Polymers encapsulated in short single wall carbon nanotubes: pseudo-1D morphologies and induced chirality.

Polymers encapsulated in short single wall carbon nanotubes: Pseudo-1D morphologies and induced chirality Sunil Kumar, Sudip K. Pattanayek, and Gerald G. Pereira Citation: The Journal of Chemical Physics 142, 114901 (2015); doi: 10.1063/1.4914463 View online: http://dx.doi.org/10.1063/1.4914463 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nanocatalyst structure as a template to define chirality of nascent single-walled carbon nanotubes J. Chem. Phys. 134, 014705 (2011); 10.1063/1.3509387 Mechanical buckling of single-walled carbon nanotubes: Atomistic simulations J. Appl. Phys. 106, 114313 (2009); 10.1063/1.3260239 Effects of tube diameter and chirality on the stability of single-walled carbon nanotubes under ion irradiation J. Appl. Phys. 106, 043501 (2009); 10.1063/1.3194784 Influence of chirality on the interfacial bonding characteristics of carbon nanotube polymer composites J. Appl. Phys. 103, 044302 (2008); 10.1063/1.2844289 Effect of chirality on buckling behavior of single-walled carbon nanotubes J. Appl. Phys. 100, 074304 (2006); 10.1063/1.2355433

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THE JOURNAL OF CHEMICAL PHYSICS 142, 114901 (2015)

Polymers encapsulated in short single wall carbon nanotubes: Pseudo-1D morphologies and induced chirality Sunil Kumar,1 Sudip K. Pattanayek,1,a) and Gerald G. Pereira2 1 2

Department of Chemical Engineering, Indian Institute of Technology, New Delhi 110016, India CSIRO Mathematics, Informatics and Statistics, Private Bag 33, Clayton South 3169, Australia

(Received 11 December 2014; accepted 16 February 2015; published online 16 March 2015) Molecular dynamics simulations are performed to investigate the stable morphologies of semi-flexible polymer chains within a single wall carbon nanotube (CNT). We characterize these morphologies with a variety of measures. Due to the different curvature inside the CNT to outside, there are increased numbers of polymer-CNT bead contacts for polymers which reside inside the CNT. A sufficiently long polymer chain first adsorbs on the exterior of the nanotube and subsequently moves inside the cavity of the nanotube. At equilibrium, the polymer configuration consists of a central stem surrounded by helically wrapped layers. Sections of the polymer outside the CNT have helical conformations (for CNTs of small radius) or circular arrangements (for CNTs of larger radius). Polymers encapsulated within the CNT have an increased chirality due to packing of the beads and this chirality is further enhanced for moderately stiff chains. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4914463]

I. INTRODUCTION

Encapsulation of a variety of molecules within carbon nanotubes (CNT) is attracting much interest not only due to the unusual mechanical,1–5 thermal,6–8 electrical,9,10 barrier,11,12 and optical properties of the CNT but also because of the enhanced properties of this pseudo-one-dimensional nanocomposite. For example, in drug delivery applications, the CNT provides a natural protective layer against undesirable degradation of the encapsulated molecules. Whereas sidewall functionalization can leave the active molecules open to attack from the surrounding environment, the carbonbased casing together with its needle-like geometry makes these ideal drug delivery devices.13 Similarly, the long term stability of metallic compounds can be greatly enhanced when encapsulated in CNT, as the carbon casing inhibits oxidization.14 Another important consequence of encapsulating molecules within CNT is that the resulting nanocomposite modifies the properties of the fillers from those in the bulk. Encapsulation of fullerenes (C60) molecules (bucky-balls) into CNT to create nano-peapods has revealed that the resultant ordered arrangement of the bucky-balls creates chiral structures which have different thermal and electrical properties to the bulk.15 Other studies have shown the encapsulation of metals in CNT can lower their bulk melting point, while encapsulation of ice within CNT leads to different morphological structures.16 Encapsulation of chainlike molecules such as DNA,17,18 graphene nano-ribbons,16,19 or polymers into CNT20–23 has also attracted interest for their capability to produce novel, functionalised nano-materials. CNT are also being investigated for the separation and/or

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]. Tel.: +91 11 26591018. Fax: +91 11 26581120.

0021-9606/2015/142(11)/114901/12/$30.00

sorting of proteins and other biological molecules and microfluidic lab-on-a-chip applications.24 To encapsulate a chain-like molecule into a narrow capillary (such as a CNT) requires a significant attractive energy between the molecule and CNT. This is because the confined space within the CNT severely limits the number of conformations (and hence entropy) of the molecule.25 This loss in entropy, on entering the CNT, must be compensated by a gain in enthalpy. Using molecular dynamics (MD) simulations, Gao et al.17 found the van der Waals interaction between CNT and DNA played the dominant role in DNA insertion. However, there still remains a competition between adsorbing on the exterior surface or being encapsulated within the CNT, so that the encapsulated DNA strand must be placed close to the entrance.18 There have been numerous theoretical studies on adsorption of polymers on the exterior surface of CNT,27–37 but much fewer on encapsulation. This is most probably due to the difficulty in getting a polymer to encapsulate within a narrow capillary. However, given the number of applications that may ensue, this is obviously an important area of investigation. Wei and Srivastava38 have studied the translocation of long polymer chains (polyethylene) through CNT channels and found that polymer chain motion into the CNT is due to a more favorable van der Waals interaction energy inside the nanotube. Through a combination of molecular dynamics simulations and theory, they estimated the translocation time for a long polyethylene chain (105 units) to be about 1 µs. Yang et al.39 carried out molecular dynamics simulation to study the diffusion of single alkane molecules into a CNT. They considered various chain lengths of alkane molecules and different CNT tube diameters at 300 K. They found that long polymer chains followed a cylindrical trajectory along the internal curved surface of the CNT. However, they did not investigate the organization of the polymer chain in detail. Li et al.40 and Zheng et al.41 have investigated the

