Materials perspective on Casimir and van der Waals interactions

Interactions induced by electromagnetic fluctuations, such as van der Waals and Casimir forces, are of universal nature present at any length scale between any types of systems. Such interactions are important not only for the fundamental science of materials behavior, but also for the design and improvement of micro- and nanostructured devices. In the past decade, many new materials have become available, which has stimulated the need for understanding their dispersive interactions. The field of van der Waals and Casimir forces has experienced an impetus in terms of developing novel theoretical and computational methods to provide new insights into related phenomena. The understanding of such forces has far reaching consequences as it bridges concepts in materials, atomic and molecular physics, condensed-matter physics, high-energy physics, chemistry, and biology. This review summarizes major breakthroughs and emphasizes the common origin of van der Waals and Casimir interactions. Progress related to novel ab initio modeling approaches and their application in various systems, interactions in materials with Dirac-like spectra, force manipulations through nontrivial boundary conditions, and applications of van der Waals forces in organic and biological matter are examined. The outlook of the review is to give the scientific community a materials perspective of van der Waals and Casimir phenomena and stimulate the development of experimental techniques and applications.

Figure 1

Schematic representation of dispersive interactions induced by electromagnetic fluctuations for (a) the dipolar vdW force between atoms and molecules, (b) the Casimir-Polder force between atoms and large objects, and (c) the Casimir force between large objects. For small enough separations one can neglect retardation effects due to the finite speed of light $c$, which corresponds to the vdW regime. For large enough separations, retardation effects become important, which is characteristic for the Casimir regime.

Figure 2

Schematic representation of first-principles methods for the (a) exact formulation of the electronic correlation energy $E_c$ from the adiabatic connection fluctuation-dissipation theorem; (b) formulation based on coupled dipolar fluctuations, such as the many-body dispersion (MBD) methods ([575]; [8]); (c) $E_c$ using two-point functionals obtained by approximating the nonhomogeneous system with a homogeneouslike response; and (d) fragment-based correlation energy obtained from multipolar expansions.

Figure 3

Various types of materials with calculated cohesion energies and errors as compared to available reference data. The top row of values shows typical errors of atomistic vdW methods compared to benchmark bending energies, while the bottom row of values shows contributions of the vdW energy to the binding energy of the corresponding materials.

Figure 4

(a) A graphene layer is an atomically thin sheet of honeycomb carbon atoms. Zigzag (green) and armchair (blue) graphene nanoribbons can be realized by cutting along the specified edges. Carbon nanotubes can be obtained by folding along the chirality vector ${\overrightarrow{C}}_h$, determined by the indices $n$ and $m$ and the lattice unit vectors ${\overrightarrow{a}}_{1,2}$. The chirality index $(n,m)$ uniquely specifies each nanotube with two achiral examples shown: (b) zigzag $(m,0)$ and (c) armchair $(n,n)$. Alternatively, folding a zigzag nanoribbon along the axial direction results in an armchaired nanotube, while folding an armchair nanoribbon gives a zigzag nanotube.

Figure 5

(a) Casimir pressure between two graphene sheets at separation $d=30\mathrm{n}\mathrm{m}$ as a function of temperature for different values of the gap $\mathrm{\Delta }$. Adapted from [299]. (b) Relative thermal correction $\delta F(\%)=[F(T)-F(T=0)]/F(T=0)$ for the graphene-Si plate interaction at separation $d=100\mathrm{n}\mathrm{m}$. Adapted from [70]. (c) Relative deviation $\delta F(\%)=[F_{dd}(T)-F_{pt}(T)]/F_{pt}(T)$ for graphene-graphene (red) and graphene-Au plate (blue) interactions, where $F_{dd}(T)$ is the Casimir force calculated via the density-density correlation function and $F_{pt}(T)$ is the Casimir force calculated via the polarization tensor. Adapted from [303]. Casimir graphene-graphene force normalized to $F_0=-3e^2/(32\pi d^4)$ with and without spatial dispersion in the graphene conductivity at $T=0\mathrm{K}$ as a function of (d) the gap $\mathrm{\Delta }$ and (e) the chemical potential $\mu$. From [167].

