Torsional Forces Mediated by Surfactant ... - ACS Publications

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Torsional Forces Mediated by Surfactant Aggregates on Carbon Nanotube Junctions Dirk Müter* and Henry Bock Department of Chemical Engineering, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, United Kingdom S Supporting Information *

ABSTRACT: Surfactant-mediated interactions acting along the line of shortest contact between carbon nanotubes have been investigated by many authors, but the surfactant-mediated torsion that arises in the case of angled tubes have so far been ignored. Here we show for the first time that a strong torsional force originates from the central surfactant aggregate that forms at the crossing between nonparallel nanotubes. Our dissipative particle dynamics simulations demonstrate that this torque pulls the tubes into a parallel arrangement. The torque increases strongly with decreasing angle between the tubes. This trend is due to the growth of the central aggregate which not only provides more molecules able to mediate the force but also increases the “lever arm” on which the force acts. Together with the strong surfactant-mediated attraction acting along the line of shortest contact between the tubes, the torsion increases the difficulty of nanotube dispersion, but could have a positive effect on carbon nanotube materials in which the adsorbed surfactant micelles are intended to bind the tubes together.

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INTRODUCTION Carbon nanotubes (CNT) have been in the focus of many scientists over more than two decades now due to their supreme mechanical and electronic properties. A huge range of technical applications has been proposed, ranging from the field of biomedicine,1,2 over sensors3,4 to composite materials.5,6 In the field of materials design CNTs are promising candidates for lightweight and yet strong materials. However, although carbon nanotubes can be produced to some centimeters in length nowadays,7 a macroscopic material would have to rely on a network of CNTs. Extrapolating the nanoscopic properties of a single nanotube seems tempting, but for a network of CNTs the overall mechanical properties are likely dominated by the joints. Due to the low interaction between the individual nanotubes, these joints are weak, thus reinforcing them is the prime objective for creating strong materials of CNTs. In the parallel arrangement, cohesion is present over the entire tube length. This leads to large CNT bundles that do not disperse in aqueous solution. Ultrasonication8 is usually employed to disrupt the bundles and disperse the CNTs. Without a stabilizing agent, e.g., surfactants,9,10 the CNTs would rebundle because of the van der Waals interaction between them.11−13 It has been shown analytically that this interaction creates a torque that turns the tubes into alignment for any angle between them.14,15 To prevent rebundling, a large variety of molecules has been tested, including SDS,16−20 SDBS,21,22 sodium cholate (SC),23,24 flavin mononucleotides (FMN),25 proteins,26 DNA,27 copolymers,28 etc. Because of the hydrophobicity of the nanotubes’ surface, the hydrophobic tail groups of the surfactants or copolymers adsorb onto the nanotubes, while the hydrophilic head groups point into the solvent. Repulsion between the head groups then keeps the nanotubes dispersed.29 © 2013 American Chemical Society

Strikingly, surfactants can also be utilized to bind nanotubes together as shown through atomistic30 and mesoscale simulations by Angelikopoulos et al.31,32 and Müter et al.33 In the latter simulations, surfactants adsorb predominantly onto the junction of two nanotubes and form a “central” aggregate. This central aggregate mediates an attractive force between the tubes which acts along the line of closest distance and over a large range of tube−tube separation distances. It was found that reduction of the angle between the tubes leads to growth of the central aggregate and an increase of the surfactant-mediated force.33 Here, we show that an additional surfactant-mediated force exists that creates a torque. This torque pulls the tubes into parallel alignment and is therefore important for carbon nanotube dispersion and for carbon nanotube materials. We present a detailed analysis of the distribution of this surfactant-mediated force along the nanotubes and investigate how the angle and tube−tube separation distance affect it. Furthermore, we discuss the link between the internal structure of the central aggregate and the forces it mediates between the tubes.

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MODEL AND SIMULATION The details of the model and the simulation have been laid out in our previous publications.34,35 Therefore, only the key elements will be discussed in this section. The surfactant molecules are modeled by a chain of five hydrophilic head beads (H5) and five hydrophobic tail beads (T5) to represent an amphiphilic molecule (H5T5). The bonds between two neighboring beads in the chain are simulated via a harmonic Received: December 12, 2012 Revised: April 11, 2013 Published: April 11, 2013 5585

dx.doi.org/10.1021/jp3122209 | J. Phys. Chem. B 2013, 117, 5585−5593

The Journal of Physical Chemistry B

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potential Φbond(r) = εbond(r − rbond)2 with rbond as the bond length and εbond as the strength parameter of the potential. Interactions between all beads in the system are modeled using the Lennard-Jones (LJ) (12,6) potential: ⎡⎛ σ ⎞12 ⎛ σ ⎞6 ⎤ ϕLJ(r ) = 4ε⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝r⎠ ⎦ ⎣⎝ r ⎠

For tail−tail interactions, the full potential is used while for all other combinations (tail−head and head−head) the potential is truncated at its minimum (rcut = 21/6σ) leading to the WCA (Weeks−Chanders−Andersen) potential where all forces are purely repulsive. At this level of coarse-graining, the carbon nanotubes represent smooth cylinders which interact with the beads via the LJ potential shifted to the surface of the tubes. Because of the hydrophobicity of carbon nanotubes, the full LJ potential is employed for interactions with hydrophobic beads, but is again truncated to the WCA potential for hydrophilic beads. For both cases, the strength of the interaction with the tubes is controlled by a multiplicative factor εCNT. The various parameters and settings used in this simulation are summarized in Table 1.