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affect of chirality, radius, temperature, and surface chemical modification of the CNT on encapsulation of polymer chains into CNTs using molecular dynamics simulations. Experimental evidence of conformation of long chain molecules in confined volumes such as nanotubes has been reported recently. Okada et al.42 found that a sufficiently long ssDNA chain takes up a helical conformation inside the CNT. Multichain polymer systems residing within nano-pores have also been studied experimentally. Chen et al.43 have studied the structure of polystyrene b-PBD block copolymers inside Anodized Aluminum Oxide (AAO) nano-pores (of 200 nm diameter) and found the formation of nano-rods. However, Hou et al.44 have reported on the structure of poly(styreneb-2-vinylpyridine) diblock copolymer in AAO nano-channels where they found the formation of spiral-like, double helical nano-structures inside the nano-pores. Whilst it is possible for a polymer chain to move within the cavity of a nanotube, there is still the open question of whether this is a stable state. In this study, we therefore address this point and consider the conditions for stable, equilibrium conformations of the encapsulated polymer chain. As we shall see, the cooling pathway must also be carefully controlled. A direct quench from high temperature does not lead to encapsulation, rather a gradual lowering of the temperature (simulated annealing) is required. Another question that has not been addressed in the literature is the conformation alignment and arrangement of the polymer within the CNT. Is the morphology ordered and if so what are these ordered structures? Pickett et al.26 have shown that when a simple system consisting of hard spheres is confined to a cylinder, chiral arrangements of the spheres naturally result. This is only due to steric interactions between spheres. Previous work on confinement of buckyballs in CNT15 has also shown a number of chiral arrangements of the bucky-balls. Thus, we investigate this in some detail. From an application point of view, the various morphological conformations of the polymer chains in the CNT, including its chirality, are expected to impart different intrinsic properties to the CNT-polymer nano-composite. Hence, we study the conformation of both flexible and semi-flexible polymer chains inside single wall carbon nanotubes of various radii during stepwise cooling45 using molecular dynamics simulations. It is found that certain polymers prefer to be inside the CNT rather than on the exterior surface due to higher enthalpy of interaction among the beads inside the nanotube. We calculate the radial distribution function g( χ), bond order parameters between polymer segments and the CNT axis, SPN (ξ), a chiral order parameter, individual energy contributions, and torsion angle distributions to characterize the encapsulation and alignment of the polymer chain. We find that the flexibility of the chain and radius of the nanotube play an important role in the structure and placement of the polymer inside the nanotube at a fixed value of polymer-CNT bead interaction energy. II. SIMULATION DETAILS

We implement constant temperature molecular dynamics simulations46,47 in this study. In a simulation box, we put a

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CNT and the required number of polymer chains. A polymer chain is made up of N beads—most simulations contain single chains, but a few have multiple chains with different N values. A CNT consists of a network of hexagonally orientated carbon atoms. As defined by Al-Haik et al.,48 there are two principal unit vectors which define the underlying hexagonal lattice, denoted a1, a2. Two coefficients (p, q) are used to define the chirality of a CNT. In our case, we have the tube axis aligned with one of the principal vectors of the hexagonal lattice and so, we have a (p, 0) structure or more commonly called zigzag geometry (see Figs. 1 and 2 of Al-Haik et al.48). The value of p is related to the radius of the CNT. A CNT is generated in three dimension by using VMD (visual molecular dynamics)49 free software. The coordinates of the beads of CNT formed are then used in the MD simulations. The polymer chain is a generic semi-flexible chain. Each bead represents a united atom of mass m. Bonded (or stretching) potentials, bond-angle potentials along the backbone, torsional potentials, and non-bonded interaction are present between two beads of the polymer or CNT. Only nonbonded interactions are present between beads of the polymer chain and CNT. We allow for the possibility of epitaxial alignment between the CNT and the polymer chain. The nonbonded interaction potential between two different beads i and j is assumed to follow a Lennard-Jones (LJ) potential. This potential, E LJ , is given in dimensionless form as E LJ (r i j ) ≡

ELJ (r i j ) 1 1 = 4ϵ i j * 12 − 6 + , ϵ pp ,rij rij -

(1)

FIG. 1. Schematic of polymer configuration to describe bond order parameter SPN (ξ) defined in Eq. (6): bi = 21 (li + li+1); l i is bond vector, cosα i b .b = |b i| |bCNT | . i

CNT


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J. Chem. Phys. 142, 114901 (2015) FIG. 2. Organization of polymer chains (N = 25) on the exterior of CNTs of various radii. (a)-(c) Snap-shots of the chain on nanotubes of radius (a) 1.9 Å, (b) 5.8 Å, and (c) 11.5 Å. (d) Variation of bond order parameter SPN with radius of the nanotube. Error bars represent the standard deviation from 12 runs for the same parameters. The various non-bonded interaction parameters for polymer and nanotube are ϵ pp = 0.009, ϵ pn = 0.006, ϵ nn = 0.004 eV, and polymer chain stiffness kϕ = 0.09 eV at temperature T = 300 K. The chain was initially at 800 K and gradually cooled to 300 K in the steps of 50 K, with 1000 ps at each temperature step. Other simulation parameters are similar as shown in Table I.

where r i j denotes a dimensionless distance between two beads i and j and ϵ i j denotes a dimensionless interaction energy, which is either ϵ pp or ϵ pn depending on the type of bead corresponding to i and j. All un-scaled distances are made dimensionless by dividing them by the equilibrium polymerpolymer bead separation distance σpp, which corresponds to the separation at which the polymer-polymer LJ potential is zero. The LJ potential between polymer beads on the same chain is only calculated for beads which are separated by four or more beads. In addition, the cutoff distance between two beads for the LJ interaction is taken to be 2.5σpp. σnn corresponds to the separation between two CNT beads at which the LJ potential is zero, while σpn is the distance at which the polymer-CNT potential is zero and we use √ σpn = σppσnn. The bonded (stretching) interaction potential between two bonded beads i and j of species β (polymer or CNT) is given by 2 EBonded, β 1 = k s r i, j − r eq , ϵ pp 2

(2)

where the dimensionless bond vector joining ith and jth bead is r i, j for CNT or polymer. We note that for a polymer chain, the distance will be given by r i,i+1. The equilibrium separation between two adjacent beads is denoted by r eq. k s is the dimensionless spring constant of the harmonic potential 2 . The harmonic (cosine) bond angle defined as k s = k s /ϵ ppσpp potential is given by EBondangle, β 1 = k θ (cos θ i − cos θ 0)2, ϵ pp 2

(3)

where θ i is the bond angles between two successive bond vectors, ri−1,i = ri−1 − ri and ri,i+1 = ri − ri+1, θ 0 is the equilibrium bond angle between adjacent bonds, and k θ is the dimensionless bond angle constant. The torsional angle potential given by Etorsion, β 1 = k ϕ (1 + cos 3ϕ), ϵ pp 2

of a bead of mass m due to the forces arising from the sum of the various potentials, E, can be integrated up to any time according to the velocity Verlet51,52 algorithm. We implement a constant number, volume, and temperature (NVT) simulation via the Nose-Hoover thermostat. A. Parameters

