Long Range van der Waals-London Dispersion Interactions

Peter Morgan – UMass, Amherst

Overview

• van der Waals-London dispersion (vdW-Ld) charge-fluctuation forces

• Often simply referred to as dispersion forces

• Origin in quantum electrodynamics

• London Dispersion (LD) forces arise from the interaction between fluctuating (induced) dipoles whose frequencies lie in the optical and UV regions of the spectrum

• vdW-Ld force is ubiquitous, acting between all types of molecules or their aggregates

• An important interaction governing nanoscience

Overview (cont’d)

• LD forces can be directed to assemble nanometer-scale structures - inorganic and biological materials

• Operate at distances longer than the interatomic covalent bond ~ few nm (long range)

• vdW-Ld component of molecular interactions has previously been poorly characterized

• Nanoscale material design unnecessarily intractable

• Also examine effects of Casimir-Lifshitz forces - repulsive dispersion forces between solids separated by a fluid (Munday, 2009).

Van der Waals interaction

• Johannes Diderik van der Waals (Nov 23, 1837 – Mar 8, 1923)

• Dutch theoretical physicist

• 1910 Nobel Prize in physics.

• vdW is an intermolecular force

• Three kinds – Dipole - dipole interaction (Keesom force)

– Dipole - induced dipole (Debye force)

– Induced dipole – induced dipole (London dispersion force)

• All three FvdW ~ C/r6

Van der Waals interaction (cont’d) • Three terms • wvdW(r) = -CK/r6 - CD/r6 - CL/r6

= [-(u12α2 + u2

2α1) - u12 u2

2/3kT - 3α1α2hν1ν2/2(ν1 + ν2)] / [(4πε0)2r6]

• where CK = Keesom constant CD = Debye constant CL = London (dispersion) constant

• Treat molecules as simple harmonic oscillators • vdW forces significantly reduced in a solvent (Petrache,

2006) • Generally non-additive (Podgornik, 2006)

Dispersion Forces

• Aka London forces, charge-fluctuation forces • Always present (e.g., non-polar molecules)

– unlike Keesom and Debye which require polar molecules

• QM in origin (electron shell polarization) – c.f. Casimir force due to vacuum polarization

• Can be repulsive or attractive • Long range, ~10-1000nm (French, 2010) • Interaction energy, wD(r) ~ I1I2α1α2/[(I1 + I2)r6]

– where Ii = ionization energy, αi = polarizability (Parsegian, 2005)

• Proportional to Z (size, mass) of atom/molecule • Dependent on mutual orientation of molecules • Play a role in adhesion, surface tension, adsorption,

molecular structure, wetting (e.g., liquid He flows out of a beaker).

Interaction potential

Interaction energy of the argon dimer. The long-range part (r > 5Å) is due to London dispersion forces.

Dispersion force contribution

Molecule pair % of the total energy of interaction

Ne-Ne 100

CH4-CH4 100

HI-HI 99

HCl-HI 96

HBr-HBr 96

H2O-CH4 87

HCl-HCl 86

CH3Cl-CH3Cl 68

NH3-NH3 57

H2O-H2O 24

Contribution of the dispersion force to the total intermolecular interaction

energy (more non-polar, greater the dispersion force contribution)

Approaches

• Theoretical – Van der Waals interaction (vdW-Ld) – Kramers-Kronig analysis – Density Functional Theory (DFT) – Lifshitz theory (QED) – Hamaker coefficients – DLVO – Graded interface models (van Benthem, 2006)

• Numerical – Simulations (Gecko-Hamaker database)

• Experimental (see, e.g., French, 2007) – VUV reflectance spectroscopy – VUV spectroscopic ellipsometry – Valence electron energy loss spectroscopy (VEELS)

Hamaker Constants

• Hamaker (1937) theory explains the van der Waals forces between objects larger than molecules

• Hamaker constant A given by • A = π2Cρ1ρ2

– ρ1 and ρ2 are the number of atoms per unit volume – C is the dispersion coefficient in the particle-particle pair interaction.

• Lifshitz theory (Dzyaloshinskii et al, 1961) – Macroscopic continuous media

• A = 3

4kT(

𝜀1−𝜀3

𝜀1+ 𝜀3) (𝜀2−𝜀3

𝜀2+ 𝜀3) +

3ℎ

4𝜋 (

𝜀1(𝑖𝜈)−𝜀3(𝑖𝜈)

𝜀1(𝑖𝜈)+ 𝜀3(𝑖𝜈))(𝜀2(𝑖𝜈)−𝜀3(𝑖𝜈)

𝜀2(𝑖𝜈)+ 𝜀3(𝑖𝜈))

∞

𝜈1dν + h.o. terms

• First term = Keesom/Debye. Second term = dispersion • Actually a function A = A(ν, d) where d = separation.