Figure 1. Snapshot of the system, produced with the VMD software package,41 along the z-axis for a tube−tube separation of d = 4.875σ and an in-plane angle of α = 45° between the tubes.

Here, we study the force between the tubes mediated by the surfactants. In vector notation, this force is given by the canonical ensemble average of the sum of all individual bead/tube forces:

Table 1. Parameters and Settings Used in This Worka surfactants tail beads = 5 rbond = 1.2σ nanotubes rCNT = 1.0σ simulation temp = 0.7ε/kB DPD ξ = 1.0 Nmolecules = 300, 1200 equilib = 3 × 107Δt a

head beads = 5 σ = 1.0 εbond = 4.0ε ε = 1.0 εCNT = 2.5ε d = 3.75−6.25σ Δt = 0.01 rcut(L-J) = 2.5σ DPD rcut = 2.5σ rcut(WCA) = 21/6σ box-z = 200σ box-x × box-y = 10000σ2 prod = 2 × 107Δt

N Nbeads

F = ⟨∑

∑

f(i , k)⟩

i=1 k=1

where the first sum runs over all molecules i and the second over all beads k in i. As will become clear below, it is most convenient to discuss the force acting on the upper nanotube. Because this tube is aligned with the y-direction in all simulations, and because the bead−tube interactions act along the line of closest distance, their y-component f y(i,k) is always zero. Consequently, Fy is always zero too. This leaves f x(i,k) and fz(i,k) the only nonzero force components. It is interesting to investigate the distribution of these forces along the upper nanotube. Therefore, we bin the bead−tube forces into segments of length Δ = 2σ along the tube, i.e., in the y-direction:

See text for further explanations.

We place the nanotubes into the lower part of the simulation box leaving the upper part as “bulk”. To study the dependence of the forces between the tubes on their mutual angle α in the x−y-plane (see Figure 1), α is set to a range of different values: 90° (perpendicular case), 67.5°, 45°, 22.5°, 9°, and 0° (parallel case). As the volume of the simulation box is to be kept constant for all angles, its extension in the x- and y-directions has to be adjusted to ensure continuation of the tubes into the next periodic image. The simulations are performed in the canonical ensemble (N,V,T). The resulting bulk surfactant concentration is approximately 0.3 times the critical micellar concentration (Ccmc is 5.2 × 10−5σ−3) for all simulations if not stated otherwise. We use the dissipative particle dynamics approach (DPD) to control the temperature. The theoretical background of DPD can be found in ref 36. A molecule is considered “adsorbed” if itself or a molecule in the same cluster is less than 1.5 σ away from the nanotube surface. If a tail bead of one molecule is less than 1.5 σ away from at least one tail bead of another molecule, these two molecules are regarded as part of the same cluster. Finally, any cluster that is adsorbed on both nanotubes simultaneously is considered a central aggregate. We find that, on angled tubes, the central aggregate is always constituted by only one cluster. On parallel tubes, the term “central aggregate” may be confusing because more than one aggregate can exist that is adsorbed on both tubes.

N

Nbeads

F(y) = ⟨∑

∑

i=1

k=1

f(i , k)⟩;

y − Δ/2 < y(i , k) ≤ y + Δ/2

The number of segments is chosen here as a trade-off between a detailed distribution and good statistics. We also determine the histogram of directly adsorbed hydrophobic beads along the tube to connect the forces acting on the tube to the actual spatial arrangement of the surfactants. The one-dimensional histogram of adsorbed beads is defined as Nads

n(y) = ⟨∑ 1⟩;

y − Δ/2 < y(k) ≤ y + Δ/2

kads

where the sum runs over all directly adsorbed hydrophobic beads (kads), i.e., all hydrophobic beads that are less than 1.5σ away from the surface, in the system.