All energy parameters of the MD simulations are rescaled with respect to the van der Waals interaction energy parameter ϵ pp and lengths are rescaled with respect to σpp. The equilibrium separation between two beads, r eq, and the spring constant of the harmonic potential, k s , are required to define bonded interactions potential between two beads. The harmonic cosine bond angle potential requires a rotational constant, kθ , and equilibrium angle, θ 0. The torsional angle potential requires a torsional constant, kϕ . The values we use for all these (rescaled) parameters are shown in Table I. The values chosen for the various parameters correspond to polyethylene chains.50 Most simulations are performed with these values although in a few simulations, where we seek to determine the effect of chain flexibility, we vary the value of kϕ . The chain lengths, N, which we have used for our simulations (depending on requirements) are N = 25, 500, or 1000. For example, to understand the reason for a polymer chain going inside the nanotube, we have used N = 25 initially for a single chain simulation. A short chain is expected to form a monolayer without any folds on the interior or exterior of a nanotube after adsorption. The entropy (after adsorption) of the short chain is expected to be same at the interior/exterior and so the reason for encapsulation of the short chain can be determined. Subsequently, we study longer chains (N = 1000) to determine the reasons that these longer chains move inside the nanotube. B. Parameters studied

(4)

where ϕ is the torsion angles between three successive bond vectors, ri−1,i = ri−1 − ri, ri,i+1 = ri − ri+1 and ri+1,i+2 = ri+1 − ri+2, while k ϕ is the dimensionless torsional constant. The total potential energy of the polymer/nanotube composite is a summation of the all the potentials detailed above: E = EBonded, β + EBondangle, β + Etorsion, β + ELJ . The atomic potential parameters between surface atoms to polymer atoms used here is the DREIDING50 potential. The equation of motion

We define the probability of finding beads at distance χ from the centre of the CNT through the radial density distribution function53 in a cylindrical geometry of unit length as g( χ). This is defined mathematically as g( χ) ≡

n( χ) , 2π χ∆ χ ρ

(5)

where n( χ) is the number of beads lying within the cylindrical shell located between χ and χ + ∆ χ and ρ is the total


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TABLE I. DREIDING force field parameters. Reduced units Parameter σ req ϵ ks kθ kϕ θ0

Real units

Values for polymer

Values for nanotube

Values for polymer

Values for nanotube

1 0.42 1 463 12 186 0 10.08 1.92

1.08 0.38 0.48 694 68 293 5 12.60 2.09

3.62 Å 1.53 Å 0.009 eV 30.34 eV/Å2 16 eV 0.09 eV 1.92 rad

3.89 Å 1.39 Å 0.004 eV 45.52 eV/Å2 25.25 eV 0.11 eV 2.09 rad

number of polymer beads present in the system. The bond order parameter54 between polymer chain segments and the CNT axis, SPN (ξ), is evaluated to determine the alignment of polymer chain segments on the inner and outer surface of the CNT. SPN (ξ) is calculated from the angle between sub-bond vectors, bi, of the polymer chain and axis vector, bCNT, of the CNT as shown in Fig. 1. The bond order parameter is defined as 2 1 −2 3cos α j − 1 ΣN ⟨ ⟩, (6) N − 2 j=1 2 where SPN (ξ) is the average bond order parameter of a polymer chain lying within the cylindrical shell located in between ξ and ξ + ∆ξ from axis of nanotube. α i is the angle between the sub-bond vectors bi and bCNT and ξ is the perpendicular distance between bi and bCNT. Typical values of SPN (ξ) are 1.0, 0.0, and −0.5 corresponding to a polymer chain whose subbond vectors are perfectly parallel, random, and perpendicular to axis of CNT.

SPN (ξ) ≡

C. Equilibration

We first placed a VMD generated CNT of required length and radius (which varies according to our requirements) in a predefined simulation box of size 180 Å × 180 Å × 180 Å and subsequently, a randomly generated polymer chain is placed near the CNT. We note that CNT is not stationary during the simulation. We ran simulations for an extended time using a stepwise cooling45 methodology. The system temperature is lowered from initial 800 K to 300 K in steps of 50 K. The simulation is run for 1000 ps at each temperature step. We ran 10–15 different simulations for each set of parameters by taking different initial conformations of the polymer chain and different initial orientations between the polymer and CNT. The observations are similar for all the cases. III. STABLE POLYMER MORPHOLOGIES A. Short chains

To understand the reason for encapsulation of a chain, we placed a short polymer chain (N = 25) near a nanotube of radius 11.5 Å using stepwise cooling from 800 K to 300 K for two different cases: (a) a chain placed near the outer surface of the nanotube (far from its mouth) and (b) a chain placed very close to the mouth of the nanotube. The length of the nanotube used is 105 Å, which is longer than the fully

stretched length of the polymer used. Assuming all C–C–C valence bond angles are 109.5◦ and all trans-configuration angles are 180◦, the fully stretched length of the chain is given by Nl sin(109.5◦/2). Hence, the length of chain with 25 beads is 30 Å. Some important parameters used are the non-bonded interaction parameters ϵ pp = 0.009, ϵ pn = 0.006, ϵ nn = 0.004 eV, and polymer chain stiffness kϕ = 0.09 eV, with other simulation parameters shown in Table I. Figure 2 shows the alignment of a short polymer chain on the exterior of a nanotube. The snapshot of evolution of the chain from the placement is shown in the supplementary material Fig. S1.62 It shows that placement of short polymer chain decides its final location (inside or outside of the nanotube). For this case, a tilted arrangement of the chain is observed. As we increase the radius of the nanotube, the polymer chain alignment varies as shown in Figs. 2(a)–2(c). These figures represent the equilibrium conformation of the chains after the stepwise cooling process has completed. We have measured SPN of the aligned polymers on the nanotubes for a number of different radii (Fig. 2(d)). The value of SPN decreases with increasing radius of the nanotube. This alignment is similar to what we have explained in our earlier publication.37 To maximize the contact energy between polymer beads, the chain wraps around the cylinder (taking a circular trajectory). However, this is at the cost of bending/torsional energy, especially for the narrower tubes. Hence, as the CNTs widen, the chains attempt to take up configurations which are more tilted with respect to the tube axis (which is the precursor to a wrapping configuration). We have observed from other simulations which we have run (not shown here) that as the length of the nanotube decreases, so that the length of the tube becomes very close to length of a polymer chain, the polymer enters the interior of the nanotube. We also observe that the polymer enters the nanotube when the chain is placed very close to opening of nanotube. This indicates the chain prefers to be inside the nanotube rather than to be on the outer surface of the nanotube. Figures 3(a)–3(c) show snapshots of the arrangement of a short polymer chain (N = 25) on nanotubes of various radii. Figure 3(d) shows variation of SPN of aligned polymers inside the nanotube as a function of nanotube radius. The chains which are inside the CNT tend to align more with the CNT axis than those on the exterior. For most of the nanotube radii, the SPN value is close to one and only for the widest nanotube (around 12 Å) this value comes down to around 0.75. This compares to a value of about 0.6, for those chains adsorbed


