1. Introduction

Recent years have witnessed an explosion of research into carbon nanotubes (CNTs) because of their unique nanostructures with remarkable mechanical, electronic, and optical properties that could lead to new applications in materials and devices. For example, CNTs are applicable to high-strength yet light-weight composite materials, electron field emission, hydrogen storage, and optical limiting [1–9]. To realize such applications, however, a number of outstanding problems have to be resolved. One of them is the requirement of fast analysis methods for industrial quality control applications. Transmission electron microscopy (TEM) or scanning electron microscopy (SEM) may be employed to characterize tube lengths, diameters, and other structural details for individual nanotubes. Yet, no investigation has been carried out for measuring the molecular weight and weight distribution of large-quantity CNTs. The molecular weight of CNTs is a fundamental characteristic which has a profound implication in many applications. Here we report, for the first time, the observation of gravitation-dependent, thermally-induced self-diffraction in CNT solutions. And we also present a demonstration how the observed nonlinear-optical effect can be exploited for directly determining the molecular weights of CNTs.

2. Experiment: methods and results

Obviously, finite-size CNTs are variable-length rods, which by virtue of shape and size can act as liquid crystals. Liquid crystals exhibit a variety of nonlinear-optical effects [10,11]. One of them is self-diffraction due to thermal lensing, resulting from heat dissipation of absorbed light radiation to the surrounding medium [11,12]. Therefore, we anticipate that CNTs possess similar behavior. And it has been confirmed by our experiment described as follows. We employed a continuous-wave laser beam from a double-frequency Nd:YAG (532 nm), He-Ne (632 nm), or Ti:Sapphire (780 nm) laser. The laser beam was focused onto a CNT solution. The CNTs used were recently developed octadecylamine (ODA)-modified multi-walled CNTs with an average length of a few microns [13]. These CNTs had high solubility (up to 13 g/L in toluene). In our experiment, the CNTs were dissolved in toluene and the solutions were contained in 1-mm-thick quartz cells. The ODA-modified CNT concentrations were in the range from 0.02 to 0.08 g/L (or 0.0014% to 0.006% in volume fraction). The CNT solution was placed at the focal point, as illustrated in Fig. 1(a). The experiment was conducted at room temperature (~295 K).

When the laser beam was normal incident onto the solution, we observed the spatial transverse variation of the transmitted irradiance at far field. If incident laser powers were low, nearly Guassian spatial profiles were recorded, as expected. As the incident laser power rose, however, the profile developed to a number of concentric rings, a typical diffraction pattern. The number of rings was increased as the incident laser power was increased. Figure 1(c) and Fig. 1(e) display the two photographs taken at 532 nm and 780 nm, respectively, with the set-up shown in Fig. 1(a). This phenomenon is similar to the one reported for liquid crystals [11,12]. We attribute it to thermally-induced self-diffraction (or self-defocusing). Due to absorption of the laser energy by the CNTs and dissipation of the heat, there is a temperature rise in the solvent. The temperature rise is determined by the heat flow equation. Under the steady-state condition, the heat flow equation has the form of

where T is the temperature, k is the thermal conductivity, ρ is the radial distance from the center of symmetry, ω is the laser beam waist, and A is a constant proportional to the absorption coefficient of the CNT solution or the input laser power. Gordon et al [14] have derived an expression for the temperature rise, shown as follows:

where P is the input laser power, α ₀ is the absorption coefficient of the CNT solution, a is the radial position at which the temperature rise, ΔT(a), is zero, and $E_n\left(x\right)=-{\int }_{-x}^{\infty }\frac{e^{-t}dt}{t}$.

The temperature rise in the solvent induces a change in the refractive index by $\Delta n\left(\rho \right)=\frac{dn}{dT}$, with $\frac{dn}{dT}$ being the thermo-optical coefficient of the solvent. As a result, different parts in the cross-section of the laser beam experience different phases, $\Delta \psi \left(\rho \right)=\frac{2\pi }{\lambda }\Delta n\left(\rho \right)L$, where L is the sample thickness, and λ is the wavelength. Thus it gives rise to self-phase modulation. If the phase difference between the two points on the observation screen is mπ, m being an even or odd integer, constructive or destructive interference occurs, respectively, resulting in the appearance of diffraction pattern.