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RESULTS AND DISCUSSION In aqueous solution, surfactant molecules adsorb onto carbon nanotubes and form aggregates. If the tubes are crossed and not 5586



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Figure 2. Force profiles along the y-axis for the upper nanotube at tube−tube separation distances of 3.75σ (a,b), 5.125σ (c,d), and 6.25σ (e,f) for selected angles as indicated in the figure. The x-component of the force, Fx(y) (a,c,e), and z-component of the force, Fz(y) (b,d,f), are shown. The gray-shaded area indicates negative values. Note that this corresponds to attractive forces for Fz(y) and Fx(y) at y < 0, but repulsive Fx(y) at y > 0.

too far apart, a central aggregate forms at the crossing. This is demonstrated by the simulation snapshot shown in Figure 1. The molecules that form this central aggregate interact with the tubes and cause forces between them. To our knowledge, in all simulation work published until now, only the z-component of this mediated force, i.e., the force along the line of shortest distance between the tubes, has been considered (e.g., refs 33, 35, and 37). This force acts on the distance between the tubes and is of great relevance. However, for nonparallel tubes an

equally important torsional component appears that so far has been ignored. Here, we show that the surfactant-mediated forces indeed lead to a torque parallel to the line of shortest distance. This torque acts on the alignment between the tubes; i.e., it tries to rotate them around the line of shortest distance. Therefore, this component of the surfactant-mediated interaction is of great importance for carbon nanotube materials and for carbon nanotube dispersion. 5587



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The torque is caused by forces that act tangential with respect to a rotation around the line of shortest distance. For the upper tube shown in Figure 1 this is the x-direction. For simplicity, we will only consider this tube. From now on we will call this tangential force the x-component of the force. Since this force leads to a torque, its dependence on the position with respect to the pivot axis needs to be considered. Because the pivot axis is the z-axis and the tube axis is parallel to the y-axis, Fx(y) provides this position dependence. In Figure 2, the x- and z-component of the force profile, Fx(y) and Fz(y), are shown for three tube−tube separation distances (3.75σ, 5.125σ, and 6.25σ) and five angles (90°, 45°, 22.5°, 9°, and 0°). It is first of all noted that all Fx(y) profiles (left column) are antisymmetric with respect to the pivot axis at y = 0 while all Fz(y) profiles (right column) are symmetric. This difference in the force profiles correlates with the symmetry of the nanotube arrangement. In the x−y-plane, the lower tube is antisymmetric with respects to y = 0 while it is symmetric in the y−z-plane (Figure 1). In the cases of parallel and perpendicular tubes the symmetry of the system is higher. In these cases, the axis of the tube under consideration (the upper in Figure 1) lies in a mirror plane parallel to the y−z-plane that divides the system into two symmetrical half-spaces. This means that, on average at any point y along the tube, the x-component of the forces on this tube originating from molecules in one-half-space are compensated by the forces originating from the other half-space. Thus, we expect Fx(y) = 0 for all y for the parallel and perpendicular arrangements. The simulations for these cases are consistent with this conclusion. For simplicity we omit their results in Figure 2. Both, the x- and z-components of the force on the tube are nonzero only in a small region around the center of the tube− tube crossing. This region increases in size with decreasing angle, which has been attributed to the growth of the central aggregate.33 In order to investigate the origin of Fx(y), we have plotted a histogram of the number of hydrophobic beads adsorbed on the upper nanotube in Figure 3. The high number of adsorbed beads around y = 0 indicates the extent of the central aggregate which forms around the crossing and stays there permanently. All other adsorbed aggregates can move freely along the tubes leading to a uniform density that only depends on the amount adsorbed, which in turn depends on the bulk surfactant concentration. Adsorbed aggregates that approach the central aggregate are repelled by the steric repulsion between the outward facing hydrophilic head groups. This results in a depletion of hydrophobic beads around the central aggregate. Instead, we find a higher number of head groups in this region (not shown). Thus, the two symmetric dips in the local number of adsorbed hydrophobic beads serve as a good indicator for the extent of the hydrophobic core of the central aggregate. It can be seen from Figure 3 that the central aggregate grows with decreasing angle. Comparing the extent of the central aggregate along the tube with the forces in Figure 2, we conclude that at low concentrations both the x- and z-components of the force solely originate from the central aggregate. Recalling the definition of the central aggregate, i.e., the aggregate that is adsorbed on both tubes, this conclusion also holds for the parallel case, because more than one aggregate can be adsorbed on both tubes. Thus, in the parallel case we have an infinite number of central aggregates that are contributing to the forces. Therefore, in the parallel case the forces must always

Figure 3. Profiles of the number of adsorbed hydrophobic beads on the upper nanotube for selected angles and a tube−tube separation distance of d = 5.875σ. For all angled cases the central aggregate is localized between the two deep density drops around the central high density region (see text).

