Nematic Droplets in Aqueous Dispersions of Carbon Nanotubes
Aqueous dispersions of exfoliated, bile-salt stabilized single-wall carbon nanotubes exhibit a first order transition to a nematic liquid-crystalline phase. The nematic phase presents itself in the form of micron-sized nematic droplets also known as tactoids, freely floating in the isotropic host dispersion. The nematic droplets are spindle-shaped and have an aspect ratio of about four, irrespective of their size. We attribute this to a director field that is uniform rather than bipolar, which is confirmed by polarization microscopy. It follows that the ratio of the anchoring strength and the surface tension must be about four, which is quite larger than predicted theoretically but in line with earlier observations of bipolar tactoids. From the scatter in the data we deduce that the surface tension of the coexisting isotropic and nematic phases must be extremely low, that is, of the order of nN/m.
pacs:61.30.Hn, 61.30.Dk, 82.70.Dd
Carbon nanotubes or CNTs are colloidal particles with a very large aspect ratio, typically in the range from many tens to hundreds up to even thousands. Hence, it is not surprising that, provided they are properly stabilized against aggregation, fluid dispersions containing CNTs exhibit an Onsager-type isotropic-nematic transition CNT1 (); CNT2 (); CNT3 (); CNT4 (); CNT5 (); CNT6 (). This happens at concentrations in excess of a critical value that depends on the aspect ratio of the rods. The relevant concentration scale here is the volume or packing fraction because the driving force for the spontaneous alignment is the anisotropic volume exclusion between the particles. For the nematic to become stable the volume fraction of CNTs should be in excess of a few times the reciprocal of some average of their aspect ratios Onsager49 (). It follows that the nematic transition must occur at very low concentrations of, say, one per cent of CNTs. This, by and large, is in agreement with experimental observation, allowing for instance for the effects of polydispersity CNT1 (); CNT2 (); CNT3 (); CNT4 (); CNT5 (); CNT6 ().
Often, before isotropic-nematic phase separation occurs on a macroscopic scale in dispersions of elongated colloidal particles, the nematic phase establishes itself in the form of droplets called tactoids. Tactoids have been observed in many dispersions, such as tobacco mosaic virus Bernal41 (); Maeda97 (), boehmite rods vanBruggen98 (), poly(butyl glutamate), self-assembled chromonics Lavrentovich05 (), fd virus Fraden01 (), f-actin Tang07 (), and vanadium pentoxide Kaznacheev02 (); Sonin98 (). These droplets have in common their unusual elongated, spindle-like shape. This shape can be explained by the preferential planar anchoring of the nematic director at the interface with the isotropic phase. The competition between surface tension and the elastic deformation of the bipolar director field that accommodates this preferential planar anchoring plausibly determines the optimal aspect ratio (Figure 1). If the director field of a tactoid is indeed bipolar with field lines connecting two boojum surface defects, then its shape, as described, e.g., by the aspect ratio or the tip angle, depends on its physical dimensions because the surface and bulk elastic energies scale differently with droplet size. This seems to be the case in all systems investigated so far. Indeed, information on the ratio of (an average of) the elastic constants and the surface tension can be obtained from the measured relation between, say, the aspect ratio and length of the droplets Kaznacheev02 (); Kaznacheev03 (); Prinsen03 (); Prinsen04 (); Tang07 ().
In this letter, we show that aqueous dispersions of surfactant-stabilized carbon nanotubes deviate from the usual picture of a bipolar director field. The tactoids that we observe in these dispersions are also quite elongated, but display a uniform director field as evidenced by polarization microscopy. This ties in with our finding that the aspect ratio of the tactoids is independent of their size, at least for the one decade range of sizes present in our samples. Theoretically, the aspect ratio of a tactoid is independent of its size if it is dictated by the surface tension anisotropy, i.e., the ratio of the anchoring strength and the surface tension Prinsen03 (). This is plausible if the director field is uniform and as a result of that the anchoring conditions at the surface of the drops are suboptimal. It is because the interfacial and anchoring free energies both scale with the area of a drop that its shape can only be a function of the ratio of the anchoring strength and the surface tension. From our observations we find that the anchoring strength is about four times larger than the bare surface tension. We put forward that the surface tension must be very small indeed, possibly as low as 1 nN/m.
We prepared our CNT tactoid dispersions from an aqueous suspension of single-wall carbon nanotubes (furnished by Elicarb batch K3778) and dispersed by bile salts, at the respective concentrations of 0.5% w/w CNTs and 0.5% w/w bile salts. To exfoliate the CNT bundles, sonication was applied to the suspension for a period of three hours. A purification process by selective centrifugation was then performed on the CNT suspensions. After removing CNT aggregates by centrifugation at low speed (30 min, 3500 rpm), the longest carbon nanotubes exhibiting some entanglements and defects were removed by ultracentrifugation (45 min, 45000 rpm). A second ultracentrifugation (180 min, 45000 rpm) was applied to the CNT suspensions to obtain the nematic liquid crystalline phase, which appeared as a black pellet on the bottom of the centrifugation tube. Finally, CNT tactoids were obtained by diluting the nematic phase up to the coexistence region with the isotropic phase. Samples of a few m thick were prepared at the isotropic-nematic phase coexistence between cover slip and glass slide. The CNT tactoids were observed by optical microscopy at different magnifications between crossed polarizers.