By the virtue of the Huygens principle under the cylindrical symmetry, one can calculate the electric field of the transmitted beam by

where ρ ^’ is the radial distance on the observation screen which is at a distance of z from the focal point, E(0) is the field at ρ ^’=0, and J₀(x) is the zeroth-order Bessel function. By employing both Eq. (2) and Eq. (3), we numerically simulate the irradiance distribution. Figure 2 shows excellent agreements if k=0.18 Wm^-1K^-1. This value is slightly greater than the thermal conductivity for toluene (0.13 Wm^-1K^-1, Ref. [15]). The larger thermal conduction in the toluene solution of CNTs is expected since multi-walled CNTs are known to have metallic properties and excellent thermal conductivity with their longitudinal thermal conductivity even exceeding the highest measured in-plane thermal conductivity of graphite [16]. The Z-scan measurement in the inset of Fig. 2(a) is a piece of evidence that shows a negative sign for the nonlinear refraction, consistent with the thermo-optical property of the solvent. Figure 3 illustrates the linear dependence of the diffraction pattern on the CNT concentration with no diffraction pattern for the pure solvent. This indicates that CNTs play a crucial role in the absorption of light radiation and the heat transfer to the solution.

The photographs presented in Fig. 1(c) and Fig. 1(e) show nearly perfect circular rings for the diffraction pattern when they are observed with the sample lying horizontally and the laser beam propagating vertically, as displayed in Fig. 1(a). The multi-walled CNTs in the solution sample are a few microns in length, and comprise of over ten millions of carbon atoms in each CNT, which can be regarded as a “supermolecule”. Thus, the gravitation effect on these “giant-molecule” tubes should not be negligible. When the sample lies horizontally and the laser light travels vertically, there is no variation in the CNT concentration along the transverse directions of the laser beam; and, hence, the gravitational effect does not manifest itself in the diffraction patterns. However, it is a different case when the sample is erected in the vertical direction and the laser beam transverses across the sample in the horizontal direction, as shown in Fig. 1(b). Indeed, we have observed distorted diffraction patterns as shown in Fig. 1(d) and Fig. 1(f). The diffraction rings appear to be compressed in the top half of the rings and stretched in the bottom half.

 

Fig. 1. Gravitation-dependant, thermally-induced self-diffraction in carbon nanotube solutions. (a) and (b) Schematic diagrams showing two experimental set-ups. (c) and (d) Diffraction patterns recorded at 532 nm with the set-ups shown in (a) and (b), respectively. (e) and (f) Diffraction patterns observed at 780 nm with the set-ups shown in (a) and (b), respectively. The input laser powers used are ~100 mW.

Download Full Size | PDF

 

Fig. 2. Far-field distribution of the transmitted irradiance measured at 780 nm with various laser powers. The squares, triangles, circles, and stars are the experimental data obtained from the set-up in Fig. 1(a). The linear transmittance of the carbon nanotube solution is 85.2%. The half angle is defined as the ratio of the radial distance, ρ ^’, on the observation screen to the distance of z. The solid lines are the numerical simulations using Eqs. (2) and (3) described in the text. The inset in A shows a Z-scan measured at 780 nm with an aperture in front of the detector.

Download Full Size | PDF

To quantitatively describe the gravitational effect on the diffraction, we need a mathematical expression for the distribution of CNTs suspended or dissolved in a liquid under the influence of the gravity. We speculate that CNT solutions are similar to the Brownian suspension, in which the Brownian particles obey Maxwell-Boltzmann distribution, like ideal gas molecules. For an ideal gas in the gravitational field, the molecules distribute exponentially proportional to their altitude [17]. Therefore, it is postulated that the CNTs arrange themselves in a similar way, that is, the nanotube number, N, depends exponentially on the vertical height. This can be described by the Boltzmann distribution law, $\text{N}={\text{N}}_0\exp \left(\frac{{\text{M}}_{\mathrm{tube}}gx}{{\text{k}}_{\text{B}}T}\right)$, where M_tube is the mass of the CNTs, g is the gravitational constant, k_B is the Boltzmann constant, N₀ is the CNT concentration at x=0, and the x-variable denotes the vertical distance from the center of symmetry for the laser beam. Because of this vertical gradient in the CNT concentration, there is the x-dependence of the diffraction. Thus, the diffraction pattern is distorted from cylindrically symmetrical rings, which is confirmed by our observation in Fig. 1(d) and Fig. 1(f).

 

Fig. 3. Far-field distribution of the transmitted irradiance recorded at 780 nm and an incident power of 9 mW with various carbon nanotube concentrations. The squares, triangles, circles, and stars are the experimental data obtained from the set-up in Fig. 1(a). The half angle is defined as the ratio of the radial distance, ρ ^’, on the observation screen to the distance of z. The solid lines are the numerical simulations using Eqs. (2) and (3) described in the text.