Figure 6

Electromagnetic pressure on each nanotube in a double-wall CNT system as a function of separation. The inset shows calculated EELS for several armchair $(n,n)$ and zigzag $(n,0)$ nanotubes as a function of frequency in eV. The attraction is strongest between two concentric armchair nanotubes due to the presence of strong overlapping low-frequency peaks in the spectra. The notation $(m,0)@(n,n)$ corresponds to $(m,0)$ as the inner tube and $(n,n)$ as the outer tube.

Figure 7

(a) Energy band structure of a CI calculated via a generic two-band tight-binding model ([243]). The low momentum limit of the energy band structure is consistent with the Dirac Hamiltonian in Eq. (15). (b) Real part of ${\sigma }_{xy}(\omega )$ (left panel) and ${\sigma }_{xy}(i\omega )$ (right panel). (c) Real parts of ${\sigma }_{xx}(\omega )$ (left panel) and ${\sigma }_{xx}(i\omega )$ (right panel) correspond to the energy band structure from (a). The calculations are performed with $C=1$, $\mathrm{\Delta }=0.25t$, and $\epsilon =2.25t$ ($t$ denotes the hopping integral for the employed lattice model; here it is taken to be equal to the frequency bandwidth). The conductivities are in units of $\alpha c/(2\pi )$. The analytically found ${\sigma }_{xx}(\omega )$ and ${\sigma }_{xy}(\omega )$ are in excellent agreement with numerically evaluated Kubo expressions. From [499].

Figure 8

(a) Casimir interaction energy $E(d)$ between two CIs in units of $E_0(d)=-\hslash c{\alpha }^2/(8{\pi }^2{d}^3)$ as a function of $\overline{d}=td/\hslash c$ for different values of $C_{1,2}$ and $|\mathrm{\Delta }|=t$. (b) Phase diagram $({\omega }_e/{\omega }_R,\theta /2\pi )$ for the interaction energy between two semi-infinite TI substrates for all separation scales. The parameters ${\omega }_e$ and ${\omega }_R$ correspond to the strength and location of the Drude-Lorentz oscillator, respectively. The repulsion for large separations at $T=0\mathrm{K}$ is given by the red region (outlined by the red line), while the repulsion for all separations at high $T$ is given by the green region (outlined by the green line). Here $\theta /2\pi =(n+1/2)$ and ${\theta }_1=-{\theta }_2=\theta$ for the substrates. Repulsion is observed for a large range of Chern numbers. Adapted from [499].

Figure 9

Casimir force per unit area $A$ between a metallic semispace and an anisotropic metallic-based planar magnetic metamaterial with a weak Drude background. The filling factors $f_{\parallel }$ and $f_{\perp }$, parallel and orthogonal to the vacuum-metamaterial interface, account for the fraction of metallic structure contained in the metamaterial. The SRR Drude parameters are ${\mathrm{\Omega }}_D=2\pi c/\mathrm{\Lambda }=1.37\times 10^{16}\mathrm{rad}/\mathrm{s}$ and ${\gamma }_D=0.006{\mathrm{\Omega }}_D$ (corresponding to silver) and its Drude-Lorentz parameters are ${\mathrm{\Omega }}_e/{\mathrm{\Omega }}_D=0.04$, ${\mathrm{\Omega }}_m/{\mathrm{\Omega }}_D=0.1$, ${\omega }_e/{\mathrm{\Omega }}_D={\omega }_m/{\mathrm{\Omega }}_D=0.1$, and ${\gamma }_e/{\mathrm{\Omega }}_D={\gamma }_m/{\mathrm{\Omega }}_D=0.005$. The Drude parameters for the metallic semispace are $\mathrm{\Omega }=0.96{\mathrm{\Omega }}_D$ and $\gamma =0.004{\mathrm{\Omega }}_D$. Temperature is set to zero, and the distance between the bodies is fixed at $d=\mathrm{\Lambda }$. From [504].