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FIG. 3. Organization of a polymer chain (N = 25) in the interior of a CNT of various radii. (a)-(c) Snapshot of the chain over nanotube of radius (a) 1.9 Å, (b) 5.8 Å, and (c) 11.5 Å. (d) Variation of bond order parameter SPN with nanotube radius. Error bars represent the standard deviation from 12 runs for the same parameters. The various non-bonded interaction parameters for polymer and nanotube are ϵ pp = 0.009, ϵ pn = 0.006, ϵ nn = 0.004 eV, and polymer chain stiffness kϕ = 0.09 eV at temperature T = 300 K. The chain was initially at 800 K and gradually cooled to 300 K in the step of 50 K, keeping the simulation run for 1000 ps at each temperature step. Other simulation parameters are similar to Table I.

on the exterior. The main reasons for this are the different curvatures (concave on the inside versus convex on the outside, see discussion in the following two paragraphs regarding this) and the slightly smaller radius of curvature of the interior of the CNT compared to the exterior of the CNT. For the system described above, the entropy of the adsorbed polymer being same, the contact energy is expected to be the driving force for its entrance into the CNT. We have determined the various energetic components: (a) nonbonded energy of the polymer chain (EPP), (b) non-bonded energy between polymer and CNT (EPN ), (c) bonded energy, (d) bond bending energy, and (e) torsion energy in polymer chain (Etorsion) for the two cases (adsorption on the exterior and in the interior of CNT). Most of the energies (EPP, Ebond, Ebondangle, and Etorsion) for the two cases are almost equal but EPN differs considerably. Figure 4 plots the variation of EPN of the polymer as a function of nanotube radius for the two cases. EPN is more favorable for the polymer inside the nanotube and especially so for the narrower tubes. The reason for this is that polymer beads are surrounded by an array of CNT beads when the polymer resides inside the CNT. On the other hand, if the polymer is on the exterior, CNT beads are only beneath it. When the chain is inside the CNT, because of the difference in curvature (concave on the inside versus convex on the outside), the beads of the polymer chain do not register exactly with the CNT beads. While beads on the exterior are found to follow the zigzag structure of the CNT surface, those on the inside do not (see Figs. 5(a) and 5(b)). Since polymer chains which are absorbed on the exterior only feel a significant attraction from CNT beads directly beneath and adjacent to them, they tend to get as close as possible to those CNT beads. However, polymer chains which enter the interior of the CNT feel an attraction from CNT beads from all around (full 360◦ arc). There is not the one-to-one mapping between polymer beads and CNT beads (as for the exterior). In turn, this means there is a higher number of polymer bead—CNT bead contacts. To verify this, we have calculated the number of contacts between a polymer bead and CNT bead, denoted

QC (δ), within a distance δ of each other. QC (δ) is defined as i=N QC (δ) = Σi=1 qi (δ),

(7)

where qi (δ) is defined as the number of CNT beads present within a spherical shell of thickness △δ at a distance δ from a polymer bead. Fig. 5(c) shows the variation of QC (δ) with the distance δ. This results in a higher gain in enthalpic energy for polymers absorbed inside the CNT compared to those adsorbed on the exterior. In summary, shorter chains will enter the interior of the CNT, in preference to being adsorbed on the exterior surface because of the larger enthalpic energy gain, which results from the different curvatures (concave on the inside compared to convex on the outside). There is an entropy cost for entering the CNT (due to confinement by the walls) but the enthalpy gain is sufficient to offset this loss. This

FIG. 4. Comparison of variation of E PN of short polymer chain (N = 25) on inner and outer surface of CNT. Error bars represent the standard deviation from 12 runs for the same parameters. The various non-bonded interaction parameters for polymer and nanotube are ϵ pp = 0.009, ϵ pn = 0.006, ϵ nn = 0.004 eV, and polymer chain stiffness kϕ = 0.09 eV at temperature T = 300 K. Other simulation parameters are similar to Table I.


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FIG. 5. Arrangement of polymer (N = 25) at exterior and interior surface of CNT of radius 11.5 Å and length 47 Å. (a) Snapshot of polymer chain at exterior, (b) snapshot of polymer chain at interior, and (c) variation in number of contacts between beads of polymer and CNT with the distance of polymer bead from the bead of CNT.

polyethylene epitaxy onto the exterior of the curved surface of CNT is observed for all cases studied. Further comparisons between polymers within the interior and on the exterior of various nanotubes are shown in the supplementary material Fig. S2.62 B. Long chains

For a long polymer with similar sized nanotubes to those used in Sec. III A, the polymer is expected to fill the space available inside the nanotube. This will be due to not only the polymer-CNT energy but also the polymer-polymer energy. We simulate a polymer with N = 1000 beads near a short CNT of length 47 Å and radius 11.5 Å at 300 K. Other simulation parameters are similar to those used previously and are listed in Table I. We evaluate EPP throughout the adsorption/encapsulation process to understand how it contributes to the final polymer configuration. Snapshots at various stages during the simulation are shown in Fig. 6. Initially, the polymer chain is located outside the CNT and

