Impact of nuclear vibrations on van der Waals and Casimir interactions at zero and finite temperature

Abstract

Recent advances in measuring van der Waals (vdW) interactions have probed forces on molecules at nanometric separations from metal surfaces and demonstrated the importance of infrared nonlocal polarization response and temperature effects, yet predictive theories for these systems remain lacking. We present a theoretical framework for computing vdW interactions among molecular structures, accounting for geometry, short-range electronic delocalization, dissipation, and collective nuclear vibrations (phonons) at atomic scales, along with long-range electromagnetic interactions in arbitrary macroscopic environments. We primarily consider experimentally relevant low-dimensional carbon allotropes, including fullerenes, carbyne, and graphene, and find that phonons couple strongly with long-range electromagnetic fields depending on molecular dimensionality and dissipation, especially at nanometric scales, creating delocalized phonon polaritons that substantially modify infrared molecular response. These polaritons, in turn, alter vdW interaction energies between molecular and macroscopic structures, producing nonmonotonic power laws and nontrivial temperature variations at nanometric separations feasible in current experiments.

Fig. 1. RMB model of molecular response.

A collection of atoms with electronic polarization response modeled as Gaussian basis functions f_p(x) interact via long-range EM fields ${\mathbb{G}}_{\text{env}}$. The individual electronic response of each atom arises from the coupling of valence electronic and phononic excitations via short-range interactions, represented schematically: For every atom p, a nuclear oscillator of mass m_Ip with dissipation b_Ip is connected to nuclear oscillators of other atoms q via anisotropic spring constants ${\mathbb{K}}_{\mathit{pq}}$, and to an electronic oscillator of mass m_ep with dissipation b_ep and isotropic spring constant k_ep; only the electrons couple directly to long-range EM fields with effective charge q_ep.

Fig. 2. Impact of phonons at large separations of a fullerene from the gold plate.

(a) Representative polarizability of an individual atomic constituent of a fullerene molecule suspended above a gold plate by a surface-surface gap z, comparing the full (including phonons) α (blue) and purely electronic α_e (green) polarizabilities. (b) RMB free energy integrand Φ(iξ) as a function of imaginary frequency ξ corresponding to $\mathcal{F}$ (blue) and $\mathcal{F}$_e (green) at a fixed z = 1 nm. (c) RMB power laws for $\mathcal{F}$(0) (blue), $\mathcal{F}$(300 K) (red), and $\mathcal{F}$_e(0) (green). Inset: Energy ratios $\mathcal{F}$(T)/$\mathcal{F}$_e for the fullerene at zero (blue) or room (red) temperature.

Fig. 3. Nonmonotonicity and temperature deviations due to phonon-induced nonlocal response in carbyne at short distance from the gold plate.

(a) Polarizability as a function of imaginary frequency for the middle atom (0) in a 500-atom-long carbyne wire, comparing α (blue) to α_e (green). (b) Imaginary frequency integrands for $\mathcal{F}$ (blue) and $\mathcal{F}$_e (green) at z = 1 nm via RMB (solid) or CP, without (fine dashed) or with (coarse dashed) artificial smearing. (c) RMB (solid) and CP, without (fine dashed) or with (coarse dashed) artificial smearing, interaction power laws of a 500-atom-long carbyne wire parallel to a gold plate, for $\mathcal{F}$(0) (blue), $\mathcal{F}$(300 K) (red), and $\mathcal{F}$_e(0) (green). Inset: Free energy ratios $\mathcal{F}$(300 K)/$\mathcal{F}$(0) as functions of z via RMB (solid) or CP, without (fine dashed) or with (coarse dashed) artificial smearing.

Fig. 4. Phononic versus purely electronic response and vdW interactions in graphene.

Magnitude of the Fourier space susceptibility ∣χ(iξ, k)∣ of a pristine (undoped) graphene sheet with rectangular unit cell 3.9 nm × 3.4 nm, obtained via either (A) RMB or (B) macroscopic, random-phase approximation (RPA) (41) models. (C) Power law of interaction free energy for a graphene sheet suspended above gold plate by a vacuum gap z, at zero (blue) or room (red) temperature, comparing RMB (solid) results to macroscopic RPA (dashed) predictions without doping. The inset shows the interaction free energy ratios $\mathcal{F}$(300 K)/$\mathcal{F}$(0) as a function of z.

Fig. 5. Graphene versus BN interaction power laws at short separations.

RMB power laws for the vdW interactions of parallel graphene (solid) or hexagonal BN (dashed) rectangular supercell with a gold plate, without phonons (green) or with phonons at zero (blue) or room (red) temperature.