be normalized by the tube length. Furthermore, in the parallel case, adsorbed aggregates are not restricted in their movement along the tubes which leads to the uniform density seen in Figure 3. Accordingly, the force on the tube is also uniformly distributed (Figure 2). It is important to point out that in the concentration range (0.3 times the critical micellar concentration) used for the simulations discussed above, the central aggregate is well developed and that there are micelles adsorbed along the tubes (see Figure 1 for an example). The average number of hydrophobic beads and therefore the number of molecules adsorbed per tube length differs for the different angles due to the necessary changes of the shape of the simulation box which changes the length of the tubes in the simulation box. However, in all cases the average number of adsorbed beads outside the central aggregate is larger than zero (Figure 3). This shows that there are other adsorbed aggregates besides the central aggregate. These aggregates do not contribute to the mediated forces as the forces in Figure 2 are only nonzero in the region that is covered by the central aggregate. We will return to this point to see if increased adsorption at very high concentrations can change this behavior. It is interesting to investigate the role the two species of beads, i.e., hydrophilic head beads and hydrophobic tail beads, play in creating the forces. In Figure 4, the contributions from hydrophilic head beads (a,b) and hydrophobic tail beads (c,d) to Fx(y) and Fz(y) are plotted for a tube−tube distance of d = 5.125σ. We immediately see that the contribution of the hydrophobic tail beads is about a factor of 10 larger than the one from the hydrophilic head beads. This may be expected as only the hydrophobic beads have an attractive interaction with the tubes and dominate the contact between the central aggregate and the tubes. It is surprising that the hydrophilic heads directly contribute to the forces at all. Inside the central aggregate this head-groupmediated force in the x-direction is attractive, while just outside it is repulsive. The same inversion of the force direction is observed for the z-component. The repulsive region originates 5588



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Figure 4. Force on the upper nanotube at a tube−tube distance of d = 5.125σ separated into contributions from the hydrophilic head groups (a,b) and hydrophobic tails (c,d). The x-component is shown on the left (a,c) while the z-component (b,d) is plotted on the right. Clearly visible is a small direct contribution from the head groups which changes sign at the border of the central aggregate. The sum of the contributions of both the heads and tails leads to panels (c) and (d) in Figure 2. The gray-shaded area as in Figure 2.

crossing is depicted for thin slices through the system parallel to the x−y- and y−z-planes. As indicated by the markings in the figure, due to the strong adsorption the tail beads form pronounced layers around the surfaces of the nanotubes. On top of the surface layer, a second and then a third layer can be seen. In three dimensions these layers bend concentrically around the nanotubes. The bead density reaches particularly high values where the layers from the two nanotubes intersect. Thus, the 3-dimensional structure of the hydrophobic core of the central aggregate is best understood as an interference pattern generated by superposition of the density layers formed around the two tubes. To analyze the force−structure correlations, we calculate the force profile F*x(y) and the local density of tail beads in a thin slab through the center of the crossing and parallel to the x−yplane, i.e., for beads with a z-coordinate between −0.25σ and 0.25σ (Figure 6). The definition of the force profile F*x(y) is identical to that of Fx(y), but F*x(y) is calculated only from a subset of the hydrophobic beads forming the central aggregate. Therefore, Fx(y) and F*x(y) are expected to be different, but F*x(y) is much better suited to discuss the correlation between density and force. As highlighted in Figure 6a, oscillations in Fx(y) correlate with the oscillations in the density close to the nanotubes surface.

from head groups of molecules belonging to the central aggregate that are confined between the tubes. Since the interaction of the head groups with each other and with the tubes is purely repulsive, an attractive mediated tube−tube force must originate from head groups “on top” of the tube−tube junction. There must be a hole that exists at least temporarily and allows the head groups to interact directly with the tubes. We frequently observe these holesindeed such a conformation is shown in Figure 1. The other striking feature of Fx(y) (and also Fz(y)) are strong oscillations. As the results in Figure 4 show, these oscillations originate solely from the hydrophobic tail beads. Oscillating forces are commonly observed in simulations38,39 and experiments40 for confined thin layers of strongly adsorbing molecules. Both conditions, i.e., strong adsorption and confinement, are fulfilled by the tail beads that are adsorbed near the crossing of the nanotubes. The strong oscillations in Fx(y) are caused by intense ordering in all three spatial dimensions. This ordering is akin to layering found in simpler systems,38−40 but because of the strong curvature of the tubes and their mutually angled orientation, ordering of the tail beads in the hydrophobic core of the central aggregate is much more complicated. In Figure 5 the density of the hydrophobic beads around the nanotube 5589



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Figure 5. Local density of hydrophobic tail beads around the nanotube crossing in the y−z-plane for a perpendicular case (a) and in the x−y-plane for an angled case (b). The local densities are calculated in a 0.5σ-thick slab running trough the center of the nanotube crossing. The adsorbed layers on both tubes are marked in different colors. The superposition of the layers forms an interference pattern causing the bead density to be particularly high where two layers intersect. Note that, to show the structure of the angled case in part (b) more clearly, the distance between the tubes was set to zero, i.e., the tube axes cross.