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early on attempts to cover the exterior surface, which can be seen from the decrease in Eppo. As the simulation progresses, parts of the polymer migrate to the entrance of the nanotube and then enter the interior. The movement to the CNT entrance (when the polymer is adsorbed on the exterior) is a random process, but the movement inside the nanotube is driven by the rapid decrease in polymer-CNT and polymer-polymer energy as can be seen from Fig. 7. Figure 7(a) plots the variation of polymer-nanotube bead contact energy for chains which reside inside or outside the nanotube (represented by Epno and Epni, respectively) and Fig. 7(b) plots the variation of polymer-polymer contact energy for chains which reside inside and outside the nanotube (represented by Eppo and Eppi, respectively). It can be seen from Fig. 7(a) that by the middle of the simulation, Epno and Epni decrease rapidly, which is the initial driver for chain absorption. As time proceeds, Epni is even lower than Epno, but at around 1000 ps, rearrangement of the chain within the nanotube results in a dramatic decrease in Eppi (see Fig. 7(b)). Around this time, the polymer beads form a multilayer inside the nanotube of radius 11.5 Å. This multilayer formation vastly increases the number of polymer-polymer contacts and consequently decreases Eppi. Finally, the polymer-nanotube bead contact energies Epni and Epno per bead are roughly the same, but Eppi ≪ Eppo presumably due to a more favorable packing of the beads within the tube. Thus, we now analyze the arrangement of the polymer beads within and near the CNT. The evolution of polymer arrangement can be seen from the variation of two parameters: g( χ) and SPN (ξ). The positional distribution of beads from the surface can be observed from g( χ) and the directional alignment of polymer segments along the axis of the nanotube can be obtained from SPN (ξ). Figure 8 shows the variation of g( χ) with distance from the center of the nanotube at various stages during the simulation. A high value of g( χ) corresponds to a high number of polymer beads at the location χ. At 200 ps (black curve), there are only two main peaks, one inside and one outside, which is an adsorbed monolayer. As time proceeds, the layer on the exterior remains as one single layer (corresponding to a sharp peak at χ ≈ 15). Meanwhile, inside the nanotube, three different layers (corresponding to three sharp peaks) gradually evolve at χ ≈ 9, 4 and zero. The sharp peaks would indicate the polymer beads inside the nanotube form a semi-crystalline structure. Clearly, this is in response to the polymer-polymer interaction energy. To understand the directional alignment of the polymer beads, we monitor SPN (ξ) as a function of the distance from the axis of the nanotube, ξ, which is plotted in Fig. 9. Three distinctly different SPN (ξ) values are observed at the four peak locations. Near the central axis (ξ = 0), the value of SPN (ξ) is close to one, which indicates alignment of

FIG. 6. Snapshots of a polymer chain with N = 1000 beads on inner and outer surface of CNT of radius 11.5 Å and length 47 Å at different time intervals. All other parameters and cooling process are same as described in Fig. 2.


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FIG. 7. Comparison of variation of energy per bead with time for polymer chain of N = 1000 beads on inner and outer surface of CNT of radius 11.5 Å and length 47 Å (a) E PN and (b) E PP. E PNo and E PNi are energy for interaction of polymer-CNT beads outside and inside of nanotube, respectively. E PPo and E PPi are energy for interaction of polymer-polymer bead outside and inside of nanotube, respectively. All other parameters and cooling process are same as described in Fig. 2.

sections of the polymer is parallel to the axis of the nanotube. The first (ξ = 4) and second (ξ = 9) layers of polymer beads, near the inner surface of nanotube, yield an SPN (ξ) value of about 0.25. This corresponds to a helical alignment of polymer segments. For the section of the polymer on the outer surface, the SPN (ξ) value is about −0.4. We recall that for a purely circular trajectory of the polymer, an SPN (ξ) value of −0.5 results. So, the alignment of the polymer chains which make up the adsorbed monolayer on the CNT exterior wraps around the CNT in (close to) a circular pattern. We next simulate a polymer of N = 1000 near short CNTs of length 47 Å and variable radii (from 3.8 Å to 11.5 Å). Snap-shots of the final equilibrium polymer configuration, after a total of 11 000 ps are shown in the Fig. 10. To obtain better visualization of the arrangement of polymers near the nanotube, we have shown sections of the polymer inside the nanotube (in the middle column of panel in Fig. 10) and sections of the polymer outside the nanotube (in the last column of panel in Fig. 10). For a CNT of radius 3.8 Å, the space available is just sufficient for a single stem of polymer (about 40 beads) to be inside and the rest of the polymer beads is arranged in a helical conformation (SPN = 0.59) on the exterior surface of the nanotube. In spite of the small radius of this CNT, it is still energetically favorable for the chain to enter and form a single strand along the tube

axis. On increasing the nanotube radius to 7.7 Å, about 336 polymer beads are accommodated inside the nanotube, which are arranged as one centrally straight stem (SPN = 0.96) and a layer of helically oriented beads of SPN = 0.60. The remaining polymer beads are accommodated outside the nanotube in a helical conformation with an associated SPN value of 0.45. On increasing the radius of nanotube to 9.5 Å, about 536 beads are accommodated inside and rest of the beads located outside the nanotube. The section of the polymer present inside the nanotube has two different alignments: a straight stem (SPN = 0.96) present at the center of the nanotube and a layer of helically arranged beads (SPN = 0.63) present adjacent to the inner surface. The section of the polymer present outside the nanotube has helical alignment (SPN = −0.21). The large difference in the order parameters of the polymer arranged on the outside of the nanotubes of radii 7.7 Å and 9.5 Å indicate a significant difference in the orientation of the polymer. The snapshots (Figs. 10(b) and 10(c)) indicate that sections of the polymer chain present outside the nanotube have changed their long helical orientation (on nanotubes of radii 7.7 Å) to a nearly circular orientation (on nanotubes of radii 9.5 Å). The absence of the circularending penalty the chain would experience. The bending penalty is inversely proportional to the square of the nanotube radius,37 and is thus large for thinner nanotubes. Further increase of the nanotube radius to 11.5 Å results in about 574 beads accommodated inside the nanotube in two different conformations: one central, straight stem (SPN = 0.96) and two layers of helical

FIG. 8. Radial distribution function g (χ) for polymer chain near the surfaces of a CNT of radius 11.5 Å and length 47 Å. χ represents the distance from central axis of the nanotube outwards. The results plotted here correspond to simulation in Fig. 6.

FIG. 9. Bond order parameter SPN (ξ) for polymer chain alignment on both the inside and outer surface of a CNT of radius 11.5 Å and length 47 Å. The value of ξ represents the distance of three bead polymer segments from the axis of CNT. The results plotted here correspond to simulation in Fig. 6.


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increased CNT length of (60 Å). The observations from these simulations are similar to those observed for the CNT of length 47 Å. This indicates the circular arrangement of the polymer beads is not specific to particular nanotube dimensions, but a general observation. In general, we do not find a circular arrangement of the chains inside the nanotube, since we never simulate nanotubes with such large diameters. 1. Effect of chain flexibility

FIG. 10. Arrangement of single polymer chains of N = 1000 beads near a short CNT (length = 47 Å) of various radii. The columns headed “inside” and “outside” indicate the arrangement of sections of the chain inside and outside the nanotube, respectively. The numbers in the curved brackets (first column) indicate the number of beads inside the CNT (first number) and outside (second number). Values of the SPN are as follows: (a) SPN = 0.96 and 0.59 at axis of nanotube and exterior surface, respectively, (b) SPN = 0.96, 0.6, and 0.45 at nanotube axis, layer at interior surface, and exterior surface, respectively, (c) SPN = 0.91, 0.63, and −0.21 at axis of nanotube, layer at interior surface, and exterior surface, respectively, and (d) SPN = 0.96, 0.21, 0.20, and −0.36 at axis of nanotube, 1st layer, 2nd layer at interior surface, and exterior surface, respectively. All other parameters and cooling process are same as described in Figure 2.