Example geometries i. between two infinite parallel surfaces

W = -A/12πD2 per unit area

ii. between a sphere and a surface

W = -AR/6D

iii. between two spheres

W = -AR1R2/[6D(R1 + R2)]

where A = π2Cρ1ρ2 = Hamaker constant

DLVO Theory

• Derjaguin and Landau (1941), Verwey and Overbeek (1948)

• Combines van der Waals attraction and the electrostatic repulsion due to double layer (DL) of counterions

• – W(D)A = Attractive vdW interaction (~1/r6)

– W(D)R = Repulsive DL interaction (screened Coulomb)

• For two spheres, radius R – WA = -A/(12 π D2)

– WR = 2 π ε R ξ2 exp(- κD)

Coulomb screening force • Two spheres of radius a with constant surface charge Z separated

by a center-to-center distance r in a fluid of dielectric constant ε containing a concentration n of monovalent ions, the electrostatic potential takes the form of a screened-Coulomb or Yukawa repulsion

• where λB is the Bjerrum length, • κ − 1 is the Debye screening length, which is given by κ2 = 4πλBn, • β = 1/kT is the thermal energy scale at absolute temperature T. • Bjerrum length is the separation at which the electrostatic

interaction between two elementary charges is comparable in magnitude to the thermal energy scale, kT and is given by

λB = e2/4πεε0kT

DLVO Interaction Potentials

van der Waals, Double Layer and Net potential interaction for the DVLO model

Double Layer model

Double layer model. Depending on the nature of the solid, there may be another double layer (unmarked on the drawing) inside the solid.

DLVO Model

Inner (Stern) and outer (diffuse) ionic layers of the DLVO model

Molecular model

1. Inner Helmholtz Layer 4. Solvated ions 2. Outer Helmholtz Layer 5. Peculiar adsorptive ions 3. Diffuse layer 6. Solvent molecule

Optical Spectra

Magnitude of the imaginary part of the dielectric function ε"(ω) vs photon energy, showing anisotropy in the radial (left) and axial (right) directions.

Base-pair sequence

Base-pair composition of oligonucleotides that will be studied with VUV spectroscopy, spanning the whole range of sequences from (GC) homopolymer to (AT) homopolymer.

vdW interaction for CNT

Van der Waals interaction between single-walled carbon nanotubules (SWCNT) – see, for example, Rajter, 2007

Example - Quantum levitation

• Application of the Lifshitz theory of vdWLd interactions when polarizability of the intervening medium between bodies is intermediate between those of the interacting materials, the prediction is for repulsive force rather than attractive.

ε(ix)1 > ε(ix)3 > ε(ix)2

• where ε(ix), are the respective dielectric functions of the interacting materials at imaginary frequencies

• Gives rise to quantum levitation

• Note that dielectric ε = ε’ + iε’’

References • van Benthem, K., Graded Interface Models for more accurate Determination of

van-der-Waals – London Dispersion Interactions across Grain Boundaries, Phys. Rev. B 74, 205110, 2006, http://prb.aps.org/pdf/PRB/v74/i20/e205110

• Dzyaloshinskii, E., E. M. Lifshitz, and L. P. Pitaevskii, The General Theory of van der Waals Forces, Adv. Phys., 10 [38] 165–209, 1961

• French, R.H. et al, Long Range Interactions In Nanoscale Science, Reviews Of Modern Physics, 82:2, 1887-1944, 2010

• French, R. H. et al, Optical Properties and van der Waals–London Dispersion Interactions of Polystyrene Determined by Vacuum Ultraviolet Spectroscopy and Spectroscopic Ellipsometry, Aust. J. Chem. 60:251–263, 2007

• Hamaker, H. C., The London – van Der Waals attraction between spherical particles, Physica 4, 1058–1072, 1937

• Landau, L. D., L.P. Pitaevskii, and E.M. Lifshitz, Electrodynamics of Continuous Media. Vol. 8, 1st ed, Butterworth-Heinemann, 1984

• Mahanty, J., and B. W. Ninham, Dispersion Forces, Academic Press London, 1976

References (cont’d) • Munday, J.N. et al, Measured long-range repulsive Casimir-Lifshitz forces, Nature

vol. 457. pp. 170-173, www. nature.com/nature/ journal/v457/ n7226/pdf/nature07610.pdf, 2009

• Munday, J. N. et al, Measurements of the Casimir-Lifshitz force in fluids: The effect of electrostatic forces and Debye screening, Phys. Rev. A 78, 032109, 2008

• Parsegian, V. A., Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists, Cambridge University Press, 2005

• Parsegian, V. A., and B. W. Ninham, Application of the Lifshitz theory to the calculation of van der Waals forces across thin lipid films, Nature 224:1197-1198, 1969

• Petrache, H.I. et al, Salt screening and specific ion adsorption determine neutral-lipid membrane interactions, Proc Natl Acad Sci U S A 103:7982-7987, 2006, www.pnas.org/content/103/21/7982

• Podgornik, R. et al, Non-additivity in Van der Waals interactions within multilayers, J. Chem. Phys., Vol. 124, 044709, 2006

• Rajter, R. F. et al, Calculating van der Waals - London Dispersion Spectra and Hamaker Coefficients of Carbon Nanotubes in Water from ab initio Optical Properties, Journal of Applied Physics 101:054303, 2007