Computational Study of Some Aspects of Chemical Optimization of a

ARTICLE pubs.acs.org/JPCC

Computational Study of Some Aspects of Chemical Optimization of a Functional Magnetically Triggered Nanocontainer Tomasz Panczyk,*,† Philip J. Camp,‡ Giorgia Pastorin,§ and Tomasz P. Warzocha|| †

Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30239 Cracow, Poland School of Chemistry, The University of Edinburgh, West Mains Road, Edinburgh, EH9 3JJ, U.K. § Department of Pharmacy, National University of Singapore, S4 Science Drive 4, 117543, Singapore Department of Chemistry, Maria Curie-Sklodowska University, pl. M. Curie-Sklodowskiej 3, 20031 Lublin, Poland

)

‡

ABSTRACT: This work concerns the properties of a nanodevice composed of a carbon nanotube and coreshell magnetic nanoparticles bound to the nanotube tips via alkane chains of various lengths. A model nanocontainer is analyzed using Monte Carlo simulations involving an extended potential function that includes magnetic, dispersion, and screened-electrostatic interactions between the nanoparticles, appropriate nanoparticlealkane chain interactions, and bond bending, stretching, and torsional interactions within the alkane chains. The energy of the system is determined along coordinate linking double-capped, semiuncapped, and double uncapped states. The transitions between these states in the absence of an external magnetic field depend sensitively on the activation barriers, which in turn can be controlled by adjusting the nanoparticle-shell material, its thickness, and the length of the alkane linkers. It is shown that the molecular parameters must be chosen carefully in order that the complete nanodevice possesses the desired properties, i.e., that the system is in a double-capped state under ambient conditions and that the uncapped states can be accessed by exposure to an external magnetic field. Specifically, the alkane linkers must consist of no more than two carbons, and the nanoparticle shell material must have a moderate Hamaker constant (e.g., alumina).

1. INTRODUCTION Drug delivery systems based on the use of nanoparticles offer many advantages over standard drug administration methods. The most pronounced is the ability to target specific locations in the body and, at the same time, reduce the quantity of drug needed to attain a particular concentration in the vicinity of the target. In this way, the concentration of the drug at nontarget sites is minimized, thus limiting severe side effects. This could lead to a reassessment of some pharmaceutically suboptimal but biologically active molecular entities that were previously considered undevelopable.1,2 Though targeted delivery of drugs is still an emerging technology, there are many products already approved for clinical trials. Most of them are the so-called firstgeneration nanotherapeutics (e.g., liposomes and polymers) that implement nontargeted delivery systems.2,3 It is anticipated that nanotechnology will yield a medical breakthrough in the near future due to the development of nextgeneration nanosystems for “smart” targeted drug delivery.2 Such a targeted delivery system should combine two features: the capability of loading and releasing molecular cargos in response to specific stimuli and the ability to attack specific targets due to the presence of recognition moieties.4 Promising candidates facilitating targeted delivery are functionalized carbon nanotubes (CNTs)49 and functionalized magnetic nanoparticles (MNPs) that can be directed or held in place by means of an external r 2011 American Chemical Society

magnetic field (gradient).1,1014 Each of these candidates, as well as many other ones,1518 is being developed extensively, but in fact the combination of CNTs and MNPs in a single multifunctional nanodevice may lead to something greater than the sum of its parts. MNPs have been successfully deposited on the sidewalls1921 and in the interior2224 of CNTs. In addition, hard FePt magnetic nanoparticles25 have been engineered and terminally fused to CNTs using plasma-enhanced chemical vapor deposition.26 The functionalization of CNT sidewalls with FePt MNPs using organic linkers has also been reported.27 There are, however, no reports concerning a selective binding of MNPs at the very tips of CNTs using an organic linker. The attachment of MNPs to the tips of CNTs using organic linkers leads to composite objects that may function as drug delivery systems. Theoretical studies have focused on identifying the properties required of a magnetically functional nanocontainer (NC).2830 Such a system should be able to switch between capped and uncapped molecular structures upon exposure to an external magnetic field. Earlier work has determined how the properties of the NC depend on the sizes of the MNPs, the length and diameter of the CNT, the mutual anchoring and alignment of Received: July 8, 2011 Revised: August 31, 2011 Published: September 01, 2011 19074

dx.doi.org/10.1021/jp2064563 | J. Phys. Chem. C 2011, 115, 19074–19083

The Journal of Physical Chemistry C the MNPs to the tips of the CNT, and the strength of the external magnetic field required to trigger uncapping.2830 Although a lot is already known about the properties of such NCs, there are still some substantial gaps in our knowledge, especially when it comes to specific chemical details such as the appropriate lengths of the alkane linkers, possible alternatives to alkane linkers, the precise composition of the coreshell MNPs, and the various possible ways of functionalizing the sidewalls and tips of the CNT. These are crucial questions as we move toward an experimental realization of a functioning, composite MNPCNT system. This contribution addresses how some of the above factors may affect the function of such systems as magnetically controlled nanocontainers. Using a suitably extended model, we achieve new physical insights into the properties of the system and show how they can be modified by adjusting the lengths of the linkers and the shell material and shell thickness of the MNPs. The results obtained here confirm and augment previous conclusions2830 concerning the application of the NC as a magnetically controlled drug delivery system. That is, drug molecules could be encapsulated in the interior of a CNT capped by the MNPs. Uncapping of the CNT can be achieved through interaction with a magnetic field, and hence drug release can be triggered in a target site at will. Experimentally, the encapsulation of guest molecules within a CNT and the subsequent capping of the CNT have already been implemented.31 In that study, the caps were fullerene molecules, which were noncovalently attached to the CNT, and the guest molecule was hexamethylmelamine. The release of hexamethylmelamine was confirmed using UV spectroscopy after removal of the fullerene caps due to a change of solvent from water/ethanol to CH2Cl2. The experimental realization of a “carbon nanobottle” function strongly supports the theoretical predictions concerning a functional NC. An important advantage of NCs is that the uncapping is triggered by an external physical stimulus that is compatible with physiological conditions, while the nanobottles work by hydrophobic interactions that would be difficult to control and trigger in vivo. The predicted properties of composite MNPCNT systems presented in this work and in previous studies2830 highlight the essence of a magnetically controlled NC. The particular systems being investigated help us to identify the essential prerequisites for a functional device and therefore direct us to a more general specification for actual systems possessing desirable properties. Indeed, the coupling of MNPs to the CNT tips using very short alkane linkers may turn out to present experimental difficulties. Nonetheless, the specific choice of alkane linkers (made for simplicity) does not compromise the robustness of the molecular model, and we may therefore draw many important conclusions concerning the key molecular characteristics of a functional NC. The most important features of the specific molecular models studied here may be realized in a variety of ways. For instance, it may turn out that, chemically, it is more feasible to use alternatives to alkane linkers; maybe polypeptides or DNA strands could be used to confer synthetic selectivity and control. Both CNTs and MNPs (particularly with an oxide shell) can be easily modified to present reactive functional groups on their surfaces.5,6,8,1214,32 In any case, the precise chemical realization of a composite MNPCNT NC offers an enticing prospect for future research. This article is organized as follows. Section 2 (Methods) provides a description of the model NC and the force field used in this study. Section 3 (Results and Discussion) is divided into

