When being mixed in a solution, filler aggregates are subjected to shear stresses imparted from the medium (e.g., solvent or polymer melt). Therefore, the flow of the medium in response to an external force (e.g., through the rotation of a mixer blade, or cavitation in ultrasonication) generates the local shear stresses that are ultimately responsible for dispersion. A mixing process can be interpreted as the delivery of a mechanical energy into the solution to separate the aggregates. The opposing factor against separation is the binding energy which holds the aggregates together. Considering the above two factors, one can establish the mixing criteria for effective aggregate separation as: the supplied energy (or, to be more precise, the local energy density) from the chosen mixing technique to be greater than the binding energy of the CNT aggregates (to be more precise, the energy per local volume of the contact). On the other hand, to retain the morphology of individual CNTs, the supplied energy should also be lower than the amount required to fracture a nanotube. Hence, an ideal aggregate separation technique should supply an energy density between the binding energy of the aggregates (lower limit), and the fracture resistance of individual nanotubes (upper limit). In the following sections, the relative energy scale associated with each of the three factors will be evaluated. We will also elucidate how one can predict the likely morphology and microstructure of the CNTs-polymer composite resulted from a particular mixing technique.

#### 2.1. Binding Energies Holding the Nanotube Aggregates

We will first evaluate the binding forces associated with carbon nanotube network and clusters. Real forms of CNTs vary significantly depending on their synthesis methods. They have as-produced lengths ranging from hundreds of nanometers to tens of microns, and often contain defects in their walls, hence are not completely straight over their contours. Furthermore, they are rarely present in isolated entities but instead are found in bundles or clusters.

Figure 1 shows two typical examples of CNTs of different morphologies: (a) illustrates the arc-discharged single-wall nanotubes (SWNTs), where each of the fine strands consists of tens or even hundreds of micron-long SWNTs; and (b) shows the common commercially available multi-wall nanotube (MWNT) clusters. Although CNTs see a large variation in their physical attributes across different types, the interaction forces between pairs of nanotubes can be estimated through theoretical models to be illustrated below.

Using a continuum model by integration over the tubular surfaces of two parallel (10,10) SWNT, Girifalco

et al. [

11] calculated the cohesion energy per unit length between the SWNT pair to be

ca. −0.095 eV/Å; the associated energy is −0.36 eV/Å for a bundle formed by the same SWNTs. The equilibrium distance between the tube axes was calculated to be

ca. 18 Å, or 2.5 times the radius of the (10,10) SWNT. Since the pair cohesion energy per unit length is much greater in magnitude than the room temperature thermal energy (

ca. 0.025 eV), ‘unzipping’ of a SWNT from an existing bundle is unlikely, despite of the fact that tubes in a separated state have higher configurational entropy. The computer simulation [

11] also provides a solution to the equilibrium spacing (

H_{c}) of nanotube sections in close proximity. This spacing (outer-wall to outer-wall) is weakly dependent on the diameter of nanotubes; and has a value of 3.14 Å, similar to the inter-plane spacing of graphite.

One can generalize the analysis of van der Waals (VdW) attraction by introducing the relevant Hamaker constant, A

_{H} [

12]. By modeling each CNT as a meso-scale rod continuum, one can determine the interaction energy associated with different pair configurations through classical solutions (these can be found in text books, like [

13]). Hamaker constant is experimentally shown to be

ca. 60 × 10

^{−20} J (3.8 eV) for the van der Waals attraction between an outer-wall of the MWNT and the metallic surface at a vacuum of 10

^{−3} Pa. One would expect the A

_{H} between the side walls of CNTs to be slightly lower, but not much different from the value reported above. With the knowledge of A

_{H}, one can determine the van der Waals energy between a pair of parallel CNTs. This is analogous to the solution which describes the van der Waals interaction energy between two parallel mesoscopic cylinders of length

L, diameter

d = 2r, separated by a gap H is:

for

(1)

Taking A

_{H} = 3.8 eV and

H_{c} = 3.4 Å to estimate the V

_{//} per unit length, for the (10,10) SWNT (diameter of 1.36 nm), Equation (1) yields an attractive potential V

_{//} /L ~ −0.09 eV/Å. This matches the pair interaction value V

_{//} /L ~ −0.095 eV/Å calculated in [

11].