Figure 6. Local density of hydrophobic tail groups around the nanotube crossing in the x−y-plane for an angle of 45° and a separation distance of 0σ (as in Figure 5b). The inset in both parts of the figure shows the x-component of the force F*x(y) exerted by the hydrophobic beads on the tube that is aligned with the y-axis (see figure). The local density and the force profile are plotted on the same scale to identify structure−force correlations. Both density and force plots refer to a 0.5σ thick slab which is centered: (a) at the center of the crossing of the nanotubes and (b) 2σ below the crossing. In part (a) the force plot “replaces” the tube parallel to the y-direction in the local density plot. In part (b) the local density plot is cut along the axis of this tube. It can easily be seen (green markings) that the oscillations in the force originate from the region between the tubes and correlate with the occurrence of local maxima and minima in the density. An attractive contribution to the force originates near the hydrophobic−hydrophilic interface of the aggregate. This force pulls the tubes toward parallel alignment.

Over a high-density region, the force switches from repulsive to attractive. In their equilibrium position, beads adsorbed in the wedges formed by the tubes do not exert a force on the tubes. However, due to thermal motion beads move along the tubes. If they move further into the wedge, they move into greater confinement, leading to a repulsive force on the confining tubes which pushes them apart. In the opposite direction, beads move into lesser confinement resulting in an attractive force on the tubes. Thus, the repulsive branch of the

oscillation is always located at greater confinement. For larger values of y the density maxima become less pronounced because the distance to the tube that causes the lateral ordering is greater. As structural order decreases, so do the force oscillations. An interesting and important change occurs if the slab is moved by −2σ in the z-direction (Figure 6b). It is now centered 2σ below the crossing. The key feature of this force profile is that the forces are almost exclusively attractive 5590



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(negative for y < 0 and positive for y > 0); i.e., these forces pull the tubes together. Thus, the attractive character of Fx(y) in Figures 2 and 4, which is particularly visible for larger y where the oscillations have decayed, is generated near the hydrophobic−hydrophilic interface of the central aggregate. The origin of this attractive force is found in the interplay between aggregation and adsorption. While a bulk micelle of the surfactant has a spherical shape, the adsorbed central aggregate is deformed to a more “rectangular” shape as the surfactant molecules try to cover more of the nanotubes’ surface. This deforms the aggregate which results in a restoring force. This restoring force acts on the tubes and tries to pull them into a position where adsorption can be maximized at minimal deformation. If the position of the tubes is fixed at the pivot point (or pivot axis for d > 0), which is the case when we consider Fx(y) only, the most favorable configuration is the parallel tube arrangement, because in this case adsorption on both tubes stretches the aggregate in the same direction. As the restoring force results from stretching and deformation of the hydrophobic−hydrophilic interface, it is expected that the largest impact on Fx(y) originates near this interface. The attractive contribution to Fx(y) has important consequences for the torque it creates around the pivot axis y = 0. The torque results from the antisymmetric nature of Fx(y) with respect to y = 0 (Figures 2 and 4). For a force F acting at a position r away from the pivot axis the torque can be written as

Figure 7. z-Component of the torque acting on the upper nanotube as a function of the angle between the tubes for selected separation distances as indicated in the figure. The torque increases strongly with decreasing angle and pulls the tubes into the parallel arrangement.

one moves away from the pivot axis, it becomes more and more attractive (as discussed above) until it vanishes where the central aggregate ends. Because of the longer lever arm length y, the attractive regions of Fx(y) dominate the torque and cause it to be always negative. Since adsorption depends strongly on the bulk concentration, we investigate if such a concentration dependence also exists for the torque. In Figure 8, Fx(y) and Fz(y) are plotted

M=r×F

By use of the force profiles Fx(y) and Fz(y), the individual components of the torque can be obtained y

Mx =

∑ ry(y)·Fz(y) − ∑ rz(y)·Fy(y), −y y

My =

−y y

∑ rx(y)·Fz(y) − ∑ rz(y)·Fx(y), −y y

Mz =

y

and

−y y

∑ rx(y)·Fy(y) − ∑ ry(y)·Fx(y) −y

−y

where the ri(y) are the components of r(y). The complexity of these equations is greatly reduced by taking the symmetry of the system into account. First, the upper nanotube only extends into the y-direction (Figure 1), leaving ry(y) = y as the only nonzero component of the vector r for all y. Second, Fz(y) is an even function, while ry(y) is odd. Thus, ∑y−yry(y)·Fz(y) = 0. This leaves only one nonzero torque component: y

Mz =

∑ Fx(y)·y

Figure 8. Profiles of the x- and z-components of the force acting on the upper nanotube for an angle of 22.5° and a separation distance of d = 5.125σ and the corresponding profile of the average number of adsorbed beads at a high bulk concentration, C = 5 × cmc. For better resolution, the bin size has been reduced to Δ = 1σ. Note that the absolute number of beads is plotted. The vertical lines mark the extension of the central aggregate. While the adsorbed micelles engulf the whole nanotube, the central aggregate sits essentially between the tubes leading to the lower number of adsorbed hydrophobic beads. Both force components have contributions from adsorbed micelles just outside the central aggregate which oppose the force exerted by the central aggregate. Although the contributions are small, this leads to a considerable countertorque because of the longer lever arm.