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Figure 1. Model nanocontainer analyzed in this study. The MNPs have a coreshell structure with magnetic-core radius c and total radius s being the sum of c and the shell thickness. The linkers are composed of several CH2 groups, each anchored on the surface of an MNP at the point a and on a tip of the CNT at point b. The point b0 denotes a carbon atom that is used for the calculation of the angle between the first segment of the linker and the vector bb0 on the CNT. The linker bond lengths and bond angles are denoted by ri and θi, respectively. The angle defining the orientation of the magnetic moment μ with respect to the radial vector of the anchoring point a is denoted by ϕ.

three subsections concerning the properties of the NC in the absence of an external magnetic field (including the effects of the lengths of the linkers, the shell material, and the shell thickness), its behavior in an external magnetic field, and the time scale for structural transformations in zero field. The paper is concluded in Section 4 with a summary.

2. METHODS The structure and properties of the NC are studied using MC simulations. The model is shown in Figure 1. It differs from previously studied models2830 in two crucial respects: first, the MNPs possess a coreshell architecture which will modify the interactions; and second, the length of the alkane linkers will be altered systematically. An MNP with a magnetic core of radius c is covered by a protective and stabilizing layer of thickness (sc). We assume that the layer is a material like alumina or silica, both of which are commonly used for the electrostatic stabilization of MNPs in aqueous media.12 The core is assumed to be a homogeneously magnetized sphere of ferromagnetic material with a high anisotropy constant, such as magnetite or cobalt. An MNP is bound to each end of the CNT via a (CH2)n linker, where n is to be varied. The anchoring point on the MNP surface, a, is not chemically resolved in the model, but it is assumed that the MNP may rotate freely about the aCH2 bond. The anchoring point b is a carbon atom located on a tip of the CNT; the point b0 is introduced to define the angle b0 bCH2 characterizing the 19075

dx.doi.org/10.1021/jp2064563 |J. Phys. Chem. C 2011, 115, 19074–19083

The Journal of Physical Chemistry C

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Table 1. Lennard-Jones Parameters in the OPLS UnitedAtom Representation33,34 group

ε (kJ mol1)

Table 2. Values of the Parameters in Equations 18 and 20 Used in Calculations

σ (Å)

C

0.4395

3.75

CH CH2

0.4814 0.4988

3.80 3.93

parameter

orientation of the linker with respect to the CNT. The mutual alignment of the points b on each end of the CNT represents an important parameter of the NC, which we define as the angle β with respect to a rotation about the cylinder axis of the CNT (not shown in Figure 1). The linkers (CH2)n are defined by the number of segments n, the bond lengths ri, and the bond angles θi. As shown in Figure 1, each MNP has a permanent magnetic moment μ fixed within its body frame such that it creates an angle ϕ with the radial vector of the point a on the surface of the MNP. It was determined in refs 2830 that the angle ϕ at each end of the CNT plays an important role in magnetically triggered uncapping of NCs. The total energy of the NC is a sum of several contributions. The linkers contribute to the total energy through the bond stretching potential Ustretch ¼

Kr ðri re Þ2 2

ð1Þ

where Kr = 9.65 106 K nm2 and re = 0.154 nm33 and the bond bending potential Ubend ¼

Kθ ðθi θe Þ2 2

ð2Þ

where Kθ = 6.25 104 K rad2.33 The bond angle θe takes on different values: θe = 2 rad for bond angles within the chains, and θe = π for (i) the terminal angle defined by the MNP center, the point a, and the adjacent CH2 group and (ii) the angle defined by the points b0 and b and the adjacent CH2 group. Additionally, when the linker has at least four segments, there is an additional torsional-energy contribution given by Utorsion ¼ V1 ð1 þ cos γÞ þ V2 ð1 cos 2γÞ þ V3 ð1 þ cos 3γÞ

ð3Þ where γ is the dihedral angle, and the parameters have the following values: V1 = 2.95 kJ mol1, V2 = 0.59 kJ mol1, V3 = 5.83 kJ mol1.33 The bonded interactions between carbon atoms within the CNT are not accounted for, and the CNT is treated as a rigid body. This is a good approximation for a CNT of chirality (45,0) and with length 169 Å and diameter 35.2 Å, as used here. The force field uses several types of nonbonded interactions. The magnetic interactions consist of the interaction of each dipole with the applied magnetic field B UB ¼ μ 3 B and the dipoledipole interactions given by 3ðμ1 3 rÞðμ2 3 rÞ μ μ μ Udip ¼ 0 1 33 2 r5 4π r

ð4Þ

ð5Þ

where |μ1| = |μ2|; μ0 is the magnetic permeability of free space; and r is the length of the separation vector r between the MNPs.

value

20 Avac J) SS (10 vac AWW (1020 J) 20 Avac J) CC (10 20 ASS (10 J) ACC (1020 J) 0 ACC (1020 J)

14.0 (alumina)a, 6.5 (silica)a 3.7a 50.0 (metallic cobalt)a 4.26 (alumina)a, 0.634 (silica)a 26.5b 11.1 (alumina)b, 20.4 (silica)b

σ core material (Å)

2.22 (atomic diameter of cobalt)

σ shell material (Å)

3.48 (alumina)c, 3.36 (silica)c

c (Å)

45

s (Å)

50, 55

a

Determined from the Lifshitz theory.43 b Determined using the mixing rules. c Determined from the density of the materials.