It is noted that, the van der Waals effect alone is not sufficient to account for the different bundling and clustering morphologies present. Analyzing Equation (1), one can note that V_{//} increases with diameter, which predicts that thick nanotubes should pack more favorably in a parallel configuration to maximize the van der Waals interaction. In practice, the opposite is found to be the case, where SWNTs and double-walled nanotubes exist in tightly packed parallel bundles, whereas thick multi-walled nanotubes (MWNTs) usually exist in clusters with a crossed mesh configuration. To form tight bundles, the neighboring nanotubes have to be in close proximity (<1 nm spacing) to each other, and align parallel with respect to the bundle axis. This means that nanotubes in a bundle will need to ‘adjust’ their conformations to confront the packing requirements co-operatively. Inevitably, there is an associated strain energy of deforming (bending) a nanotube to follow the contour of its neighbors; the magnitude of this strain energy scales with the tube diameter to a fourth power, within a classical model of beam bending. Hence for MWNTs, the energy cost resulted from bending much exceeds the energy savings in forming parallel bundles. Although most MWNTs are not closely packed in order to retain their original contour shapes to minimize the strain energy, they are formed into networks and are subjected to van der Waals binding at the contact junctions. The energy associated with each contact can be modeled by two perpendicularly crossed rods of radius r and gap size H:

for

(2)

For a tube diameter of 100 Å (10nm), spaced by 3.4 Å (0.34 nm),

~ −10 eV per contact is obtained. This value is lower than, but comparable to what was calculated in [

14],

~ −15 eV per contact, which was calculated based on the surface integral of graphene layers for a similar rolled configuration. The esults from [

14] also showed a linear dependence of

on the diameter

d for all MWNTs. From the

estimate, we can see that the energy associated with each contact is orders of magnitude greater than the room-temperature thermal energy. The number of contacts which can form between neighboring CNT pairs increase dramatically with the length of nanotubes. Therefore, the binding of a clustered network of long MWNTs is extremely strong, where each contact acts effectively as a physical cross-link fixing the network.

**Figure 1.**
Scanning electron micrographs showing: (**a**) Single-wall nanotubes (SWNTs), grown by arc-discharge, with inset showing a bundle of closely-packed SWNTs; (**b**) multi-walled nanotubes (MWNTs), grown by chemical vapor deposition. The van der Waals (VdW) interaction between a pair of SWNTs or a pair of MWNTs is modeled by a pair of rods with parallel and perpendicular configuration respectively.

**Figure 1.**
Scanning electron micrographs showing: (**a**) Single-wall nanotubes (SWNTs), grown by arc-discharge, with inset showing a bundle of closely-packed SWNTs; (**b**) multi-walled nanotubes (MWNTs), grown by chemical vapor deposition. The van der Waals (VdW) interaction between a pair of SWNTs or a pair of MWNTs is modeled by a pair of rods with parallel and perpendicular configuration respectively.

To convert the values of these ‘contact’ energy into the more macroscopically meaningful quantity of energy density (ε), the following analysis is applied. For SWNTs, the energy density ε required to separate a pair of parallel tubes of length

L and diameter

d, bound by an attractive potential energy V

_{tot}, is

ε ~ |V_{tot}|/(Ld)^{2}. Using L = 1 μm, d = 1 nm and V

_{tot} ~V

_{// }~ −10

^{3} eV (here, the van der Waals binding energy per unit length is taken as −0.095 eV/Å), ε ~ 100 MPa is obtained. For MWNTs networks, one needs to have additional information about the number density of nanotube contacts in the network,

c_{t} (in units of number per cubic meter). Then, the energy density required to separate the raw MWNT network is

ε ~ c_{t}|V_{tot}|. One can estimate

c_{t} by looking at the spacing ζ between the direct crossing junctions of the neighboring nanotubes visible on the surface of a cluster (e.g., in a scanning electron micrograph). Then,

c_{t} ~ 1/ ζ

^{3}. Taking the spacing between MWNT junctions to be ζ ~ 100 nm (estimated from the SEM image of MWNTs with diameters of 60–100 nm in

Figure 1 of reference [

15]) and

~ −100 eV [

14] for a 80 nm diameter MWNT, we obtain

ε ~ 16 kPa. The above two examples demonstrate the vastly different energy levels, with many orders of magnitude difference, required to separate the CNTs of various as-grown morphologies.

