Supracolloidal Reaction Kinetics of Janus Spheres
Qian Chen, Jonathan K. Whitmer, Shan Jiang, Sung Chul Bae, Erik Luijten, Steve Granick*
*To whom correspondence should be addressed. E-mail: sgranick@uiuc.edu
Published 14 January 2011, Science 331, 199 (2011) DOI: 10.1126/science.1197451
This PDF file includes: Materials and Methods Figs. S1 to S3 References Other supplemental material for this manuscript includes the following: Movies S1 to S6
Method and Materials:
Experiments Fluorescent latex particles of sulfate polystyrene (1 μm in diameter, F-8851 from Invitrogen, Inc.) are made hydrophobic on one hemisphere through successive deposition of titanium (2nm) and gold (15 nm) thin films, onto which are formed self-assembled monolayers of n-octadecanethiol (Sigma-Aldrich). This process produces spheres which are hydrophobic on one surface region and negatively charged elsewhere. Using epifluorescence microscopy (63× air objective with a 1.6× or 2.5× post magnification, N.A. = 0.75), we track and identify each three-dimensional shape. This allows
2 visualization of the cluster evolution. The experiments are performed at room temperature in water with additional salt (NaCl).
Theoretical Methods The relative free energy of various helical structures (4) as a function of Janus balance (Fig. 3A) is calculated by considering the orientational freedom of a single Janus particle. This particle i is taken to be far from the ends of a chain (as particles near the end have considerably more rotational freedom) and all its nearest neighbors j are fixed with their director perpendicular to the helical axis, their hydrophobic side facing inward. This is a plausible first approximation, since deviations of neighboring particles from their average orientation that would increase the rotational freedom of the central Janus sphere i would also decrease the rotational freedom of other Janus spheres in the helix. The central particle interacts only with its nearest neighbors j through an angular square well attraction, and only attractive orientations of the particle are permitted,
ˆ ˆ i.e., cos1 rij ·di . We compute the orientational entropy from a Monte Carlo integral over the corresponding rotational phase space.
Collective distortions of the helix are the next correction to the free-energy difference between 3(0,1,1) and BC helices. We calculate the contribution of vibrational modes to this difference following the approach of Ref. 5, using a representative N = 24 chain length. This involves obtaining the eigenvalues of the Hessian matrix of coordinate derivatives, which yield the spring constants and normal modes associated with each
3 structure (up to the six zero modes associated with overall translation and rotation). The vibrational partition function for each mode is then
Z m dqm e
2 m qm
2 kBT
2
(1) ,
m
yielding the free energy per particle, f 1 1 2 . log Z m log m N N m m
(2)
Here m ranges only over the rigid modes, which have nonzero spring constant λm. For nonrigid modes, the degrees of freedom must be integrated explicitly (5). Here this is unnecessary, as the two structures considered each have only rigid modes aside from global translation and rotation of the particle aggregate. The vibrational correction reduces the free-energy difference between the BC helix and the 3(0,1,1) helix, although the latter remains the thermodynamically favored state at α = 90◦. The calculation is independent of interaction strength, as the internal energy of all conformations is identical; in reality a change in relative orientation of two Janus spheres may affect their pair energy even when their hydrophobic sides continue to face each other, e.g., because of the change in proximity of their charged hemispheres. This