Carbon atoms and terminal CH groups belonging to the CNT interact also with the CH2 linkers according to the standard Lennard-Jones potential provided that they are separated by more than three successive bonds. The relevant parameters are calculated using the conventional LorentzBerthelot mixing rules and the input parameters for unmixed interactions in the OPLS united-atom representation33,34 collected in Table 1. Because we consider the MNPs as having coreshell structures, their interactions with other components of the NC require a detailed description. We assume that the shell is composed either of silica or alumina. Thus, when the NC is in an electrolyte solution the MNPs develop a negative charge according to the mechanism of double-layer development. This process protects an ensemble of NCs against agglomeration (favored by the magnetic and dispersion forces) by inducing an electrostatic repulsion between the MNPs. Without such a stabilizing factor, the MNPs might form chainlike clusters, rings, or 3D networks depending on the concentration.35,36 Therefore, it is necessary to account for the effects of the electrostatic interactions in the force field and the calculations. 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.3741 It involves only the macroions, whose Coulomb repulsion is screened exponentially by the surrounding counterions and salt ions. The electrostatic interaction between two charged MNPs is described by the screened Coulomb interaction given by the Yukawa term37 UYuk ðrÞ ¼

ðZe0 Þ2 expð2ksÞ ekr 4πεε0 ð1 þ ksÞ2 r

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 k involves contributions from the counterions and the salt ions. The salt contribution is measured by the ionic strength I (in mol L1).37 Assuming charge neutrality between macroions and counterions, the inverse Debye length can be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1000e20 ðZF þ 2INA Þ k¼ ð7Þ ε0 εkB T 19076



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where F is the macroion concentration (in mol L1) and NA is Avogadro’s number. Because the NC is intended to work in a biological environment, we will consider only the physiological ionic strength, I = 0.145 mol L1,42 and the concentration F of about 104 mol L1. Because of the relatively high ionic strength, the Debye length k1 is less than 10 Å for Z of the order of 102, and so the electrostatic repulsion between two MNPs associated with the same NC is normally negligible. It will be very important in an ensemble of NCs at high concentrations. A crucial component of the force field is the dispersion interactions between the coreshell MNPs across an aqueous electrolyte solution. According to the Hamaker theory, the interaction energy between two spherical particles of radii a1 and a2 immersed in a continuum solvent at a centercenter distance r can be written in the form Uða1 , a2 , rÞ ¼ A12 f ða1 , a2 , rÞ

" 1 σ 6 r2 7rða1 þ a2 Þ þ 6ða21 þ 7a1 a2 þ a22 Þ 37800 r ðr a1 a2 Þ7

þ

r 2 þ 7rða1 þ a2 Þ þ 6ða21 þ 7a1 a2 þ a22 Þ ðr þ a1 þ a2 Þ7

r þ 7rða1 a2 Þ þ 7a1 a2 þ ðr þ a1 a2 Þ7 2

U ðiiÞ ðrÞ ¼ ASC ½f ðs1 , c2 , rÞ f ðc1 , c2 , rÞ

6ða21

U ðiiiÞ ðrÞ ¼ ASC ½f ðc1 , s2 , rÞ f ðc1 , c2 , rÞ U ðivÞ ðrÞ ¼ ACC f ðc1 , c2 , rÞ

ð15Þ

Summing these four contributions gives the net interaction between the two coreshell particles UðrÞ ¼ ASS f ðs1 , s2 , rÞ þ ðASC ASS Þ½f ðs1 , c2 , rÞ þ f ðc1 , s2 , rÞ þ ðASS 2ASC þ ACC Þf ðc1 , c2 , rÞ

ð16Þ

This is easily shown to have all the right limits: if all the Hamaker constants are equal to A, then U(r) = Af(s1,s2,r); if ASS = ASC = 0 (shells index-matched with the solvent), then U(r) = ACCf(c1,c2, r); and if ASC = ACC = 0 (cores index-matched with the solvent), then U(r) = U(i)(r). The interaction between a colloidal particle and a single atom can be obtained by analogy with eq 8 by using an appropriate Hamaker constant and the function " # 2a3 σ 3 ð5a6 þ 45a4 r 2 þ 63a2 r 4 þ 15r 6 Þσ 6 f ða, rÞ ¼ 1 9ða2 r 2 Þ3 15ða rÞ6 ða þ rÞ6

ð9Þ

and

" 1 2a1 a2 2a1 a2 þ f ða1 , a2 , rÞ ¼ 2 6 r 2 ða1 þ a2 Þ2 r ða1 a2 Þ2 !# r 2 ða1 þ a2 Þ2 þ ln ð10Þ r 2 ða1 a2 Þ2

where σ is the Lennard-Jones diameter of the atoms in the colloidal particle. Now consider the total interaction between particles with cores of radii c1 and c2 made of material C and shells of thicknesses (s1c1) and (s2c2) made of material S, which is the sum of four terms: (i) the shellshell interaction; (ii) the interaction between the shell of particle 1 and the core of particle 2; (iii) the interaction between the core of particle 1 and the shell of particle 2; and (iv) the corecore interaction. These are evaluated in turn. (i) The total interaction between two particles each made entirely of material S is ASS f ðs1 , s2 , rÞ ¼ U ðiÞ þ ASS f½f ðs1 , c2 , rÞ f ðc1 , c2 , rÞ þ ½f ðc1 , s2 , rÞ f ðc1 , c2 , rÞ þ f ðc1 , c2 , rÞg

ð17Þ where a is the radius of the colloidal particle; r is the centercenter separation distance; and σ is the Lennard-Jones diameter of atoms creating the colloidal particle. The total interaction energy between a coreshell particle and a single atom across the continuum solvent takes the following form UðrÞ ¼

where ASS is the Hamaker constant for interactions of shell material across water; the terms in square brackets represent the coreshell interactions; and U(i) is the shellshell interaction being sought. This is given by ð12Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 144εASS f ðs, rÞ 144εASS f ðc, rÞ þ 144εACC f ðc, rÞ