#### 2.2. Energy Density Delivered from Mixing

Separation of nanotubes is usually performed in a solution phase using shear-mixing, [

16,

17,

18] or ultrasonication, [

19,

20,

21,

22]. These processes are both governed by the transfer of local shear stress which breaks down the aggregates. It is therefore intuitive to suggest that complete separation of nanotubes would require the shear energy densities delivered to the bundle/cluster to exceed the binding energy (which can be treated as an ‘activation energy barrier’) of the system. Although the exfoliation state achieved may only be a temporary one, it greatly assists the surface adsorption of interfacial molecules (such as surfactants and compatible solvent molecules) which may subsequently stabilize the dispersion of CNTs [

23]. During mixing, the level of energy density delivered into a solution is equivalent to the shear stress attainable. Shear stress (σ

_{s}) is defined as the product of fluid viscosity (η) and fluid strain rate (

),

i.e.,

. We next consider the magnitude of the energy density that can be delivered by a mechanical mixing or a sonication technique.

Mechanical shear-mixing through stirring or extrusion can be performed in both low viscosity solvents (e.g., water or organic solvents with or without dissolved polymers), or highly viscous polymer melts. Hence the

η values employed in shear mixing can span from 0.01 Pa∙s to 10 Pa∙s. The fluid strain rate (

) for the common melt shear mixing is dependent on the rotational speed of the mixer blade (ω in units of rad/s), and the geometry of the mixer and the container. For a typical Couette (concentric cylinder) shear-mixing geometry,

, with

R being the radius of the container, and

h the spacing between the leading edge of the mixer blade and the inner wall of the container. The standard Couette mixing conditions in reference [

15] have yielded a strain rate of 500 s

^{−1}; experimental setups of different geometries could obtain a fluid strain rate as high as 4,000 s

^{−1}[

24]. Therefore, using a viscous polymer melt such as PDMS uncross-linked polymer (η = 5.6 Pa∙s), the shear stress imparted by the mixing medium is below 20 kPa. Note that if one wishes to shear-mix in a low-viscosity solvent (such as water, toluene or chlorophorm), the shear stress delivered to the CNT clusters will drop to below 50 Pa, offering very little hope of achieving dispersion. The simple message of this estimate is: shear mixing is only suitable for dispersion of MWNT clusters in high-viscosity polymer melts.

Compared to mechanical shear mixing, ultrasonication uses a very different mechanism in delivering the shear stress for dispersing aggregates. Cavitation occurs in a low-viscosity fluid above a certain ultrasonic intensity in the low-pressure regions of the travelling wave. Once created, the cavitation bubbles collapse causing an extremely high strain rate in the fluid in the proximate regions of bubble implosion. A strain rate of up to 10

^{9} s

^{−1} [

25,

26] is produced. The distribution of cavities is controlled by the geometry of the sonicator and the sonication settings, and is inhomogeneous throughout the solution [

27]. Taking a typical low viscosity solvent of 0.1 Pa∙s, the localized shear stress imparted in the vicinity of an imploding bubble can approach 10

^{8} Pa (100 MPa).

#### 2.3. Fracture of CNTs during Mixing

The shear stress imparting on the surface of a CNT can induce a pulling effect (a tensile force) on the nanotube. As a result, dispersion methods supplying high energy input can also induce fracture of CNTs. Mixing-induced fracture is of particular relevance in ultrasonication, where literature studies have confirmed the breakage of CNTs during sonication, with an example illustrated in

Figure 2 [

28]. Here, we will review a theoretical model for sonication scission [

20].

**Figure 2.**
Histogram of length distribution for d = 10–20 nm MWNTs after the (**a**) 1 h, and (**b**) 10 h horn sonication treatment. Gaussian profiles are fitted to both histograms for better statistical analysis. The corresponding SEM images of the MWNTs after sonication treatment are shown as insets.

**Figure 2.**
Histogram of length distribution for d = 10–20 nm MWNTs after the (**a**) 1 h, and (**b**) 10 h horn sonication treatment. Gaussian profiles are fitted to both histograms for better statistical analysis. The corresponding SEM images of the MWNTs after sonication treatment are shown as insets.

Let us consider a simple potential-flow description of bubble implosion dynamics which is based on the radial solvent flow around a bubble (

i.e., an inverse-square dependence between the fluid velocity

V(S) and the distance

S from the bubble center). The radial fluid velocity at a distance

S from the bubble is estimated by

, where

R_{i} is the characteristic bubble radius. It is expected that most nano-filaments, when located close to the imploding bubble (typical sizes of tens of micron), will re-orient themselves to align with the radial flow field. The final configuration of the filament with respect to the imploding bubble and its associated flow field is schematically illustrated in

Figure 3(a). Within this framework, an affine estimate is used to calculate the stress that is exerted on a suspended filament by the viscous forces transmitted from the solvent.