Figure 10

Top: Normalized Casimir force as a function of separation between a gold semispace and a 2D metasurface made of close-packed (lattice constant $a=11.24\mu \mathrm{m}$) ${\mathrm{LiTaO}}_3$ spheres of radius $r=5.62\mu \mathrm{m}$, calculated via scattering theory. The optical response of ${\mathrm{LiTaO}}_3$ is described by a single-resonance Drude-Lorentz model $\epsilon (\omega )={\epsilon }_{\infty }[1+({\omega }_L^2-{\omega }_T^2)/({\omega }_T^2-{\omega }^2-i\omega \gamma )]$, where ${\epsilon }_{\infty }=13.4$, the transverse and longitudinal optical phonon frequencies are ${\omega }_T=26.7\times 10^{12}$ and ${\omega }_L=46.9\times 10^{12}\mathrm{rad}/\mathrm{s}$, and the dissipation parameter is $\gamma =0.94\times 10^{12}\mathrm{rad}/\mathrm{s}$. The gold Drude parameters are taken as $\hslash {\mathrm{\Omega }}_D=3.71\mathrm{eV}$ and ${\mathrm{\Omega }}_D{\gamma }_D^{-1}=20$. Bottom: Effective permittivity and permeability of the close-packed ${\mathrm{LiTaO}}_3$ spheres as a function of (a) real and (b) imaginary frequencies as calculated by the Maxwell-Garnett effective medium theory. From [618].

Figure 11

Top: (a) Schematic of the experimental configuration used to measure the Casimir force between a gold-coated sphere and a gold nanostructured grating. One of the many nanostructures used in the experiment is shown [scanning electron microscopy images in (b)–(d)]. The radius of the Au sphere is $R=151.7\mu \mathrm{m}$. Bottom: Plane-grating pressure as a function of sphere-grating separation for a lamellar grating with period $p=250\mathrm{nm}$, tooth width $w=90\mathrm{nm}$, and height $h=216\mathrm{nm}$. Experimental measurements of sphere-grating force gradient divided by $2\pi R$ (dots with error bars), and plane-grating pressure computed with the proximity force approximation (dashed lines), exactly using scattering theory (solid line). From [265].

Figure 12

Top: Schematic configuration to measure the Casimir-Polder potential probed by a BEC diffracted from a plasmonic nanostructured being excited by an external laser field in the Kretschmann configuration. Bottom: Atom-grating potential landscape arising from the combination of a repulsive evanescent-wave potential and the Casimir-Polder attraction. The external laser field has a power of $P=211\mathrm{mW}$. The evanescent field from the grating modulates the repulsion, and the attractive Casimir-Polder potential is the strongest on top of the gold stripes. At a distance of 200 nm, the potential is laterally modulated with an amplitude of $\mathrm{\Delta }E/k_B=14\mu \mathrm{K}$. From [555], and [39].

Figure 13

Schematic illustration of numerical methods recently employed to compute Casimir interactions in complex geometries: (a) scattering methods where the field unknowns are either incident or outgoing propagating waves (left) or vector currents $\mathbf{J}(\mathbf{x})$ defined on the surfaces of the bodies (right), and the resulting energies are given by Eq. (21); (b) stress-tensor methods in which the force is obtained by integrating the thermodynamic Maxwell stress tensor over a surface surrounding one of the bodies utilizing the fluctuation-dissipation theorem.

Figure 14

Results illustrating the range of validity of PFA and PWS approximations in the sphere-plate geometry, involving a sphere of radius $R$ separated from a semi-infinite plate by a surface-surface distance $d$. The exact Casimir energy is computed by application of the scattering method. (a) Ratio of the exact and PFA energies as a function of $d/R$ for perfectly metallic conductors. Numerical data (dots) are compared to the first correction of the PFA (dashed lines), described in the text, and also to a fit performed via Padé approximants (solid curves), in which the energy ratio is given by $\mathcal{E}/{\mathcal{E}}_{\mathrm{PFA}}=1+(1/3)(1-60/{\pi }^2)d/R+(8/100)(d/R{)}^2\log (d/R)$. From [52]. (b) Ratio of the PWS and exact Casimir energies as a function of the static permittivities of the sphere and plate, for various ratios $d/R$ (blue curves). The red (steepest) and black (least steep) curves represent equivalent results for the plate-plate and sphere-plate geometries in the limit of infinite separations. From [55].