−y

This component is nonzero for all cases except for the parallel and perpendicular arrangement because in these special cases Fz(y) = 0 ∀ y due to symmetry. In Figure 7 this component of the torque Mz is plotted as a function of the angle between the tubes. For all cases studied here the torque is negative; i.e., it rotates the tubes toward each other into alignment and it increases strongly with decreasing angle. On first sight this may be surprising that the torque is always negative since Fx(y) shows strong oscillations (Figure 2). Intriguingly, for the z-component the oscillations in Fz(y) indeed lead to oscillations in the total force Fz = ∑y−yFz(y) (Supporting Information, Figure S1). While the oscillations in Fx(y) decay as

together with the average number of adsorbed beads n(y) on the upper nanotube at a very high concentration of C = 5Ccmc 5591



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(corresponding to N = 1200 molecules). The angle between the tubes is 22.5° and the separation distance is 5.125σ. At this high concentration, the tubes are completely covered as indicated by the high values of n(y) for all y. The strong oscillations in the density demonstrate that there are individual micelles adsorbed outside the central aggregate which are almost immobile because of the high coverage and the immobility of the central aggregate. In contrast to the simulations discussed above, there is a region (20σ < d < 45σ) outside the central aggregate where small forces Fx(y) and Fz(y) can be found. The x-component, Fx(y) produces a torque that counteracts the torque of the central aggregate. It is produced by micelles adsorbed right next to the central aggregate and on different tubes so that their hydrophilic groups interact repulsively via the gap between the tubes. Although these forces are small, the long lever arm causes them to contribute a considerable torque. In the present case, this contribution compensates more than 30% of the central aggregate’s torque. While this is a very large contribution, the central aggregate still dominates the torque. A rough estimate of the torque for decreasing angle α suggests that there is no crossover at any angle, i.e., there appears to be no α where the counter torque from the steric repulsion compensates the torque originating from the central aggregate (Appendix A). Thus, as soon as a central aggregate is formed, it will always facilitate rebundling via coalescence and alignment of carbon nanotubes even at high concentrations.

central aggregate, which means that the torque remains negative for all tube−tube angles. However, this might be different for a different surfactant. For the purpose of building a network of CNTs, the tendency toward alignment of the nanotubes caused by the torque might seem discouraging, but the internal stresses this would cause have the potential to improve material properties. Our results also have an impact on the development of improved dispersion systems for carbon nanotubes. While surfactants may stabilize an already existing dispersion, our results show that as soon as the tubes come close enough, the formation of a central aggregate will facilitate rebundling by attraction (Fz) and alignment (Mz). Consequently, surfactants that can form such central aggregates should be avoided. The findings presented in this work complete the description for the forces occurring in surfactant stabilized carbon nanotube networks. This description can now be utilized to simulate such networks on a larger scale with the goal of assessing their macroscopic properties and to establish the feasibility of their application.

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APPENDIX A Two distinct regions in the system at high concentration contribute to the total torque. For all systems studied here, the torque generated by the central aggregate rotates the tubes into alignment. For fully covered tubes, a region adjacent to the central aggregate exists where steric repulsion between the adsorbed surfactant aggregates causes a torque that opposes alignment. Since the simulation of systems with very small angles is challenging, because of the resulting large system sizes, we estimate the α-dependence of the torque for small α. The requirement of small angles is necessary because the discrete nature of the adsorbed surfactant micelles and ordering effects near the central aggregates make some of the assumptions below unsuitable for large angles. The region where we expect steric repulsion is limited by a distance, b, between the tubes beyond which adsorbed surfactant molecules can no longer interact. The lower boundary is the distance, a, at which the central aggregate starts. It is reasonable to assume that the distance between the tubes, which the central aggregate can span, is limited. Indeed, we find for this distance 8σ < a < 9σ. The torque generated by the central aggregate can be written as