beads with (SPN = 0.2). The section of the polymer chain present outside the nanotube is aligned circularly, with the centers of these circles lying along the axis of the nanotube. The calculated SPN value for this section of the polymer is −0.36. We therefore surmise that the arrangement of the sections of the polymer outside the nanotube may be helical (thin nanotubes) or circular (thick nanotubes) depending on its radius. The circular structure is a prelude to the toroid structure and it has been shown the toroid has an optimal radius proportional55,56 to L 1/5(κ/k BT)2/5, where L is the contour length of the semi-flexible chain and κ is a bending constant. So, when the nanotube radius becomes comparable to this value, a circular arrangement of the chains result. We have also run simulations similar to those just described above, with the same parameters, but with an

In our previous work,37 we were able to show that the torsional potential (and associated constant, kϕ ) has a similar form to the potential of a semi-flexible polymer chain, with bending constant κ.55,56 So, by increasing kϕ , we can change the chain’s flexibility from a flexible chain (low kϕ ) to a stiff chain (high kϕ ). This in turn will result in different conformations of the chain and may also effect the location of the chain (in the interior or on the exterior of the CNT). Figure 11 shows snap shots of polymer chains with different flexibilities (kϕ ) on CNT of radius 11.5 Å and length 47 Å at 300 K. The system is cooled in a stepwise manner to a temperature of T = 300 K from 800 K as stated earlier. From the simulations described in Fig. 10, a 500 bead chain will fill the interior of the nanotube of these dimensions. At the lowest value of kϕ = 0.02 eV, the polymer is completely encapsulated within the CNT. Because the effective bending constant of the chain is so small, the chain can easily deform and fill the available volume and so minimize the enthalpy. The SPN value for this configuration is −0.0034 which indicates a random configuration of the chain. At a value of kϕ = 0.09 eV, the chain remains encapsulated but now its configuration becomes helical and the chain trajectory is much more aligned with the CNT axis (SPN = 0.3765). At larger value of k ϕ = 0.35 eV, the polymer resides both inside and outside the CNT with a helical alignment inside (SPN = 0.2671) and circular alignment outside (SPN = −0.3488). At the largest k ϕ = 0.52 eV, the bending of the chain within the nanotube is too costly and so the polymer adsorbs to the exterior. Here, the alignment is clearly circular with an SPN value of −0.4481. The parameter kϕ controls the chain’s capability to bend (flexibility). At low values, the chain bends easily and hence, the chain is readily encapsulated within the CNT. At large values, the chain is prohibited from sharp bends and hence, it is difficult to be completely encapsulated within the CNT. In this case, it can wrap around the nanotube’s exterior. C. Mixtures of short and long chains

We next consider the effect of polydispersity in a mixture of chains. The important point to recognize is that as the temperature of the solution is lowered, one passes from good solvent conditions to poor solvent conditions. When in the poor solvent, the chain condenses out of solution. For an infinite length chain, the temperature when one passes from good to poor solvent is the Θ-temperature. However, this temperature is chain length dependent and given by57 ( ) 1 1 1 1 1 = + , (8) √ + Tc Θ B N 2N


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FIG. 11. Snapshot of arrangement of polymer chain (with N = 500) on CNT of radius 11.5 Å and length 47 Å at (a) kϕ = 0.02, (b) kϕ = 0.09, (c) kϕ = 0.35, and (d) kϕ = 0.52 eV. The system is cooled in steps to a temperature of T = 300 K from 800 K as stated earlier. The various non-bonded interaction parameters for polymer and nanotube ϵ pp = 0.009, ϵ pn = 0.006, ϵ nn = 0.004 eV, N = 500, and other simulation parameters are similar as shown in Table I.

where Tc is the critical temperature (at which a finite length chain passes from a good to poor solvent) and B is a positive constant. This implies shorter chains have a lower critical temperature and condense out of solution further away from the Θ-temperature. We have determined the Tc of monodisperse polymers through simulation using the following method. We maintain the total number of polymer beads (at 1000) in a simulation box but vary the number of independent chains in the monodisperse mixture. Four sets of simulations were run with chains of length N = 25, 50, 100, and 500. The temperature of the system was initially set at 1000 K and reduced stepwise with a step size of 25 K each. The system was run for 1000 ps at each temperature step. The interaction parameter was set to ϵ pp = 0.009 eV. We then determined the temperature at which the polymers form a stable globule. The critical temperature, Tc for chains of length N = 25, 50, 100, and 500 are 500 K, 650 K, 775 K, and 850 K, respectively. Thus, as predicted by Eq. (8), our simulation results yield lower Tc ’s for shorter polymer chains. To observe the effect of polydispersity, we ran simulations containing one long chain (N = 500 beads) and 20 short chains (N = 25 beads each) with a CNT (L = 47 Å and r = 11.5 Å). In any given simulation, the value of k ϕ , for both shorter and longer chains, is chosen to be the same. Fig. 12 shows snap-shots of the equilibrium conformations of the polymers. (Each case in Fig. 12 corresponds to a different value of kϕ .) If the polymers are flexible (Fig. 12(a)), the long polymers is found inside the nanotube, while the shorter chains are found adsorbed on the exterior of the CNT. Since we are implementing the stepwise cooling methodology, the long chain condenses out of solution first (i.e., at a higher

temperature) and rapidly moves towards the CNT, due to preferable CNT-polymer interactions. Once the long chain adsorbs on the exterior of the CNT, it quickly moves inside the nanotube (as described previously). Meanwhile, the short chains are still in a good solvent and fluctuate in solution. As the temperature decreases further, the critical temperature for short chains is achieved and these chains condense out of solution and onto the exterior of the CNT. They cannot move inside the CNT as there is no free volume available. At higher kϕ values of 0.09 and 0.35 eV (Figs. 12(b) and 12(c)), a similar description of the condensation and adsorption process occurs. However, due to the larger kϕ value, there is a greater alignment of the chain segments, as has been explained previously. At the largest kϕ value of 0.52 eV, the long chain is found wrapped on the exterior of the nanotube and the short chains inside. Even though the longer chain condenses out of solution first, because of its large stiffness, it does not collapse inside the nanotube. Hence, the available free space inside the nanotube is filled, at a later time, by the short chains.

