Article pubs.acs.org/JPCC
Van der Waals Torque Coupling between Slabs Composed of Planar Arrays of Nanoparticles R. Esquivel-Sirvent*,† and George C. Schatz‡ †
Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, D.F. 01000, México Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States
‡
ABSTRACT: We present a theoretical study of the van der Waals torque between plates that are composed either of anisotropic material or of arrays of nanoparticles. The torque is calculated within the Barash and Ginzburg formalism in the nonretarded limit, and is quantified by the introduction of a Hamaker torque constant. Calculations are conducted between anisotropic slabs of materials including BaTiO3 and arrays of Ag nanoparticles, and it is found that measurable torques occur for 200 nm slabs separated by 15 nm. Depending on the dielectric function, in particular the zero frequency contribution, the Hamaker torque constant can be big enough to induce the orientation of arrays of nanoparticles.
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can be generalized using transfer matrix techniques.15 Also, in the nonretarded limit, it was shown that the van der Waals torque can be induced and modulated using external magnetic fields.16 The generation of van der Waals torque in the self-assembly of nanoparticles is possible if there is an optical anisotropy. Theoretical and experimental results were reported by Sung17 describing the birefringence of two-dimensional L-shaped nanoparticles. In elongated noble metal nanoparticles, the optical anisotropy was studied as a function of particle shape and composition.18 Orientational dependent optical properties were achieved by Henzie et al.19 with arrays of aligned pyramidal shaped nanoparticles of Au embedded in siliconbased polymer matrix. This work correlated the geometry of the particle with the optical response of the array. Similarly, Oates20 measured the optical anisotropy of randomly nucleated arrays of Ag nanospheres and nanorods. In another direction, the template-directed orientation of rodlike molecular assemblies arising from anisotropic van der Waals interactions between the assembly and crystalline surfaces has been studied experimentally. In this case, the substrate anisotropy provides a torque that overcomes the rotational Brownian motion near the surface.21 The self-assembly of nanoparticles is determined by the interplay of several interactions such as dispersive, electrostatic, and depletion forces, for example. For anisotropic nanoparticles, such as prisms, periodic structures can be achieved.23 However, the production of a net orientation of the particles,
INTRODUCTION The Lifshitz theory of van der Waals interactions was originally derived for optically isotropic systems.1 For anisotropic systems, besides having an attractive force between bodies due to fluctuating electromagnetic fields, there is also a van der Waals torque. This torque arises when the optical axes of two anisotropic bodies are not aligned. This was shown by Kats,2 who studied the dispersive torque between isotropic plates separated by a cholesteric liquid crystal applying the finite temperature Green function approach developed by Dzyaloshinski.3 Similar work was done by Parsegian and Weiss who also derived the van der Waals torque for anisotropic birefringent bodies.4 The van der Waals response of nematic liquid crystals was also worked by Smith and Nonham.5 A comprehensive theoretical study of the van der Waals torque in the retarded and nonretarded limit was done by Barash and Ginzburg,6 who extended the finite temperature Green function approach to anisotropic media. In a more modern approach, the van der Waals torque can be explained in terms of angular-momentum transfer and an orientational dependent zero-point energy.8,9 Munday10,11 et al. recently proposed an experiment to observe the van der Waals torque between two barium titanate plates immersed in ethanol. In this case, the retarded van der Waals force was repulsive. By balancing the repulsive force and the weight of the plates, one of the plates will rotate freely due to the dispersive torque, opening the possibility of frictionless bearings. Another experiment was proposed by Chen and Spence using a torsional pendulum giving an estimate on the experimental errors expected.13 Also, the possibility of aligning carbon nanotubes using dispersive torques in the retarded and nonretarded limits has been considered.14 Optical anisotropy leading to a van der Waals torque can be achieved using multilayered systems. The theory for the torque © 2013 American Chemical Society
Received: January 17, 2013 Revised: February 15, 2013 Published: February 20, 2013 5492
dx.doi.org/10.1021/jp400581j | J. Phys. Chem. C 2013, 117, 5492−5496
The Journal of Physical Chemistry C
Article
⎛ ε cos2(θ) + ε sin 2(θ) (ε − ε ) sin(θ) cos(θ) 0 ⎞ 2⊥ 2⊥ 2∥ ⎜ 2∥ ⎟ ⎜ ⎟ 2 − + ε ε θ θ ε θ ε ( ) sin( ) cos( ) sin ( ) 0 2∥ 2∥ 2⊥ ⎜ 2⊥ ⎟ 2 ⎜ ⎟ cos (θ) ⎜⎜ ⎟⎟ ε 0 0 ⎝ 2⊥⎠
i.e., alignment of the prism tips, cannot be explained. A possible mechanism could be the dispersive torque acting on the prisms. On the basis of the formalism of Barash and Ginzburg, we present a theoretical calculation of the van der Waals torque between optically anisotropic arrays of nanoparticles and anisotropic crystals, showing that the main contribution to the torque comes from the low frequency behavior of the dielectric tensor. The influence of torque on the self-assembly of nanoparticles is also discussed.
The dielectric functions are local and depend only on frequency. We are using Matsubara’s frequency representation. That is, the frequencies are along the imaginary axis (ω → iζn), where ζn = 2πκTn/ℏ, with κ being the Boltzmann constant, T the temperature, and n a positive integer.24 For example, the frequency dependent dielectric function will be changed according to ε(ω) → ε(iζn).25 Since the optical axes are not aligned, the van der Waals torque will make the plates rotate to minimize the energy of the system. In this case, the Helmholtz free energy of the system depends not only on the separation but also on the angle θ between the optical axes. That is,