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CONCLUSION We have examined the spatial distribution of the surfactantmediated force between two carbon nanotubes and its dependence on the angle and distance between the nanotubes using the dissipative particle dynamics approach. For all angles between the tubes other than 0° (parallel tubes) and 90° (perpendicular tubes), we find torsional forces Fx(y) along the tubes. At low surfactant bulk concentrations, these forces are solely caused by the central aggregate that is permanently adsorbed at the tube−tube crossing. These forces oscillate strongly as one moves along the tubes away from the center and show an overall attractiveness. The oscillations are caused by strong ordering inside the hydrophobic core of the central aggregate while the attractiveness is a result of a restoring force opposing the stretching of the aggregate, which itself is caused by the interplay between aggregation and adsorption. As a consequence of the attractiveness of the force, the resulting torque is always negative; i.e., it always turns the tubes into alignment. The torque increases very strongly with decreasing angle between the tubes as a result of the size increase of the central aggregate. This trend is similar but much stronger compared to the angle dependence of Fz reported in ref 33. We have analyzed the force contributions from the hydrophilic head groups and hydrophobic tail groups individually and found that, although most of the attraction between the tubes is caused by the tail groups, the head groups can directly mediate an attraction too, despite interacting solely repulsively with the tubes. Large bulk surfactant concentrations cause the tubes to be fully covered in micelles. At small angles the head groups of the micelles adsorbed adjacent to the central aggregate may come close enough to interact with each other. This interaction is repulsive and weak. However, because the repelling micelles are located so far away from the pivot axis, the repulsion makes a large contribution to the torque. Nevertheless, we estimate that this contribution never compensates the attraction from the

M1 = 2

∫0

y1

yFx ,1(y) dy

where y1 is the length of the central aggregate along the tubes. Assuming that Fx,1(y) can be represented by its average Fx,1, we obtain for the torque M1 ≈ 2Fx ,1

y1

∫0

y dy = Fx ,1 y12

To calculate y1 we assume that the distance between the tubes d = 0. This is a simplification for the sake of the argument, but will not change the key result. We obtain a y1 = α 2 sin 2

()

Analogously we find M 2 ≈ 2Fx ,2

∫y

1

5592

y2

dy y =

1 Fx ,2(y2 2 − y12 ) 2



Article

(10) Coleman, J. N. Adv. Funct. Mater. 2009, 19, 3680−3695. (11) Sun, C.; Yin, L.; Li, F.; Lu, G.; Cheng, H. Chem. Phys. Lett. 2005, 403, 343−346. (12) Girifalco, L.; Hodak, M.; Lee, R. Phys. Rev. B 2000, 62, 13104− 13110. (13) Ke, C.; Zheng, M.; Zhou, G.; Cui, W.; Pugno, N.; Miles, R. N. Small 2010, 6, 438−445. (14) Mitchell, D. J.; Ninham, B. W. J. Chem. Phys. 1973, 59, 1246− 1252. (15) Mitchell, D. J.; Ninham, B. W.; Richmond, P. Biophys. J. 1973, 13, 359−369. (16) Yurekli, K.; Mitchell, C. A.; Krishnamoorti, R. J. Am. Chem. Soc. 2004, 126, 9902−9903. (17) Uddin, N. M.; Capaldi, F. M.; Farouk, B. Polymer 2011, 52, 288−296. (18) Vichchulada, P.; Cauble, M. A.; Abdi, E. A.; Obi, E. I.; Zhang, Q.; Lay, M. D. J. Phys. Chem. C 2010, 114, 12490−12495. (19) Cardenas, J. F.; Gromov, A. Nanotechnology 2009, 20, 465703. (20) Eda, G.; Fanchini, G.; Kanwal, A.; Chhowalla, M. J. Appl. Phys. 2008, 103, 093118. (21) McDonald, T. J.; Engtrakul, C.; Jones, M.; Rumbles, G.; Heben, M. J. J. Phys. Chem. B 2006, 110, 25339−25346. (22) Angelikopoulos, P.; Gromoy, A.; Leen, A.; Nerushev, O.; Bock, H.; Campbell, E. E. B. J. Phys. Chem. C 2010, 114, 2−9. (23) Coello, A.; Meijide, F.; Rodriguez Nunez, E.; Vazquez Tato, J. J. Phys. Chem. 1993, 97, 10186−10191. (24) Lin, S.; Blankschtein, D. J. Phys. Chem. B 2010, 114, 15616− 15625. (25) Ju, S.-Y.; Doll, J.; Sharma, I.; Papadimitrakopoulos, F. Nature Nanotechnol. 2008, 3, 356−362. (26) Kurppa, K.; Jiang, H.; Szilvay, G. R.; Nasibulin, A. G.; Kauppinen, E. I.; Linder, M. B. Angew. Chem., Int. Ed. 2007, 46, 6446− 6449. (27) Di Crescenzo, A.; Aschi, M.; Fontana, A. Macromolecules 2012, 45, 8043−8050. (28) Zheng, M.; Jagota, A.; Semke, E. D.; Diner, B. A.; McLean, R. S.; Lustig, S. R.; Richardson, R. E.; Tassi, N. G. Nat. Mater. 2003, 2, 338− 342. (29) Matarredona, O.; Rhoads, H.; Li, Z.; Harwell, J.; Balzano, L.; Resasco, D. J. Phys. Chem. B 2003, 107, 13357−13367. (30) Xu, Z.; Yang, X.; Yang, Z. Nano Lett. 2010, 10, 985−991. (31) Angelikopoulos, P.; Bock, H. Phys. Chem. Chem. Phys. 2012, 14, 9546−9557. (32) Angelikopoulos, P.; Bock, H. J. Phys. Chem. Lett. 2011, 2, 139− 144. (33) Müter, D.; Angelikopoulos, P.; Bock, H. J. Phys. Chem. B 2012, 116, 14869−14875. (34) Angelikopoulos, P.; Bock, H. J. Phys. Chem. B 2008, 112, 13793−13801. (35) Angelikopoulos, P.; Schou, K.; Bock, H. Langmuir 2010, 26, 18874−18883. (36) Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 107, 4423− 4435. (37) Angelikopoulos, P.; Al Harthy, S.; Bock, H. J. Phys. Chem. B 2009, 113, 13817−13824. (38) Schoen, M.; Paris, O.; Günther, G.; Müter, D.; Prass, J.; Fratzl, P. Phys. Chem. Chem. Phys. 2010, 12, 11267−11279. (39) Günther, G.; Schoen, M. Phys. Chem. Chem. Phys. 2009, 11, 9082−9092. (40) Klapp, S. H. L.; Qu, D.; von Klitzing, R. J. Phys. Chem. B 2007, 111, 1296−1303. (41) Humphrey, W.; Dalke, A.; Schalke, K. J. Mol. Graphics 1996, 14.1, 33−38.