IV. CHIRALITY OF POLYMER MORPHOLOGIES

The chirality of a molecular structure is an important quantity which can lead to a host of different intrinsic (macroscopic) properties.58,59 For the polymers considered in this study, there are three main sources for this chirality: (1) natural chirality which derives from the chain’s flexibility and solvent quality to which the chain is exposed. In this case, the polymer (in the absence of the nanotube) would condense to a helical structure or in the presence of a nanotube

FIG. 12. Snapshot of a multichain polymer system (single long chain containing 500 beads and 20 short chains containing 25 beads each) on a CNT (radius = 11.5 Å, length = 47 Å) with (a) kϕ = 0.02, (b) kϕ = 0.09, (c) kϕ = 0.35, and (d) kϕ = 0.52 eV. First row of figures represents the side view and second row corresponds to the top view of the system. Blue colour chain represents long chains and green color corresponds to short chains.


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would adsorb in a helical conformation to the outer surface. (2) An induced chirality when polymer chains are encapsulated within the nanotube. This is due to packing of the polymer beads within the confined inner geometry of the nanotube. Previously, Pickett et al.26 have shown that when spheres are confined to cylindrical tubes, at certain packing densities, the spheres take up a chiral structure. (3) Alignment of the polymer beads with the underlying carbon atoms of the CNT. Chirality of a structure is essentially the absence of mirror plane symmetry. The degree of chirality can be quantified. For example, helical structures can be more or less chiral if the pitch of the helix is shorter or longer, respectively. The simplest measure of chirality (for a sequence of three segments) is given by Ci =

(vi−1 × vi ) · vi+1 , d 03

(9)

where the v’s represent segment vectors which make up the molecule and d 0 is the magnitude of the segment vector. This measure implies if all vectors in a molecule are coplanar, the chirality will be zero. It has previously been used by Kwiecinska and Cieplak58 for quantifying chirality in structures formed during protein folding. One advantage of this structure is that it differentiates between right-handed (Ci > 0) and left-handed (Ci < 0) molecules. In determining the segments, we do not want to capture the fine structure of the chain, i.e., the tetrahedral configuration of the carbon atoms. This tends to blur out the chiral property of the chain. So, we have implemented a chiral measure where the segment vector is made up of w monomers, i.e., vi = ri+w − ri . However, w cannot be too large for then it would sample the long-range geometry of the chain and miss the important meso-scale structure which gives rise to chirality. After some testing, we found that the optimal value of w to be a sequence of eight monomers. The number of segments in the chain is then Nsegment, where Nsegment = N/w. The chirality for the entire polymer chain is then summed over all segments which make up the chain as follows: C=

1

Nsegment  −1

Nsegments − 2

i=2

Ci .

(10)

Note, we ensured that beginning on the first monomer gave, on average, similar results to beginning on the second, third, . . . , or eight monomers. An alternative measure of chirality can also be used. This was originally given by Harris, Kamien, and Lubensky59 and modified by Pickett et al.26 which uses spherical harmonic functions. (See Eq. 9 in Harris, Kamien, and Lubensky59 and the definition of ζ in Pickett et al.26) This measure gives the relative magnitude of a chiral object but does not differentiate between objects based on their handedness. We have also implemented this measure but found the relative difference between chiral and achiral objects to be comparatively small (i.e., compared to the vector product method in Eqs. (9) and (10)). So, in this study, we just use the vector/dot product method. The first polymer whose chirality we quantified is that given in Fig. 5(a). This chain of length N = 25 gave a

chirality value of C = 0.0476. (We verified that the same polymer winding in the opposite sense gave a negative value, i.e., C = −0.0476 and that a chain forming a circular loop around the cylinder gave a zero chirality.) The monomers in this polymer tend to align with the underlying CNT carbon atoms. Thus, the chirality here is derived from the CNT lattice. (Note that even though a zigzag (p, 0) is not termed as chiral, there still are trajectories along lattice vertices which lead to chiral structures.) We now proceed to quantify the chirality of the structures in Fig. 10. As we have pointed out previously, when the polymer is encapsulated within a nanotube, it forms a central stem surrounded by layers of beads. We therefore calculate chirality for each layer. These are given in Table II. First, notice that the chirality of the central stem (outermost column in each case) has a chirality value close to zero. For the 3.8, 7.7, and 11.5 Å radius cases, it is exactly zero since the central stem consists only of a straight rod, while in the 9.5 Å case, it consists of two straight rods connected by a hairpin leading to a negligible, but non-zero, chirality. The chirality values of chains absorbed on the outside of the CNT have relatively small values. In the case of the narrowest tube, this is because the chain takes on a lamella structure, while in the case of the larger diameter tubes, it is because the chains tend to wrap circularly around the tube. The tubes with larger diameters (7.7 Å to 11.5 Å) have two or more polymer layers encapsulated. These layers (neglecting the central stem) have relatively large chirality (from 0.25 to 0.67). This value is larger than the chirality value of the portions of the chain which are absorbed to the outside of the CNT. Thus, this suggests packing of the CNTs by polymer beads leads to an enhanced chirality. This is a similar effect to that found by Pickett et al.26 when packing spheres into cylinders. However, we have a structured cylinder, so it requires further investigation as to whether this is the sole cause for the increase in chirality. We therefore infer that the chirality of the polymer is increased due to encapsulation within the CNT. Chirality values have also been calculated for the chains shown in Fig. 11 (see Table III, first two rows). Comparing, the chirality values for the polymer with N = 500 beads with N = 1000 beads at the same condition (i.e., kϕ values of 0.09 eV), it is found that the values are different. This is due to fact that the polymer with N = 500 beads, the nanotube is not completely filled. In this case, there is a completely covered layer close to the CNT surface and few segments of polymer in the 2nd layer. However, for the N = 1000 bead polymer, the CNT chosen is completely filled leading to three layers of polymer beads inside the CNT, which increases the TABLE II. Chirality values for each of the polymer chains shown in Fig. 10. Each column corresponds to a different layer starting from outside the CNT. The absence of a value indicates that such a layer does not exist at this particular CNT diameter. CNT radius (Å)

Outside

Layer 1

Layer 2

Layer 3

3.8 7.7 9.5 11.5

0.099 0.09 −0.16 −0.13

0.0 0.24 −0.21 −0.27

0.0 0.01 −0.67

0.0


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TABLE III. Chirality values for each of the polymer chains shown in Fig. 11 (first two rows) and Fig. 12 (last two rows). Each column corresponds to a different chain stiffness. kφ Fig. 11 (inside) Fig. 11 (outside) Fig. 12 (short chains) Fig. 12 (long chain)

