Platonic Micelles: Monodisperse Micelles with Discrete Aggregation Numbers Corresponding to Regular Polyhedra

Abstract

The concept of micelles was first proposed in 1913 by McBain and has rationalized numerous experimental results of the self-aggregation of surfactants. It is generally agreed that the aggregation number (Nagg) for spherical micelles has no exact value and a certain distribution. However, our studies of calix[4]arene surfactants showed that they were monodisperse with a defined Nagg whose values are chosen from 6, 8, 12, 20, and 32. Interestingly, some of these numbers coincide with the face numbers of Platonic solids, thus we named them “Platonic micelles”. The preferred Nagg values were explained in relation to the mathematical Tammes problem: how to obtain the best coverage of a sphere surface with multiple identical circles. The coverage ratio D(N) can be calculated and produces maxima at N = 6, 12, 20, and 32, coinciding with the observed Nagg values. We presume that this “Platonic nature” may hold for any spherical micelles when Nagg is sufficiently small.

Monitoring the Discontinuous Dodecamer–Icosamer Transition of a Calix[4]arene‐Derived Surfactant by Time‐Resolved Small‐Angle X‐ray Scattering

“…The N agg number can be controlled by the chemical structure of the surfactants or the solution conditions. For example, it has been reported that QACaL5 (Figure ) exhibits N agg =12 at a NaCl concentration ( C NaCl ) of 50–100 m m and N agg =20 at C NaCl ≥200 m m . This N agg change may be caused by a decrease in the volume of the hydrophilic moiety (the green part in Figure a), which is due to reduced electrostatic repulsion at higher C NaCl .…”

“…We recently found that calix[4]arene‐derived surfactants form spherical micelles with a defined, monodisperse N agg value of 4, 6, 8, 12, or 20, corresponding to the Platonic solids. We thus refer to these as Platonic micelles.…”

“…Many studies on calixarene‐derived surfactants have been carried out, and several reviews have been published, but such aggregation behavior has not been reported. We had postulated that μ N ° may not be a continuous function of N when N is sufficiently small, and this may be the reason for the appearance of micelles with Platonic numbers in N agg . The N agg number can be controlled by the chemical structure of the surfactants or the solution conditions.…”

“…The stopped‐flow device was coupled to a quartz capillary cell with a diameter of 1.3 mm, and we confirmed that homogeneous mixing was attained within 0.0025 s . The final c and C NaCl values were 0.008 g cm −3 and 290 m m , respectively, that is, C NaCl had rapidly increased from 50 to 290 m m , which was expected to result in a change in N agg from 12 to 20 . The scattering intensities were recorded with a Rayonix MX170 CCD detector with an exposure time of 0.01 s at 22 °C.…”

“…Δ ρ r and Φ ( x ) are defined as Δ ρ r =(Δ ρ c −Δ ρ s )/Δ ρ s and Φ ( x )≡3(sin x − x cos x )/ x 3 , respectively. Detailed analyses of the SAXS profiles of similar micelles were provided in previous papers …”