page_title: Lifshitz Theory of Van der Waals Force
section_title: Hamaker Constant in the Lifshitz Theory
hierarchical_section_title: Lifshitz Theory of Van der Waals Force / Hamaker Constant in the Lifshitz Theory
caption_attribution_description: English: a system of an electrical charge and a polarizable dielectric medium for the Lifshitz-Van-der-Waals wikipedia page.
context_page_description: In condensed matter physics and physical chemistry, the Lifshitz theory of van der Waals forces, sometimes called the macroscopic theory of van der Waals forces, is a method proposed by Evgeny Mikhailovich Lifshitz in 1954 for treating van der Waals forces between bodies which does not assume pairwise additivity of the individual intermolecular forces; that is to say, the theory takes into account the influence of neighboring molecules on the interaction between every pair of molecules located in the two bodies, rather than treating each pair independently.
context_section_description: The Lifshitz theory can be expressed as an effective Hamaker constant in the van der Waals theory.
  Consider, for example, the interaction between an ion of charge , and a nonpolar molecule with polarizability  at distance . In a medium with dielectric constant , the interaction energy between a charge and an electric dipole  is given by
  with the dipole moment of the polarizable molecule given by , where   is the strength of the electric field at distance  from the ion. According to Coulomb's law:
  so we may write the interaction energy as
  Consider now, how the interaction energy will change if the right hand molecule is replaced with a medium of density  of such molecules. According to the "classical" van der Waals theory, the total force will simply be the summation over individual molecules. Integrating over the volume of the medium (see the third figure), we might expect the total interaction energy with the charge to be
  But this result cannot be correct, since It is well known that a charge  in a medium of dielectric constant  at a distance  from the plane surface of a second medium of dielectric constant  experiences a force as if there were an 'image' charge of strength  at distance D on the other side of the boundary. The force between the real and image charges must then be
  and the energy, therefore
  Equating the two expressions for the energy, we define a new effective polarizability that must obey
  Similarly, replacing the real charge  with a medium of density  and polarizability  gives an expression for . Using these two relations, we may restate our theory in terms of an effective Hamaker constant. Specifically, using McLachlan's generalized theory of VDW forces the Hamaker constant for an interaction potential of the form    between two bodies at temperature  is
  with , where  and  are Boltzmann's and Planck's constants correspondingly. Inserting our relations for  and approximating the sum as an integral , the effective Hamaker constant in the Lifshitz theory may be approximated as
  We note that  are real functions, and are related to measurable properties of the medium; thus, the Hamaker constant in the Lifshitz theory can be expressed in terms of observable properties of the physical system.
caption: The nonpolar molecule is now replaced with a dielectric medium (in grey); the force experienced by the molecule can be found using the method of image charges.
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