between the NCs and the field direction will be established, and the next pulse would uncap another portion of the NCs. It is likely that in this second EMF pulse the already uncapped NCs can be taken back to the capped state according to the same mechanism. Therefore, the conclusion concerning the pulse-wave modulation of the EMF needs a closer analysis, by dedicated calculations, as it may lead to very promising results. The previously determined maps of flash uncapping21 did not reveal such scattered patterns as those in Figure 6; they were more similar to the bottom-right map in Figure 7. This is because we used a

different way of arranging the directions of the NCs and the field. After performing a run for a given set (jL, jR) the resulting configuration and orientation of the NC with respect to the field were used in the next run with an increment of 5° in one of the angles jL or jR. Thus, that approach was in fact similar to the approach assuming the relaxation of the NC in a weak field, as in Figure 7. On the other hand, the results published in ref 21 concerned exclusively the trans configuration, β = 180°, which is insensitive to the arrangement of the field direction, and thus they are still valid. Figure 6 shows maps of uncapping without distinguishing the cu and uu states. It is very interesting to compare how the occurrences of these states depend on jL, jR, and β. To that end we present Figure 8 which shows the results from Figure 6 but with the occurrence of the cu states subtracted; only the uu states are shown. 7935

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Figure 8. Maps of the alignments of magnetic moments (jL, jR) for which the flash uncapping occurs for different configurations of the anchoring points on the CNT tips, β, denoted on the axis labels. The uncapping is triggered by a short impulse of the magnetic field B. A given color represents the B value under which the NC enters the uu state. The lack of a point means that for that (jL, jR) the uncapping does not occur at all or that the NC enters the cu state. Calculations were performed for (45, 0) CNT, dMNP = 90 Å, Acc = 8  10
20 J, T = 300 K, and lCNT = 84 Å.

An analysis of Figure 8 leads to interesting conclusions. Clearly, the configurations closer to the cis one are more abundant in the double uncapped states. Thus, there is a pronounced risk of locking the NC in the uu state due to the close contact of the MNPs for a significant number of jL, jR, and β combinations. The pure trans configuration reveals some small amount of the uu states in the region where the NC should not normally be active. It leads to the conclusion that the randomly oriented field can quickly detach both the MNPs for parallel alignments of the magnetic moments, but in the majority of other alignments such a phenomenon does not occur. The occurrence of the uu states in the ramped application of the EMF is zero. If we subtract the cu states from the maps shown

in Figure 7 we get empty patterns. Simply put, all of the uncapped states in Figure 7 are cu states. Analogous maps of uncapping were determined for NCs composed of a longer CNT with lCNT = 126 Å and all other parameters unchanged. We observed that the CNT length has no influence on the occurrence of the magnetically triggered uncapping, and with almost no effect on the patterns in the uncapping maps. This is not surprising since the activation barriers for the uncapping do not depend on β and the CNT length, as follows from Figures 3 and 4. However, as was already mentioned, longer nanotubes represent a better choice since we avoid the risk of locking the NC in the uu states. Occurrence of these states is not negligible when the NCs are subjected to the 7936

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The Journal of Physical Chemistry C interaction with a strong and randomly oriented magnetic field. Therefore, this remark is important for the design of functional nanocontainers.

4. SUMMARY AND CONCLUSIONS This work gives new insights into the properties of a molecular nanocontainer proposed in our recent series of publications. We found that the configuration of the anchoring points on the CNT tips β does not significantly affect the activation barriers for uncapping but strongly affects the barriers for the capping process when the NC structure allows for a close contact of the MNPs. Then the contribution to the total energy from the dispersion and magnetic-dipole interactions becomes large, and the uu state of the NC becomes very strongly metastable. As a result, once the NC is driven to the uu state it may stay in that state for a very long time. Because the uu f cu transition might be accompanied by barriers reaching 150 kJ/mol, its occurrence without an external stimulus would be a very rare event, and thus, the NC could not work in a reversible manner. Therefore, longer nanotubes (with lengths greater than the sum of the radii of the MNPs) are preferred in the construction of functional NCs since the risk of locking the NC in the uu state is eliminated. On the other hand, that risk can be reduced if we account for the presence of stabilizing layers on the MNP surfaces. A simple analysis of the electrostatic stabilization allows us to conclude that, depending on the charge of the MNPs, the locking phenomenon indeed may be removed, but over a wide range of physically relevant charges it is only reduced to some extent. We found that the occurrence of magnetically triggered uncapping depends strongly on the way the field is applied. If the NC is allowed to relax in a weak field prior to the application of the target field strength, then only the trans configurations (β ≈ 180°) exhibit strong sensitivity and a high probability of undergoing the cc f cu transition; the cu f uu transition does not occur at all. However, if the field instantaneously increases from zero to the target value then the vast majority of the NCs undergo flash uncapping. This is due to the random initial alignments of the NCs and the field direction, and therefore, almost every alignment of the magnetic moments (jL, jR) leads to some activity, although the response is not uniform over the full range of (jL, jR). The most dominant are the cc f cu transitions occurring mainly when (jL, jR) creates antiparallellike alignments in the cc states. There is, however, a considerable probability of undergoing the cu f uu transition, especially for configurations close to cis (β = 0°) and with values of (jL, jR) giving parallel alignments in the cc state. Thus, a practical realization of magnetically triggered uncapping should involve a fast increase of the magnetic field strength, as it minimizes the influence of the NCs internal structure on the probability of the uncapping process. ’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

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

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