ð18Þ where ε is the Lennard-Jones energy parameter associated with the single atom and 144ε is the formal Hamaker constant for a single atom. Calculations were performed for two types of shell material, alumina and silica. The core material is assumed to be metallic cobalt. Table 2 collects the appropriate values of the nonretarded Hamaker constants calculated from the Lifshitz theory43 and used in the calculations. The parameters Avac SS (shell material), vac Avac CC (core material), and AWW (water) are for two identical media interacting across vacuum; they are required for the calculation of the Hamaker constants in aqueous solution using the mixing rules pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi vac vac vac vac ffi 2 A11 ¼ ð Avac 11 AWW Þ ¼ A11 2A1W þ AWW

ð11Þ

U ðiÞ ðrÞ ¼ ASS ½f ðs1 , s2 , rÞ f ðs1 , c2 , rÞ f ðc1 , s2 , rÞ þ f ðc1 , c2 , rÞ

ð14Þ

(iv) The corecore interaction is

a22 Þ

r 2 7rða1 a2 Þ þ 6ða21 7a1 a2 þ a22 Þ ðr a1 þ a2 Þ7

ð13Þ

(iii) The interaction between the core of particle 1 and the shell of particle 2 is

ð8Þ

where A12 is the Hamaker constant for materials 1 and 2 in the solvent and f (a1,a2,r) is a function only of the radii and the separation consisting of repulsive (f+) and attractive (f) terms. These latter functions are given by fþ ða1 , a2 , rÞ ¼

(ii) The interaction between the shell of particle 1 and the core of particle 2 is

A12 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi A11 A22

ð19Þ

Using these mixing rules, we can express the prefactors in eq 16 by combinations of Hamaker constants involving only interactions between two identical media interacting across another medium. This is more convenient because these types of Hamaker constants can often be calculated directly from Lifshitz theory43 and are therefore more accurate than those 19077



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determined from the mixing rules. To this end, we notice that (ASS 2ASC + ACC) = A0 CC (the Hamaker constant for interactions between cores across shell material) and that (ASC ASS) = 1/2(ACC A0 CC ASS). Using these substitutions, we finally get UðrÞ ¼ ASS f ðs1 , s2 , rÞ þ

1 0 ðACC ACC ASS Þ½f ðs1 , c2 , rÞ 2 0

þ f ðc1 , s2 , rÞ þ ACC f ðc1 , c2 , rÞ

ð20Þ

All energy contributions were computed as full pairwise additive sums without cut-offs or periodic boundary conditions since we are considering an isolated system of one CNT and two MNPs throughout. Finally, the magnetic moment of an MNP is a function of its magnetic core diameter and the saturation magnetization Ms of the constituent material μ¼

4π Ms c 3 3

ð21Þ

The saturation magnetization of cobalt is 1070 kA m1.44 The magnetic moment of an MNP with typical core radius c = 45 Å is therefore μ = 44 040 Bohr magnetons or 4.08 1019 A m2. Equilibrium sampling of the structure of the NC was achieved using the standard Metropolis MC scheme. A given degree of freedom was chosen at random, and a small trial displacement from its original value was performed. The associated change in total energy (ΔUt) was computed including contributions given above and the move accepted with probability min(1,exp(ΔUt/kBT)). Typically, a single run consisted of 107108 MC steps, where one step consists of one attempted displacement of a randomly selected degree of freedom. The most important factor controlling the properties of the NC in the absence of the external magnetic field is the total energy profile connecting the capped and the uncapped states. We can distinguish two types of transitions by analyzing the states of the left-hand side (L) and right-hand side (R) of the carbon nanotube: cappedcapped to cappeduncapped (denoted as a cc f cu transition) and cappeduncapped to uncappeduncapped (denoted as a cu f uu transition). Each transition is accompanied by an energy barrier E arising from deformations of the linkers and changes in the nonbonded interactions. The Ut profiles, or more precisely the minimum total energy paths for the transitions, have been determined using umbrella sampling and without the external magnetic field. This method requires the definition of appropriate order parameters. The most obvious choice is a pair of parameters that define the degree of capping at each end of the NC. To this end, we introduce sL (sR) as the maximum radial distance between the surface of the MNP on the left-hand side (right-hand side) and the surface of any one CH group located on the terminal ring of the left-hand (right-hand) CNT; this is given by the center center distance between the MNP and the relevant CH group minus (2s + σCH)/2, where σCH = 3.8 Å is the Lennard-Jones range parameter for the CH group within the united-atom representation.34 To perform umbrella sampling, the Hamiltonian of the system was modified by adding the term 1/2δ[(sL s0L)2 + (sR s0R)2] where (s0L,s0R) are the target values of (sL,sR). The parameter δ describes the strength of bias directing the system toward the target configuration; it took various values from 5 to 100 kJ Å2. The minimum energy path (MEP) for the transition from one

Figure 2. Minimum total energy paths (MEPs) for the transitions between fully capped (cc), semi uncapped (cu), and fully uncapped (uu) states of the NC. The profiles were determined for various lengths of the linkers and for two shell materials, alumina (top) and silica (bottom). In each case, the arrangement of the anchoring points b corresponds to the trans configuration (β = 180°) and the alignments of the magnetic moments (ϕL,ϕR) = (0,0). Calculations were performed for the temperature T = 300 K and for the shell thickness (s c) = 5 Å.

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).

3. RESULTS AND DISCUSSION a. Systems Properties without an External Magnetic Field. Figure 2 shows MEP profiles for various alkane-chain lengths, two shell materials (alumina and silica), at temperature T = 300 K, and a shell thickness (s c) = 5 Å. The MEPs have been determined for many sets of NC parameters, particularly for various combinations of the magnetic-moment alignments (ϕL,ϕR) and the arrangements of the CNT anchoring points (b) measured by the angle β. These parameters do not significantly affect the picture shown in Figure 2 because the CNT length is greater than the sum of the MNPs radii.30 Therefore, representative results for β = 180° (the trans configuration) and (ϕL,ϕR) = (0,0) are shown. Figure 2 clearly shows that a rapid change in the MEP profiles occurs upon enhancing the linker length. The activation energies with a single CH2 linker are significantly greater than those with two or more CH2 linkers, which will have a strong effect on the respective transition rates between various states of the NC. Moreover, the relations between the local and global energy minima switch from Ut(cc) e Ut(uc) e Ut(uu) for n = 1 to Ut(cc) g Ut(uc) g Ut(uu) for n = 2, 3, 4. Another interesting effect is the change in shape of the MEPs. With single CH2 segments, the ccfcu and cufuu energy barriers are “early”, occurring in the region of sL,R = 10 Å, nearer the cc and cu states, respectively. For linkers composed of two CH2 segments, the 19078