Consider the bubble with an instantaneous radius of

and wall velocity

. Near the bubble there is a segment of the filament with a length

L (with

S_{1} and

S_{2} being the starting and the ending positions of the filament) and a diameter

d, which is accelerated by the surrounding viscous (surface shear) forces. Let us take a frame moving with the instantaneous velocity of the filament and assume the tube is in a quasi-equilibrium and its center of mass moving with a speed of

V_{tube}. Surface shear stresses are created when the local velocity of the fluid flow

V_{s} is different from the velocity of the filament

V_{tube}. Assuming that the tube surface is non-slip and the characteristic length scale of the velocity decay across the cylindrical filament is of the order of its diameter

d, the local shear strain on the surface can be approximated as (

V_{s} −

V_{tube})/

d. This gives the local shear stress as

, with

η the solvent viscosity.

The quasi-equilibrium condition requires the total shear forces applied on the filament surface add to zero. Hence, between

S_{1} and

S_{2}, there must be a position

S* at which the surface fluid would have the same velocity as the filament: the fluid stagnation point on the filament surface with

V_{tube} =

V (S*). For

S_{1} <

S <

S*, the surface fluid is traveling at a greater speed than

V_{tube}, thus imposing surface shear forces acting towards the bubble center on the front section of the filament,

Figure 3(b); on the other hand, for

S* <

S <

S_{2}, the surface fluid is traveling at a lower speed than

V_{tube}, and the filament is experiencing drag forces on the surface of its rear section. The total surface shear forces acting on the front part of the filament, for

S_{1} <

S <

S* in

Figure 3(c) is:

(3)

Similarly, the surface shear forces on the rear part of the filament, for S* < S <

S_{2}, is

. Since the total shear forces applied on the tube surface add to zero, these two forces are balanced. After cancelation of factors on both sides, one obtains:

(4)

Substituting

, one can solve Equation (4), giving

.

**Figure 3.**
(

**a**) A snapshot during the cavitation process, showing a bubble of radius

R_{i} collapsing with its wall velocity

. The instantaneous velocity field of the fluid medium surrounding the bubble

V_{s} is illustrated, of which magnitude is indicated by the color-coded scale bar. The filament immersed in the flow can only have a single velocity

V_{tube} along its entire length, hence local surface shear stresses

σ_{s} are created. The variation of

σ_{s} w.r.t the position on the filament are shown in (

**b**) and (

**c**). The values of the surface stresses are normalized against the position-independent parameters, and the distance is normalized against the length of the filament.

**Figure 3.**
(

**a**) A snapshot during the cavitation process, showing a bubble of radius

R_{i} collapsing with its wall velocity

. The instantaneous velocity field of the fluid medium surrounding the bubble

V_{s} is illustrated, of which magnitude is indicated by the color-coded scale bar. The filament immersed in the flow can only have a single velocity

V_{tube} along its entire length, hence local surface shear stresses

σ_{s} are created. The variation of

σ_{s} w.r.t the position on the filament are shown in (

**b**) and (

**c**). The values of the surface stresses are normalized against the position-independent parameters, and the distance is normalized against the length of the filament.

As a result of the surface shear forces acting on opposite directions of the filament, an axial tensile force is generated. This ‘pulling’ force has a maximum at S*, and is of a magnitude equal to the net surface shear force (e.g., surface force integral between S_{1} and S*). Therefore, by knowing S*, one can re-calculate the integral in Equation (4) to determine the total force pulling in each direction; dividing this by the tube cross-section area gives the maximum tensile stress exerted on the tube:

(5)

Taking the typical literature values for the bubble size and rate of implosion (

= 10 μm, and mean

~ 10

^{8} s

^{−1}), the CNT diameter

d ~ 10 nm, the viscosity of a typical low-molecular weight solvent η ~ 0.1 Pa·s, and

S_{1} ~L ~10 μm (the maximal stress occurs for a filament positioned closest to the bubble such that

S_{1} = R_{i}), one obtains the estimate for the maximum tensile stress generated by viscous forces near the imploding bubble: σ

_{t} > 100 GPa. This is enough to break most nanotubes! However, it is also clear from the Equation (5) that the tensile stress on the tube decreases dramatically as the tube length

L diminishes, and a characteristic threshold length

L_{lim} exists for tube scission, for a set of pre-defined parameters

η,

d and

. If the value of breaking stress (ultimate tensile strength) of the nanotube is σ*, then this threshold length, after small-parameter expansion of Equation (5), is:

(6)

Tubes shorter than L_{lim} will not experience scission anymore.