Figure 2 shows an image by polarization microscopy of the kind of tactoids that we find in our carbon nanotube dispersions. The background is dark because the tactoids float in the coexisting isotropic liquid phase. The tactoids are indeed quite elongated and have the typical spindle-like shape. A perfect extinction is observed when the tactoids are aligned along the polarizer or analyzer direction, demonstrating that the director field is uniform, as shown for a typical tactoid in figure 3. A further study under confocal Raman micro-spectroscopy also confirms that the director filed is aligned along the tactoid long axis Puech (), which coincides with the main optical axis.
In figure 4 the aspect ratio of a large collection of tactoids is plotted against their length, which, within experimental error, is constant as advertised. Apparently, the mean aspect ratio of the tactoids is about four for tactoids up to 36 m in length. This result is consistent with a uniform director field in the droplets, as the aspect ratio of the drops is then dictated by the ratio of the anchoring strength and the surface tension Prinsen03 (). By applying an inverse Wulff construction Wulff01 () to the shape of a typical tactoid, we have been able to probe the anisotropy of the surface free energy, . Shown in figure 5 is , where is the bare surface tension and the angle between the surface tangent and the director field. We find that this angle dependence is consistent with the often-used Rapini-Papoular model for the surface free energy, i.e., with the ratio of the anchoring energy and the surface tension RapiniPapoular69 (). For the tactoid shown in the figure, we obtain a value for the dimensionless anchoring strength of . From the Wulff construction, we can in fact predict the aspect ratio of a nematic drop that within a Rapini-Papoular model must be equal to if Prinsen03 (). It follows from figure 4 that for our coexisting isotropic and nematic phases of CNTs in water, . This is quite larger than theoretical predictions according to which , at least for cylindrical particles interacting via a harshly repulsive potential Chen92 (); Chen95 (); Harlen99 (); vanderSchoot99 ().
Additional information that we can deduce from figure 4 can be obtained by realizing that according to recent calculations Prinsen03 (), a tactoid changes its director field from a uniform to a bipolar one if its volume is larger than , with an average of the Frank elastic constants. It has to be noted that this crossover has so far only been observed in computer simulations Trukhina09 (). It seems that even the largest tactoid in our samples with a length of 36 m, has a uniform director field. This implies that a lower bound for the ratio of the nematic of CNTs must be approximately 5 m. Interestingly, this lower bound is comparable to values found for vanadium pentoxide and fd virus Kaznacheev02 (); Prinsen03 (); Prinsen04 (), although that tactoids in these two systems do exhibit a bipolar field if larger than a few m. It is not quite clear why single-wall CNTs behave so differently from other types of elongated particle. Indeed, CNTs exhibit a phase behavior that is in good agreement with the behavior expected for bulk suspensions of rod-like particles CNT3 (). Nevertheless the structure of tactoids does result from a delicate interplay of bulk elastic and surface properties of the coexisting isotropic and nematic phases. These properties are known to quite sensitively depend on molecular details such as a the degree of bending flexibility and the type and the strength of interactions involved in stabilizing them in suspension. Another issue is also the influence of polydispersity, which is known to be large. CNTs are polydisperse and a small fraction of very long or very short particles could affect the surface properties with a little effect on the bulk behavior. Of course, this is speculative and further work will be needed to confirm whether or not the size distribution of the CNTs confined at the isotropic-nematic interface differs from that in bulk.
Finally, the scatter in the data of figure 4 potentially provides physical information because in part it must be caused by thermal fluctuations of the aspect ratio of the tactoids. The Wulff construction provides only the optimal aspect ratio but does not give an indication of its variance. Using a simple scaling Ansatz based on the Rapini-Papoular surface free energy Prinsen03 () and making use of the equipartition of free energy, we find that the standard deviation of the aspect ratio must approximately be equal to , so depend on the size of the tactoids. In figure 4 we have indicated around the estimated average aspect ratio the standard deviation presuming a surface tension of 0.5 nN/m. The prediction follows the magnitude and size dependence of the experimental scatter in the data reasonably well. Obviously, we should not over interpret this result because we have ignored any influence of the intrinsic error in the size measurement of the tactoids.
If we accept the very small value of the surface tension that we find at face value, then it is very much smaller than values in the range of tenths to tens of N/m typically found for coexisting isotropic and nematic phases in dispersions of rod-like particles Gray02 (). Values of nN/m have been found for coexisting isotropic and nematic phases but only in dispersions of colloidal platelets Vanderbeek06 (). Clearly, the scaling theory does not give the numerical pre-factor, and this could increase the found surface tension by another factor of, say, ten. Still, this by no means takes our value within the range of the other experimentally found values. One might of course conclude from this, that the presumption that the scatter in the data is dominated by thermal fluctuations must be wrong. Indeed, some flow can occur just after sample preparation, which can induce droplet alignment and some shape distortion. The latter is clearly seen in the largest CNT tactoids, see also figure 2. Explicit surface tension measurements, e.g., by the capillary rise method Vanderbeek06 () on macroscopic interfaces between coexisting isotropic and nematic phases are necessary to confirm our finding, but these are outside the scope of the present paper.
In conclusion, we find nematic tactoids of aqueous dispersions of surfactant-stabilized single-wall carbon nanotubes that have a uniform director field. Our observation accounts that the aspect ratio of about four is independent of the size of these tactoids at least for lengths up to 36 m. The crossover to a bipolar director field must occur for sizes much larger than this value, but we have not been able to confirm this.
Acknowledgements.PvdS gratefully acknowledges the hospitality and a supporting grant from Université Bordeaux 1.
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