Figure 3. Activation barriers for the cc f cu and cu f cc transitions determined from the MEPs (Figure 2) for the two shell materials alumina (Al) and silica (Si), as functions of the linker length (from one to four CH2 segments). The top and bottom figures show the results for shell thickness (s c) = 5 Å and 10 Å, respectively. In each case, the arrangement of the anchoring points b on the CNT tips corresponds to the trans configuration (β = 180°) and the alignments of the magnetic moments (ϕL, ϕR) = (0,0); other combinations of these parameters give identical results within the estimated error from (0.23 to (2.62 kJ mol1.

cc f cu and cu f uu energy barriers are “late”, with the energy maxima occurring in the vicinity of the cu and uu states, respectively. For longer linkers the MEPs are almost flat with hardly any pronounced energy barriers separating the energy minima. Figure 2 also shows that the shell material is a critical parameter affecting the properties of the NC. In the case of silica shells, the range of dispersion interactions is much smaller than that for alumina. This is due to the small value Hamaker constant of silica in water (ASS). In this case, even with the shortest linkers, the differences in total energy between the cc, cu, and uu states are small. A more detailed picture of the system properties emerges from Figures 3 and 4. Figure 3 shows how the linker lengths, shell material, and shell thickness affect the activation barriers E for the transitions between states. The barriers were determined from the MEPs as differences between adjacent maxima and minima in the total energy. Thus, we can distinguish four activation barriers, namely, E(cc f cu), E(cu f cc), E(cu f uu), and E(uu f cu). Figure 3 shows only E(cc f cu) and E(cu f cc), i.e., the barriers associated with the first stage of uncapping. The barriers associated with the second stage of uncapping, E(cu f uu) and

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Figure 4. Mean total energy of the NC in the cc, cu, and uu states as a function of the linker lengths, the two shell materials alumina (Al) and silica (Si), and two shell thicknesses 5 Å (top) and 10 Å (bottom). In each case, the arrangement of the anchoring points b on the CNT tips corresponds to the trans configuration (β = 180°) and the alignments of the magnetic moments (ϕL, ϕR) = (0,0); other combinations of these parameters give identical results within the estimated error denoted by the error bars. Calculations were performed for the temperature T = 300 K.

E(uu f cu), have almost identical values to E(cc f cu) and E(cu f cc), respectively, and are therefore omitted. Figure 4 shows the mean total energies of the system, , in the cc, cu, and uu states for various linker lengths, shell material, and shell thickness. The mean energies were determined from 10 to 20 independent, standard unbiased MC runs, each consisting of 5 105 MC steps. An analysis of the data collected in Figures 24 allows us to draw many important conclusions concerning the role of the linker length, shell material, and shell thickness on the function of the system as a magnetically controlled NC. (i). Linker Length. A functional NC must exhibit significant differences in the mean energies between the cc, cu, and uu states. In particular, the cc state should be the lowest-energy state so that the fully capped configuration is the most stable under ambient conditions. The activation barriers E(cc f cu) and E(cu f uu) should be high enough to inhibit spontaneous uncapping of the NC. The activation barriers E(cu f cc) and E(uu f cu) should be as low as possible so that the NC can return to the capped state in a reasonable time scale. Figures 24 show that only a few combinations of system parameters satisfy these general conditions; these combinations are highlighted by the shaded areas in Figures 3 and 4. Lengthening the linker chains does not improve 19079


The Journal of Physical Chemistry C the situation: with alumina shells, we observe worsening relationships between the mean energies and the activation barriers as the number of CH2 segments is increased; with silica shells, acceptable relationships occur only with linkers composed of two segments, but the differences in energies are small. Similarly, for alumina shells and two CH2 segments, the energy differences are much smaller than those with a single CH2 linker. Clearly, then, only the shortest linkers in combination with alumina shells give the relationships between the energetic parameters that are required of a functional NC, as determined in earlier work.29,30 That is, the mean energy is lower in the cc state than in the cu state by 56 and 45 kJ mol1, and the barrier for the cc f cu transition is as high as 140 and 130 kJ mol1, with shell thicknesses of 5 and 10 Å, respectively. The reverse barrier of 86 kJ mol1 allows for a spontaneous cu f cc transition in a reasonable time scale.29 The effects of increasing linker length arise from the accompanying enhancement in flexibility and the fact that in the cu and uu states the MNPs can more easily approach the sidewalls of the CNT. The MNP in an uncapped state can then interact with more CNT-sidewall carbon atoms in the sidewall than it can in the capped state with CNT-terminal ring atoms. Therefore, in the present NC design, longer linkers reduce the energy differences between capped and uncapped states and hence lead to deterioration in functionality. Nonetheless, longer linkers have one feature that might be helpful in some cases, namely, that they reduce the activation barriers for the transitions. This might be important for the design of NCs with faster relaxation to the cc state from the cu and uu states. However, such a design must be accompanied by some chemical modification of the CNT to retain absolute energetic stability of the cc state with respect to the uncapped states. This may be done in several ways, e.g., by introducing multiwalled carbon nanotubes, by functionalizing the terminal-ring carbon atoms to give favorable interactions between the MNPs and the tips of the CNTs, or by introducing steric “obstacles” to frustrate the dispersion interactions with CNT sidewalls in uncapped states. (ii). Shell Material. Bare metallic nanoparticles are strongly unstable in air and in aqueous media due to oxidation and irreversible agglomeration. Therefore, a protective and stabilizing layer is normally required to stabilize the particles in a colloidal suspension.12 Among the various possible approaches, the application of oxides as shell materials is particularly interesting because of the opportunity for electrostatic stabilization of the nanoparticles in aqueous media. Figures 24 show how an application of two selected oxides alters the energetic properties of the NC. The oxide material affects the balance of intramolecular interactions through modification of the dispersion interactions between the MNPs and the other components of the NC. The Hamaker constants ASS (Table 2) for alumina and silica differ significantly, and this leads to qualitative changes in the properties of the system and its ability to function as a NC. Using silica as a shell material gives rather weak energetic stabilization of the cc, cu, and uu states, and the mean total energies are normally lower in the uncapped states. Thus, a NC with silica shells will tend to remain in the uncapped states under normal conditions, and its exposure to an external magnetic field will not produce any interesting phenomena. Only for chains composed of two CH2 segments is mean total energy lower in the cc state than in the cu or uu states, but the difference between these energies is small. In view of the almost flat MEPs, we can expect spontaneous

