(i) The combination of short-range electrostatic repulsive forces (shown in blue) and van der Waals attraction will result in an attractive well (shown in red) potential between the two particles. However, the exact form of these functions is unknown for complex solvent and solutes such as superacids and SWCNTs. (ii) A simple phenomenological square well potential is used to capture this balance of repulsive forces and attractive forces between SWCNTs in various acids. As the acid quality decreases, the well becomes deeper. (iii) is the distance between neighboring SWCNTs in the LC phase. This simplified potential is inspired by the unusual LC order seen in “spaghetti” in SWCNT/superacid systems (Ref. 33). (Iv) A schematic of spaghetti geometry depicts the threadlike, aligned nature of these unusual LC domains.
Computational results for an athermal solvent illustrating the dependence of and on for polydisperse distributions for or and . Squares indicate shadow points, and diamonds indicate cloud points. This stands in Stark contrast to the monodisperse case where and do not vary.
The isotropic and nematic cloud points are depicted as a function of the well depth for and . When , the nematic cloud point rapidly increases to as decreases. For the case , remains constant at 0.210. The Onsager predictions for monodisperse rods are depicted as triangles (△).
Isotropic cloud curves ( vs ) as a function of for . As decreases, the isotropic cloud curves move down to lower well depths and to the right to higher rod concentration as length fractionation effects diminish. The critical value of drops with decreasing .
Nematic cloud curve for for various values of . The cloud curves increase to as attractive interactions increase, but this effect diminishes with decreasing .
SWCNT/superacid phase behavior as a function of SWCNT volume fraction and acid strength (measured by ). Black symbols denote experimental results (Ref. 17) Red symbols refer to theoretical predictions, and red open circles (○) designate the theoretical predictions for compared with experimental measurements of , denoted by black circles (●). Black/red diamonds (◇) indicate the initial system concentration for both experiments and simulations of isotropic (i)—LC phase separation. The open red triangles (△) is the Onsager predictions for for a system of monodisperse hard rods. Red line (–) represents the theoretical predictions for the isotropic cloud curve. Simulations are for , , and .
SWCNT/120 mixtures of varying were prepared and phase separated; the isotropic concentration was measured by UV-vis-NIR absorbance. These experimental measurements showed a close match with theoretical predictions ( and ) for phase separation as a function of .
(Left) Volumetric fraction of system that is nematic (denoted as ) as a function of for for , , , and . The system reaches at (noted as triangle), but the concentration must increase to in order to reach (the nematic cloud point, noted as diamond). Experimental data indicate a cloud point of but cannot detect the small isotropic regions filled with short rods that persist up to . (Right) Average length of rods in the isotropic phase (denoted as ) as a function of . Near the isotropic cloud point, approaches that of the parent population, but decreases dramatically as approaches the nematic cloud point. Thus, near the nematic cloud point, the rods in the isotropic phase tend to be the shortest rods in the parent distribution, as confirmed experimentally elsewhere (Ref. 50).
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