One limitation of the above model is that the instant shear force is linearly dependent on viscosity. However, as viscosity increases, ultrasound absorption in the medium also increases and there is a much lower probability of cavitation at higher viscosities. Equations (5) and (6) are therefore only applicable to sonication with low viscosity solvents where cavitation is permitted. Nevertheless, this analysis gives a qualitative picture of the role played by the imploding bubble parameters

and

in tube breakage. We can further re-arrange Equation (6) to the form of

, by replacing

L_{lim} with

L, the starting length of a filament. We see that the left hand side contains purely the filament parameters, the aspect ratio (

L/d), and the strength of the fiber σ*; while the right hand side is equal to the stress induced during cavitation,

σ_{son}. Therefore, we have obtained an important filament fracture resistance parameter

. If a filament has

>

σ_{son}, it will not fracture during sonication; on the other hand, when

<

σ_{son}, fracture occurs, and

L will decrease until

L =

L_{lim} as discussed above. Filaments with higher strength and lower aspect ratio are more resistant to fracture during sonication, as one would expect.

#### 2.4. Effects of Shear-Mixing and Sonication

Combining the above analysis, an energy density diagram is presented in

Figure 4, which compares the theoretical values of energy density input, filament fracture resistance, and binding energy of the CNT aggregates. The lower and upper binding energy limits for MWNTs are calculated based on a tube diameter

d = 80 nm, for nanotube junctions spaced by distances of four-diameter, and one-diameter apart, respectively. For SWNTs, the ranges of binding energies shown are determined from the energy balances between the bending strain and the van der Waals attraction. The filament fracture resistance parameter is defined as

introduced previously (which has a dimension of energy density). The span of values in

is resulted from the different ranges of strengths reported for CNTs, such that σ∗

_{MWNT} ~ 3–100 GPa, and σ∗

_{MWNT} ~ 10–100 GPa [

29,

30,

31,

32,

33]. To understand the diagram, we first focus on the mixing conditions. High speed shear mixing in a high viscosity polymer melt can deliver an energy density reaching 10

^{4} Pa (as derived in the previous section). However, this energy level is still orders of magnitude lower than an ultrasonic cavitation event. For an aspect ratio of 10, the fracture resistance of SWNTs is of a slightly higher value than the binding energy of their aggregates. Ultrasonication just delivers a sufficient stress level to separate the SWNT aggregates without causing much fracture to individual nanotubes. For MWNTs, one finds that high viscosity shear mixing is able to separate the aggregates apart. Shear mixing is a better dispersion method because it can effectively separate the MWNTs without causing damage to the filament. For an aspect ratio of 1,000, which is relevant to the as-produced CNT sources, one notices that the fracture resistance of both SWNTs and MWNTs are lower than the stress input level delivered by sonication. On the other hand, shear mixing is not able to achieve a stress level matching the binding energy density. A dilemma therefore arises here, such that for complete separation of long CNTs, the tubes will also be broken during the process.

**Figure 4.**
An energy diagram showing the theoretical capabilities and limitations of shear-mixing and ultrasonication for carbon nanotube (CNT) dispersion, using filament aspect ratios L = d of 10 and 1,000 as examples.

**Figure 4.**
An energy diagram showing the theoretical capabilities and limitations of shear-mixing and ultrasonication for carbon nanotube (CNT) dispersion, using filament aspect ratios L = d of 10 and 1,000 as examples.

Based on the above, one can predict the morphology of the CNTs immediately after mixing. It is important to emphasize that the findings stated above only hold when there exists a steady state condition during mixing/sonication. For mechanical shear mixing, it is useful to determine the time t*, required to achieve such a steady state condition. A rheological prototype has been developed to determine t* [

15], where the evolution of the solution microstructure for t < t* was also studied. The viscous medium also provides a mean to temporarily stabilize the as-produced dispersion state for subsequent composite fabrication. On the other hand, since ultrasonication both disperses and cuts the CNTs [

28], finding a good ‘recipe’ for sonication dispersion is system dependent and requires significant trial and error experimentation. Since ultrasonication is performed in a low viscosity solvent/solution, additional means are required to stabilize the as-dispersed state. Various stabilization methods of CNTs in a solution are discussed in

Section 3 below.