0.02 eV

0.09 eV

0.02

0.14

0.07 0.01

−0.03 0.10

0.35 eV

0.52 eV

0.13 0.03 −0.01 −0.17

0.036 0.04 0.07

chirality. At the two extremes, where the chain has either small stiffness or large stiffness, the chirality values are small—but for different reasons. At small stiffness, the chains are easily encapsulated within the CNT and tend to form a random structure. Even though the chains tend to be well packed, their chirality is still low. Although we have not made a systematic study of the change in chirality with ratio of bead diameter to nanotube diameter, from previous work,26 it is known chirality fluctuates significantly with this ratio. In spite of the fact that the present ratio of bead diameter to tube diameter gives a small chirality for fully flexible chains, the moderately stiff chains yield a significant chirality which implies, in general, a large chirality results from the combination of packing and chain stiffness. At the largest chain stiffness, the chains only wrap around the outside of the tube in circular rings and hence, the overall chirality is low. In the two other cases (kφ = 0.09 eV and 0.35 eV), the chirality values are significant for portions of the chain within the tube. Finally, chirality values corresponding to longer chains in Fig. 12 (Table III, last two rows) are similar to those in just discussed for Fig. 11. However, the shorter chains have small chirality values which presumably is due to the fact that these chains are quite straight (due to the fact their persistence length is comparable to actual chain length).

V. CONCLUSIONS

Using molecular dynamics simulations, we have studied the organization of long polymer chains, based on polyethylene, into and onto short CNTs. This polymer is semi-flexible which contributes significantly to the different conformations that it may take up. There is a significant van der Waals attractive energy between beads from the polymer and the CNT. Due to the different curvature on the inside (concave) versus the outside (convex), the preferred location of the polymer is inside the nanotube rather than adsorbed on the outside. Long polymer chains initially adsorb on the exterior of the nanotube before rapidly moving inside the cavity. There the chain re-arranges into a semi-crystalline structure consisting of a straight stem, aligned with the nanotube axis, and surrounded by helically wrapped layers. If there is not sufficient volume within the nanotube, any remaining portion of the chain will adsorb on the exterior of the nanotube, wrapping in a circular manner with the centers of the circles aligned with the nanotube axis. Polymer beads adsorbed on the outside of the nanotube will tend to register precisely with the underlying lattice structure to optimize the attractive energy.

While the favorable CNT-polymer bead interaction favors polymer beads being encapsulated within the nanotube, the MD simulations also showed that the polymer-polymer interaction is minimized in this configuration. This in turn causes re-organization of the chain within the nanotube to form semi-crystalline (layered) structures. For stiffer polymer chains, encapsulation within the nanotube is inhibited due to the sharp bends which must occur when filling the small cavity. As a result, the chain will wrap on the exterior. Thus, chain flexibility affects the location of the chain—at high stiffness, the polymer chains only adsorb on the exterior of the CNT. The organization of mixtures of short and long chains into a CNT using the step-wise cooling methodology has also been investigated. Because longer chains have a higher critical temperature (in going from good to poor solvent conditions), they first adsorb onto the CNT. Hence, the long polyethylene chains will be found inside the CNT, while the shorter ones are adsorbed on the outside. However, for much stiffer chains, the reverse is found where the long chains are wrapped on the outside, while the short ones are adsorbed on the interior surface. Hence, the step-wise cooling methodology, as opposed to a direct quench, can be implemented to selectively absorb longer chains from a polydisperse polyethylene mixture into the CNT. The chirality of the various stable polymer morphologies has been quantified. We have found that generally the polymers encapsulated within the CNT have a larger chirality compared to those absorbed on the outside. This increased chirality is due to the packing of the beads within the CNT. Such an affect has already been observed for spheres packed into cylinders.26 However, we find the maximum chirality enhancement occurs when the chain has a moderate stiffness. When the chains are fully flexible, the packing of the beads within the CNT is more random, while at large stiffness, the chains prefer to be wrapped in circular rings around the exterior of the CNT. The induced chirality of polymers within the nanotube depends on the ratio of tube diameter to bead diameter as well as chain stiffness. Thus, by carefully tuning these variables, it should be able to produce polymer-CNT nano-composites with a prescribed chirality. Understanding the packing of polymers within nanoconfined regions and the evolved microstructure may be roughly related to the structural organization of DNA during its packaging within a cell. It is well known60,61 that the double helical structure of DNA (which is about 2 nm in diameter) is coiled around a protein core to form a nucleosome (which is about 11 nm in diameter). This helical structure leads to compact packing of DNA within the chromosome of a cell. Thus, an understanding of the formation of these helical structures found for simple systems is a start to understanding much more complex systems such as packaging of DNA within a cell. 1J.

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Controlled synthesis of single-chirality carbon nanotubes.

Industrial-scale separation of high-purity single-chirality single-wall carbon nanotubes for biological imaging.

Bandgap renormalization in single-wall carbon nanotubes.

Empirical Equation Based Chirality (n, m) Assignment of Semiconducting Single Wall Carbon Nanotubes from Resonant Raman Scattering Data.

Synthesis of PbI(2) single-layered inorganic nanotubes encapsulated within carbon nanotubes.

Minimal inflammogenicity of pristine single-wall carbon nanotubes.

Three-dimensional polymeric structures of single-wall carbon nanotubes.

Structure of single-wall carbon nanotubes: a graphene helix.

Single Wall Carbon Nanotubes Based Cryogenic Temperature Sensor Platforms.

Preparation of metallic single-wall carbon nanotubes by selective etching.

Actin reorganization through dynamic interactions with single-wall carbon nanotubes.

Ultrafast terahertz probes of interacting dark excitons in chirality-specific semiconducting single-walled carbon nanotubes.

A Ratiometric Sensor Using Single Chirality Near-Infrared Fluorescent Carbon Nanotubes: Application to In Vivo Monitoring.

Bolometric-Effect-Based Wavelength-Selective Photodetectors Using Sorted Single Chirality Carbon Nanotubes.

Optical isomer separation of single-chirality carbon nanotubes using gel column chromatography.

Chirality-specific growth of single-walled carbon nanotubes on solid alloy catalysts.

Empirical prediction of electronic potentials of single-walled carbon nanotubes with a specific chirality (n,m).

Single-step total fractionation of single-wall carbon nanotubes by countercurrent chromatography.

Asymmetric autocatalysis initiated by finite single-wall carbon nanotube molecules with helical chirality.

Effects of prenatal exposure to single-wall carbon nanotubes on reproductive performance and neurodevelopment in mice.

Diameter selective electron transfer from encapsulated ferrocenes to single-walled carbon nanotubes.

Use of carbon nanotubes (CNTs) with polymers in solar cells.

Improved Dispersion of Carbon Nanotubes in Polymers at High Concentrations.

In situ formation of carbon nanotubes encapsulated within boron nitride nanotubes via electron irradiation.