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and fast transitions between these states in dynamic equilibrium. We can therefore conclude that silica is probably not a good shell material; in an aqueous suspension, its relatively strong interactions with water overwhelm the dispersion attractions with other components of the NC. Thus, without additional chemical modification of the CNT tips (creating hydrogen bonds, for instance) the system might not work properly as a magnetically controlled NC. Alumina has a Hamaker constant in water (ASS = 4.26 1020 J) about seven times greater than that of silica, and as shown in Figures 24, this is enough to change the properties of the NC qualitatively, at least for the shortest linkers. The relationships between all the energies with a single CH2 segment are very similar to those previously found in refs 29 and 30, and the NC will work properly without additional modifications. As already discussed, longer linkers lead to less optimal energetics, and restoring the function of the NC would involve additional chemical modification of the CNT. We conclude that a useful shell material should possess a Hamaker constant in water (ASS) greater than about 4 1020 J (or about 10kBT). This is sufficient to give the required energetic separation of the cc, cu, and uu states. Very high Hamaker constants are, however, not preferred because they would produce very strong binding in the cc state that would, in turn, require huge magnetic field strengths to trigger uncapping. Of course, not only alumina in water exhibits a value of ASS > 4 1020 J; many other inorganic materials satisfy this condition (e.g., zirconia and rutile43,45), and depending on the feasibility of the synthesis of such coreshell nanoparticles they might be good choices. The use of precious metals or carbonaceous materials as shells is questionable, though, because they exhibit very high values of ASS (3040 1020 J for metals43 and 23 1020 J for graphite46), and also electrostatic stabilization of the NCs in aqueous media would be difficult. (iii). Shell Thickness. Figures 24 show that the shell should be as thin as possible. This is due to two factors. The first is that the thicker the shell the weaker the net dispersion interactions between MNPs and the rest of the NC. This arises because the core material (cobalt) has a high Hamaker constant and hence makes a significant contribution to the total energy; the thicker the shell, the greater the separation between the core and other components of the NC, and hence there is less energetic stabilization. For thin shells of 5 Å, the contribution from cores interacting with other components of the NC across the shells and the aqueous medium is significant. Increasing the shell thickness to 10 Å significantly reduces all of the energies due to greater isolation of the cores from the rest of the system. However, this effect quickly saturates because thicker shells totally screen the contribution from the cores. Test calculations revealed that increasing the shell thickness above 10 Å does not significantly alter the results shown for 10 Å. Another factor favoring the use of thin shells is the volume of the magnetic core. The overall diameter of an MNP must be comparable to the CNT diameter. In addition, the volume of the magnetic core should be as large as possible because this dictates the dipole moment and hence the range of external magnetic fields required to trigger uncapping. Therefore, the volume occupied by the shell of an MNP should be as small as possible. b. Interaction with an External Magnetic Field. Among the systems studied here, only six fulfill the criteria necessary for function as a NC. That is, in the absence of an external magnetic field, the cc state is energetically most favorable, and there are 19080



Figure 5. Probability p of observing states with sL,R > 5 Å as a function of the magnetic field strength B. Results are shown for the systems Al1, Al2, and Si2 with shell thicknesses of 5 Å. All other parameters are the same as in previous figures.

distinct activation barriers between the cc, cu, and uu states. These are the systems highlighted by the shaded areas in Figures 3 and 4 and labeled hereafter by Al1, Al2, and Si2 for each shell thickness; Aln (Sin) means a NC with alumina (silica) shells and n CH2 segments in each linker. We now consider the important issue of how these NCs respond to an external magnetic field. In previous studies of magnetically triggered uncapping, we have used so-called “uncapping maps”.29,30 These maps indicate which combinations of magnetic-moment alignment (angles ϕL and ϕR) and arrangement of anchoring points b (angle β) produce NCs that undergo magnetically triggered uncapping at some field strength. From these previous studies, we determine that initially capped states of the Al1, Al2, and Si2 will uncap on exposure to an external magnetic field. In this work, we only consider NCs in trans (β = 180°) configurations that, in the cc state, adopt antiparallel alignments of the dipole moments. An external magnetic field of sufficient strength favors a parallel alignment of the dipole moments, and this can only happen if the NC is uncapped. An interesting and important task, therefore, is to determine the threshold external magnetic field leading to uncapping. To this end, we have calculated the probability p that uncapped states are observed in simulations started from the cc state, under the influence of an external magnetic field of strength B. More precisely, for a given value of B, we perform a simulation of 3 108 MC steps and determine the fraction of configurations p with at least one of the order parameters sL,R > 5 Å; recall from Section 2 that this is the maximum radial distance between the MNP surface and the CNT terminal ring. Note that this threshold value of the order parameter does not necessarily mean that the NC reaches the cu state; any transient configuration with sL,R > 5 Å is effectively open to typical, small guest molecules. Because the initial orientation of the NC with respect to the field direction affects the probability of uncapping in a complex manner,30 the NC is given a new random orientation every 105 MC steps. In this way, the measured probability p represents an isotropic average over the NC orientation, as would occur experimentally with a pulsed application of the magnetic field.30 Figure 5 shows how the fraction of “uncapped” states changes with the field strength in systems Al1, Al2, and Si2 with shell thicknesses 5 Å. As expected, system Si2 undergoes uncapping in