Figure 15

Selected results of Casimir interactions involving compact bodies suspended above planar objects or interacting with other compact bodies, calculated via scattering-matrix techniques in combination with spectral methods. (a) Casimir energy of a perfectly conducting vertically oriented cone, suspended above a perfectly conducting plate by a fixed distance $d$, as a function of the semiopening angle ${\theta }_0$ of the cone. From [360]. The PFA approximation is shown as the dashed line. (b) Casimir energy of a perfectly conducting wedge and plate of length $L$ and tilt angle $\pi /4$ as a function of the opening angle $\psi$. The system is suspended above a semi-infinite perfectly conducting plate by a surface-surface distance $d$. From [360]. The PFA prediction, dashed lines, is compared to the exact calculation, solid circles. (c) Ratio of the exact Casimir and PFA energies of the perfectly conducting cylinder-plate structure shown in the inset, as a function of $a/R$, decomposed into TE and TM contributions. From [188]. (d) Distance dependence of the Casimir force between perfectly conducting metallic cubes (red squares) or spheres (green circles), as computed by the BEM method of [477], divided by the corresponding PFA forces. Results for spheres are compared to computations performed using scattering-matrix methods (blue circles) which yield fast-converging semianalytical formulas, in contrast to situations involving realistic materials with dispersion such as gold, which require brute-force methods.

Figure 16

Selected results illustrating unusual Casimir effects from nonadditive interactions in complex structures. (a) Casimir force between either translationally invariant waveguides (solid lines) or cylindrically symmetric rods (dotted lines), normalized by the corresponding PFA force, as a function of their separation from adjacent sidewalls a distance $h$ apart ([486]; [369]). Configurations of either perfect-electric (blue) or perfect-magnetic (red) conductors are considered. (b) Casimir pressure between two patterned structures involving periodic arrays of gold cylinders (${\epsilon }_1$) embedded in a semi-infinite silica substrate (${\epsilon }_2$) surrounded by ethanol (${\epsilon }_3$), as a function of their surface-surface separation $d$, computed by application of both scattering and FDTD methods ([491]; [369]; [371]). Exact calculations (solid lines), PFA (dotted lines), and effective medium theory (dashed lines) are also shown. (c) Casimir force between a small, anisotropic gold particle (top panel) or a circular array of $N$ gold particles (bottom panel) and a gold plate with a hole ([335]; [370]), demonstrating repulsion for vacuum-separated metallic bodies.

Figure 17

Multilamellar array of lipid membranes immersed in water, each composed of a hydrocarbon core with two surface hydrophillic headgroup layers. Left: A realistic presentation with fluctuating positions of the membranes [for details see Kollmitzer et al. (2015b)]. Middle: A model system with a rigid array of alternating solvent ($B$) membrane ($A$) solvent ($B$) regions. Right: The Hamaker coefficient, $H(a,d)$ in [zJ], as a function of the separation between membranes $d$ and the thickness of the lipid bilayers $a$ in nm. The Hamaker coefficient is calculated based on the full dispersion spectra of water and lipids (hydrocarbons). From [456].

Figure 18

Forces and torques between anisotropic molecules and molecular aggregates. Top: $\tilde{\tau }(d,\theta )$, torque between two inclined long cylindrical molecules (above DNA, below single-walled CNTs) as a function of the closest separation $d$ and angle of inclination $\theta$, calculated within the Lifshitz theory. The angle of inclination is between the axes of the two cylindrical systems. The two Hamaker coefficients ${\mathcal{A}}^{(0)}(d)$ and ${\mathcal{A}}^{(2)}(d)$ depend on the dielectric anisotropy of the interacting materials as well as on the separation due to retardation effects, but they do not depend on mutual orientation. Bottom: $\tau (d,\theta )$, torque per unit surface area between two planar semi-infinite slabs composed of arrays of cylindrical molecules at separation $d$ with angle of inclination $\theta$ between their anisotropic uniaxial tensors. Only the surface layer of the bottom slab and one molecule of the upper slab are shown. Again, the Hamaker coefficient ${\mathcal{A}}^{(2)}(d)$ depends on the dielectric anisotropy of the interacting materials as well as on the separation due to retardation effects, but not on the orientation of the dielectric tensors. Adapted from [171].