Download Full Size | PDF

 

Fig. 4. Far-field distribution of the transmitted irradiance measured at 780 nm with various laser powers. The squares, triangles, circles, and stars are the experimental data obtained from the set-up shown in Fig. 1(b). The transmittance of the carbon nanotube solution is 85.2%. The half angle is defined as the ratio of the x’-coordinate on the observation screen to the distance of z. The solid lines are the numerical simulations using Eqs. (7) and (8) with an approximation of very narrow weight distribution described in the text.

Download Full Size | PDF

To estimate the change in the nonlinear refraction due to the gravity, we make the first-order derivative over x for the relative change in the concentration, and we have ΔN/N₀=-M_tube g Δx/(k_BT). This results in a variation in the absorption coefficient as Δα ₀/α ₀=ΔN/N₀=-M_tube g Δx/(kBT), which leads to a change in the temperature rise ΔT_g=-M_tube g Δx ΔT/(kBT), where ΔT is the temperature rise at the center. Interestingly, it has an opposite sign to ΔT for the top half of the diffraction rings, and the same sign for the bottom half, explaining why the top half is compressed (or it offset the temperature rise caused by the transverse variation of the laser irradiance) and the bottom half is stretched (or enhanced), respectively.

If the above temperature difference is responsible for the two adjacent rings, its phase difference, $\Delta \psi =\frac{2\pi }{\lambda }\frac{dn}{dT}\mid \Delta T_g\mid L$, should be π. Thus we estimate the minimum measurable M_tube to be ~8×10^-16 g (or 1×10⁷ Dalton) if Δx=ω=80 µm, λ=800 nm, L=1 mm, $L=1\mathrm{mm},\frac{dn}{dT}=-5.3\times {10}^{-4}{K}^{-1}$, T=293 K, and ΔT=5 K (calculated by Eq. (2) and Eq. (3) for a laser power of 9 mW). For M_tube <8×10^-16 g, no gravitational effect on the diffraction is seen. However, the CNT mass is computed to be 30×10^-16 g, if we assume that the tube length is 10 µm, the radius of the inner tube is 10 nm, the spacing between the two adjacent concentric tubes is 0.34 nm and there are ten walls in the multi-walled CNT. Therefore, the gravitation effect is significantly pronounced.

To obtain a more rigorous solution, we have to take the distribution of tube masses into account for the absorption coefficient. For a dilute solution or suspension, its absorption coefficient may be described by

where N_i and σ_i are the number and absorption cross section of the tubes that have a mass of M_i, respectively. By substituting the Boltzmann-Maxwell law, Eq. (4) becomes

where N_i(0) is the number of the tubes at x=0. Thus, the heat flow equation has the form of :

Eq. (6) is a nonlinear equation. For small values of the temperature rise (ΔT<5 K) or low laser powers (a few mW), we take the first-order approximation by substituting a constant background temperature, T ₀, into the right-hand side of Eq. (6). Then, the solution has the form of

with ${\rho }_i=\sqrt{(x+\frac{M_{t}g{\omega }^2}{2k_B{T}_0}{)}^2+y^2}$ and $q_i=\exp (\frac{M_i^2{g}^2{\omega }^2}{8k_B^2{T}_0^2})$ . Combining Eq. (7) with the following Huygens principle,

where x ^’ and y ^’ are the two Cartesian coordinates on the observation screen, we numerically compute the profiles of the transmitted beam along the x ^’-axis, |E(x ^’,0)|². To simplify the numerical calculation, we assume that all the CNTs have a very narrow distribution of masses, representing by a single, average mass, M_tube. It is important to note that all parameters involved in Eqs. (7) and (8) are either known or measurable except for M_tube, which has been taken as an adjustable one in the computer simulation. Figure 4 shows good theoretical fits with the molecular mass being 8×10^-15 g, or 1.3×10⁸ Dalton, for input laser powers below 10 mW. The discrepancy is found for input laser powers greater than 10 mW. It is not surprising because the first-order approximation taken in solving Eq. (6) becomes invalid for higher temperature rises. In addition, we neglect the molecular weight distribution of the CNTs by assuming that all the CNTs have an average mass.

3. Conclusion

In conclusion, we have presented the observation of thermally-induced self-diffraction in CNT solutions. We have also described a thermally-induced, self-defocusing theory, which is in excellent agreement with the experimental data. We have found that the observed self-diffraction depends on the gravitation because of heavy molecular weights of CNTs. We have proposed a theoretical model in which CNTs are assumed to distribute themselves exponentially along the height. Under the approximations of small temperature rise and a narrow distribution of CNT masses, the model is consistent with the measurements at low laser powers. Such a gravitation-dependent characteristic opens up a new avenue to investigate giant molecular weights.