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very small external fields because the barrier E(cc f cu) is only 48 kJ mol1. Systems Al2 and Al1 require stronger fields as the barriers are 93 and 140 kJ mol1, respectively. However, the threshold fields are relatively small compared to those in previous studies of bare magnetic nanoparticles, where fields of the order 312 T were necessary to trigger the uncapping process.29,30 On the other hand, Figure 5 represents the most responsive construction of the NC in terms of the alignments of the magnetic moments (ϕL,ϕR) and configuration of the anchoring points β. As already stated,30 both (ϕL,ϕR) and β affect the sensitivity of the NC to the magnetic field, and the threshold fields will move toward stronger fields for less sensitive configurations of the NC. An important conclusion arising from Figure 5 is that the capped and uncapped states are strictly separated by a threshold field. The probability of observing uncapped configurations rapidly jumps from almost zero to unity as the magnetic field reaches the threshold region. In this region, the error bars become large because of large configurational fluctuations. Away from the threshold region, we observe pure capped or uncapped states with insignificant error bars in the probability. Thus, the cc states at zero field, and the cu and uu states above the threshold field, are highly stable. All of the systems shown in Figure 5 should work perfectly well as magnetically controlled NCs, but among them the Si2 system is particularly interesting because it undergoes uncapping in weak magnetic fields. However, the cc state is not perfectly stable in zero field; with B = 0, p = 5.5 103, which means that the NCs spontaneously attain configurations with sL,R > 5 Å and thus guest molecules cannot be perfectly locked in the NC interior. On the other hand, these configurations do not correspond to cu states; they are simply triggered by thermal fluctuations of the MNP positions around the cc states. Full transitions to the cu or uu states were never observed in these simulations; in practice, the Si2 system could be still a good NC for larger guest molecules. Systems Al1 and Al2 give precisely p = 0 in zero field within the simulation period. Thicker shells (10 Å) reduce the activation barriers for transitions and/or the mean total energies as shown in Figures 3 and 4. Therefore, the threshold magnetic field strengths for uncapping will be shifted toward weaker fields, and the probability of uncapping at zero field due to thermal fluctuations will be enhanced for the Si2 system. For systems Al1 and Al2, the thermal fluctuations will not play an important role as the activation barriers to uncapping are still high. c. Time Scale for the Transitions. The microscopic simulations utilized so far cannot definitively answer whether these systems are useful or not because a significant factor controlling their behavior is the dynamics of the transitions and the associated activation barriers. It turns out that these barriers are high, and any transition will therefore be classified as a rare event; thus, the transitions cannot be directly probed by standard computer simulation techniques. However, the conclusions arising from the MEPs and the heights of the energy barriers, here and in the earlier works, are obviously valid and insightful. A valuable indicator of the system’s performance is the mean residence time τ in a given state, which is computed using the well-known expression τ1 ¼ fc expðE=kB TÞ

ð22Þ

where fc is the attempt frequency of escaping from a given state and E is the energy barrier separating that state from another one. 19081



Figure 6. Mean residence times determined from eq 22 for systems Al1, Al2, and Si2, assuming fc = 1010 s1 and temperature T = 300 K.

fc cannot be determined from MC simulations, but it can be roughly estimated by the frequency of conformational transitions in n-alkanes as the capping/uncapping processes proceed through similar sequences of events. In particular, for n-butane, fc takes values in the range 10101011 s1.47 Thus, by assuming that the attempt frequency for capping/uncapping under zero magnetic field is fc = 1010 s1 and using the activation barriers from Figure 3, we find that the mean residence times span a very wide time range, as shown in Figure 6. The estimates in Figure 6 indicate that system Si2 must be rejected as a candidate for a functional NC. The residence time in the cc state is well below 1 s, and thus a guest molecule cannot be contained for long times. However, the release time for guest molecules might be longer than 1 s because the desorption process might be activated as well, and in such circumstances an NC modeled on system Si2 type might work properly. System Al2 (with both shell thicknesses) exhibits much longer mean residence times in the cc state, around 18 days and 3.6 days with shell thicknesses of 5 and 10 Å, respectively. Both seem to be sufficient for strong containment of the guest molecules prior to release at the target site. Once uncapped, system Al2 stays in the cu state for 5.2 or 7.9 h, depending on the shell thickness. Again, these residence times seem to be sufficient to allow the successful release of guest molecules. Finally, system Al1 exhibits huge residence times in the cc state, meaning that spontaneous uncapping will never be observed on realistic time scales. However, the reverse processes are very slow as well, around 26 h. This means that after a short exposure to an external magnetic field, which may lead to flash uncapping as discussed in Section 3b and ref 30, the system Al1 would stay in the cu or uu states for more than one day, on average, prior to spontaneous transitions to the energetically favorable cc state. This feature would be useful in practical applications for drug release in living organisms since it can be triggered by short exposure to an external magnetic field.

4. SUMMARY Analysis of an extended model of a magnetically triggered NC gives important new physical insights into its potential properties and allows for their better control. Crucially, some chemical details have been considered that take us closer to a laboratory realization of a functional nanodevice, comprising MNPs and a CNT. Specifically, we have considered a realistic representation

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of the coreshell architecture of MNPs and variations in the length of the alkane chains linking the MNPs to the CNT. By deriving the appropriate expressions for the dispersion interactions and incorporating them into a molecular model of the system, we were able to draw several important conclusions about the potential properties of a functional NC. We found that the shell material is of crucial importance because, for the same core material, changing the shell material can enhance or reduce the strength of the dispersion interactions. We studied two particular materials (alumina and silica) that are commonly used as protective shells for ferromagnetic, metallic particles. Though apparently similar, these two materials have quite different Hamaker constants, and this significantly alters the properties of the NC. Using silica leads to a strong reduction in the dispersion interactions between the MNPs and the other components of the NC, which reduces the energetic stabilization of the cc state, and a reduction in the activation barriers separating the cc and cu states. As a result, the NC has no unique stable state, and its function as a NC is questionable. On the other hand, alumina shells ensure tight binding of the MNPs in the cc, cu, and uu states due to its smaller affinity for aqueous media. At the same time, the activation barriers are high enough to ensure sufficient kinetic stabilization of all available states of the NC. The flexibility of the linkers connecting the MNPs to the CNT turn out to be crucial for the function of the system as a NC. Short (less flexible) linkers lead to a good balance between the energy in the cc and cu states, while the cc state is the most energetically favorable (as required). An exception is the system comprising a single CH2 group and silica coatings for the MNPs. All systems with linkers of three or more CH2 segments exhibit unfavorable relations between the energies of the cc and cu states; specifically, the uncapped states are energetically preferred under zero or nonzero magnetic fields. Obviously, such systems cannot function as magnetically controlled NCs. The analysis of the time scale for transitions between the capped and uncapped states led to some very interesting conclusions: the range of time scales is extremely wide, from 103104 s with silica shells up to 1014 s with alumina shells and single CH2 linkers. It generally seems that NCs composed of silica-covered MNPs are not strongly stabilized in any one state. Even in the energetically favorable cc state, the thermal fluctuations can easily detach the MNPs from the CNT. The aluminacovered MNPs stick to the CNT tips more firmly, and the time necessary for spontaneous uncapping due to thermal fluctuations can be extremely long. That time can be tuned to some extent by adjusting the shell thickness; generally the thicker the shell, the weaker the binding of the MNPs, and thus the residence times in the cc and cu states are shorter. All of these new findings can be used to inform and direct the design and synthesis of magnetically controlled NCs. The computational studies reported here have been used to screen some of the most important molecular parameters, therefore bypassing expensive trial-and-error approaches of the system parameters without involving time-consuming and expensive trial-and-error approaches in the laboratory.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. 19082



