van der Waals interactions at the nanoscale: The effects of nonlocality

Abstract

Calculated using classical electromagnetism, the van der Waals force increases without limit as two surfaces approach. In reality, the force saturates because the electrons cannot respond to fields of very short wavelength: polarization charges are always smeared out to some degree and in consequence the response is nonlocal. Nonlocality also plays an important role in the optical spectrum and distribution of the modes but introduces complexity into calculations, hindering an analytical solution for interactions at the nanometer scale. Here, taking as an example the case of two touching nanospheres, we show for the first time, to our knowledge, that nonlocality in 3D plasmonic systems can be accurately analyzed using the transformation optics approach. The effects of nonlocality are found to dramatically weaken the field enhancement between the spheres and hence the van der Waals interaction and to modify the spectral shifts of plasmon modes.van der Waals | nonlocality | transformation optics | plasmonics T he van der Waals force is an electromagnetic interaction between correlated fluctuating charges on two electrically neutral surfaces (1-4). As the surfaces approach more closely, the force increases as fluctuations of shorter and shorter length scale come into play, but ultimately the force will saturate when the surfaces are so close that the even shortest wavelength charge fluctuations are included. It is this saturation with which we are concerned in this paper and to treat it, we need to go beyond the conventional description of a solid by a permittivity, «ðωÞ, that depends only on the frequency, ω. Here we recognize that the response of a solid depends on the length scale of the fluctuations and introduce a formalism using a generalized nonlocal permittivity, «ðω; kÞ (5-13), that also depends on the wave vector, k, and hence takes into account the saturation. Neglect of nonlocality leads to an unphysical diverging van der Waals force at short distances. We apply the technique of transformation optics (14-16) to solve the difficult problem of including nonlocal effects when two nanoscale bodies interact and illustrate our theory with calculations for two closely spaced nanospheres.Ultimately at a few tenths of a nanometer, just before the surfaces touch, direct contact of the charges will come into play through electron tunneling (17-25); in other words, chemical bonding will dominate the final approach. Here we are not concerned with chemical bonding, which is extensively treated elsewhere in the literature.The van der Waals force is weaker than chemical bonding effects, but plays an important role in a broad range of areas, such as surface and colloidal science, nanoelectromechanical systems, and nanotechnology (2,3,(26)(27)(28). In the classical electrodynamics picture where nonlocality is neglected, the fluctuating surface charges are located precisely on the surface and these infinitely compressed charges result in the unphysically divergent van der Waals force (1, 29). In a more realistic framework,...

Probing nonlocal effects in metals with graphene plasmons

“…Figure 3(a) also shows that an increment of the spacer thickness induces a blueshift in the dispersion relation of the plasmonic modes. For that reason, a naïve approach to account for nonlocal effects while carrying out a local calculation is to consider an effective spacer thickness, s eff , larger than the actual value, s, in a similar fashion to what has been proposed in earlier works [44,47,48]. Although this method can indeed be regarded as a somewhat naïve version of quantum-corrected boundary conditions [49,50], it can mimic the proper nonlocal calculation as illustrated in Fig.…”

Biosynthesis of zinc sulfide quantum dots using waste off-gas from a metal bioremediation process

“…We suggest that the centrifugal forces in the sucrose density gradient could be sufficient to overcome the van der Waals forces holding the agglomerations. However, while van der Waals forces are simple to calculate between parallel surfaces of >10 nm apart, a calculation between only two nanospheres <5 nm apart becomes difficult, 23 and, with the agglomerations comprising many small NPs, and additional consideration of the differential centrifugal force, the calculation is well beyond the scope of this study. An alternative explanation is that while drying (used in the EM method but not in the differential centrifugal method) the small NPs are pulled together by water surface tension force which causes them to agglomerate artifactually; hence the centrifugal method gives a more accurate representation.…”

Hybridization of singular plasmons via transformation optics

“…We should mention that the non-local effect, which has a quantum origin, is an important Figure issue when the gap distance between the particles is extremely small (below 0.25 nm) [49][50][51][52][53][54]. Our focus is not on modelling the non-local effect but on understanding the strong interaction between the particles.…”