and

y2 =

b

( α2 )

2 sin

With these results, we find for the total torque M = M1 + M 2 ≈

⎡⎛ b ⎞ 2 ⎤⎫ 1⎧ a2 ⎨Fx ,1 + Fx ,2⎢⎜ ⎟ − 1⎥⎬ 2 2⎩ ⎣⎝ a ⎠ ⎦⎭ 4⎡sin α ⎤ ⎣ 2 ⎦ ⎪

⎪

⎪

⎪

()

(1)

Thus ⎡ ⎛ α ⎞⎤ –2 M ∝ ⎢sin⎜ ⎟⎥ ⎣ ⎝ 2 ⎠⎦

(2)

As α decreases, the rhs of eq 2 monotonously increases until it diverges. Most importantly, it never changes its sign. Thus, there is no indication that, for the systems studied here, at smaller angles between the tubes steric repulsion will compensate the aligning torque of the central aggregate. However, it is quite possible that the prefactor in curly brackets in eq 1 becomes negative for strong and long-range repulsion, i.e., large and negative Fx,2 and large b. Such a case would be very interesting.

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ASSOCIATED CONTENT

S Supporting Information *

The z-component of the force acting on the upper nanotube for different angles and tube−tube separation distances (data from ref 27). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +44 (0)131 451 3074. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.M. is grateful for financial support for this project through the German Science Foundation’s (DFG) research stipend MU 3236/1 “Multiscale modelling of surfactant reinforced carbon nanotube networks”.

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REFERENCES

(1) Cveticanin, J.; Joksic, G.; Leskovac, A.; Petrovic, S.; Sobot, A. V.; Neskovic, O. Nanotechnology 2010, 21, 015102. (2) Mahmood, M.; et al. Nanomedicine 2009, 4, 883−893. (3) Capek, I. Adv. Colloid Interface Sci. 2009, 150, 63−89. (4) Cheung, W.; Chiu, P. L.; Parajuli, R. R.; Ma, Y.; Ali, S. R.; He, H. J. Mater. Chem. 2009, 19, 6465−6480. (5) Blighe, F. M.; Young, K.; Vilatela, J. J.; Windle, A. H.; Kinloch, I. A.; Deng, L.; Young, R. J.; Coleman, J. N. Adv. Funct. Mater. 2011, 21, 364−371. (6) Young, K.; Blighe, F. M.; Vilatela, J. J.; Windle, A. H.; Kinloch, I. A.; Deng, L.; Young, R. J.; Coleman, J. N. ACS Nano 2010, 4, 6989− 6997. (7) Wang, X.; Li, Q.; Xie, J.; Jin, Z.; Wang, J.; Li, Y.; Jiang, K.; Fan, S. Nano Lett. 2009, 9, 3137−3141. (8) Giordani, S.; Bergin, S.; Nicolosi, V.; Lebedkin, S.; Blau, W. J.; Coleman, J. N. Phys. Status Solidi B 2006, 243, 3058−3062. (9) Wang, H.; Zhou, W.; Ho, D. L.; Winey, K. I.; Fischer, J. E.; Glinka, C. J.; Hobbie, E. K. Nano Lett. 2004, 4, 1789−1793. 5593


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