’ ACKNOWLEDGMENT This work was supported by the Polish National Science Centre (NCN) Grant N N204 205240. ’ REFERENCES (1) Arruebo, M.; Fernandez-Pacheco, R.; Ibarra, M. R.; Santamaria, J. Nanotoday 2007, 2, 22. (2) Shi, J.; Votruba, A. R.; Farokhzad, O. C.; Langer, R. Nano Lett. 2010, 10, 3223. (3) Wagner, V.; Dullaart, A.; Bock, A. K.; Zweck, A. Nat. Biotechnol. 2006, 24, 1211. (4) Li, R.; Wu, R.; Zhao, L.; Hu, Z.; Guo, S.; Pan, X. Carbon 2011, 49, 1797. (5) Pastorin, G.; Wu, W.; Wieckowski, S.; Briand, J.-P.; Kostarelos, K.; Prato, M.; Bianco, A. Chem. Commun. 2006, 1182. (6) Pastorin, G. Pharm. Res. 2008, 26, 746. (7) Zhang, X.; Meng, L.; Lu, Q.; Fei, Z.; Dyson, P. J. Biomaterials 2009, 30, 6041. (8) Prato, M.; Kostarelos, K.; Bianco, A. Acc. Chem. Res. 2008, 41, 60. (9) Feazell, R. P.; Nakayama-Ratchford, N.; Dai, H.; Lippard, S. J. J. Am. Chem. Soc. 2007, 129, 8438. (10) Yang, X.; Zhang, X.; Ma, Y.; Huang, Y.; Wang, Y.; Chen, Y. J. Mater. Chem. 2009, 19, 2710. (11) Son, S. J.; Reichel, J.; He, B.; Schuchman, M.; Lee, S. B. J. Am. Chem. Soc. 2005, 127, 7316. (12) Lu, A.-H.; Salabas, E. L.; Schuth, F. Angew. Chem., Int. Ed. 2007, 46, 1222. (13) Claesson, E. M.; Mehendale, N. C.; Klein Gebbink, R. J. M.; van Koten, G.; Philipse, A. P. J. Magn. Magn. Mater. 2007, 311, 41. (14) Claesson, E. M.; Erne, B. H.; Bakelaar, I. A.; Kuipers, B. M. W.; Philipse, A. P. J. Phys.: Condens. Mater. 2007, 19, 036105. (15) Song, Y.-Y.; Schmidt-Stein, F.; Bauer, S.; Schmuki, P. J. Am. Chem. Soc. 2009, 131, 4230. (16) Yu, J.; Bai, X.; Suh, J.; Lee, S. B.; Son, S. J. J. Am. Chem. Soc. 2009, 131, 15574. (17) Zhu, M.; Diao, G. Nanoscale 2011, DOI: 10.1039/c1nr10165j. (18) Yuan, L.; Tang, Q. Q.; Yang, D.; Zhang, J. Z.; Zhang, F. Y.; Hu, J. H. J. Phys. Chem. C 2011, 115, 9926. (19) Georgakilas, V.; Tzitzios, V.; Gournis, D.; Petridis, D. Chem. Mater. 2005, 17, 1613. (20) Correa-Duarte, M. A.; Grzelczak, M.; Salgueirino-Maceira, V.; Giersig, M.; Liz-Marzan, L. L.; Farle, M.; Sierazdki, K.; Diaz, R. J. Phys. Chem. B 2005, 109, 19060. (21) Zhang, H.; Du, N.; Wu, P.; Chen, B.; Yang, D. Nanotechnology 2005, 19, 315604. (22) Korneva, G.; Ye, H.; Gogotsi, Y.; Halverson, D.; Friedman, G.; Bradley, J.-C.; Kornev, K. G. Nano Lett. 2005, 5, 879. (23) Kornev, K. G.; Halverson, D.; Korneva, G.; Gogotsi, Y.; Friedman, G. Appl. Phys. Lett. 2008, 92, 233117. (24) Elias, A. L.; Rodriguez-Manzo, J. A.; McCartney, M. R.; Golberg, D.; Zamudio, A.; Baltazar, S. E.; Lopez-Urias, F.; Munoz-Sandoval, E.; Gu, L.; Tang, C. C.; Smith, D. J.; Bando, Y.; Terrones, H.; Terrones, M. Nano Lett. 2005, 5, 467. (25) Sun, S.; Fullerton, E. E.; Weller, D.; Murray, C. B. IEEE Trans. Magn. 2001, 37, 1239. (26) Schaffel, F.; Taschner, C.; Rummeli, M. H.; Neu, V.; Wolff, U.; Queitsch, U.; Pohl, D.; Kaltofen, R.; Leonhardt, A.; Rellinghaus, B.; Buchner, B.; Schultz, L. Appl. Phys. Lett. 2009, 94, 193107. (27) Tsoufis, T.; Tomou, A.; Gournis, D.; Douvalis, P.; Panagiotopoulos, I.; Kooi, B.; Georgakalis, V.; Arfaoui, I.; Bakas, T. J. Nanosci. Nanotechnol. 2008, 8, 5942. (28) Panczyk, T.; Warzocha, T. P. J. Phys. Chem. C 2009, 113, 19155. (29) Panczyk, T.; Warzocha, T. P.; Camp, P. J. J. Phys. Chem. C 2010, 114, 21299. (30) Panczyk, T.; Warzocha, T. P.; Camp, P. J. J. Phys. Chem. C 2011, 115, 7928.

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Computational Study of Some Aspects of Chemical Optimization of a

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