Historical research guide · 0749 · 2019
Scattering theory of the screened Casimir interaction in electrolytes
Paulo A. Maia Neto · Felipe S. S. Rosa · Luis B. Pires · Anna B. Moraes · Antoine Canaguier-Durand · Romain Guérout · Astrid Lambrecht · Serge Reynaud
In one minute
Paulo Maia Neto’s group in Rio de Janeiro and Serge Reynaud’s at the Laboratoire Kastler Brossel in Paris ask what the Casimir interaction does when the gap between two surfaces is filled with salt water instead of vacuum. Salt water carries free ions, and ions move to blot out slowly changing electric fields — the same screening that stops charged surfaces in a cell from feeling one another beyond a nanometre or so, a distance called the Debye length. The authors rebuild the calculation in the scattering language, where the force comes from fluctuations bouncing back and forth between the two surfaces, and add the extra channel that ions make possible: longitudinal waves, running along the field direction, coupled to the ordinary transverse ones at each surface. The answer splits cleanly. Fast fluctuations are left alone by the ions. At zero frequency one part is screened away over the Debye length, and a second part, carried by transverse-magnetic modes, is not screened at all and sets the interaction at long range.
Connection
Chapter 2 holds that the vacuum is a real medium whose fluctuations do mechanical work, and the obvious objection to using that in a wet, salty, room-temperature world is that ions should short the effect out. This paper answers the objection with an exact calculation: part of the zero-frequency interaction is screened, but a transverse-magnetic part survives no matter how salty the solution, and it is the part that dominates at long distance. That is the theory a colloid, a lipid membrane or an optical-tweezers measurement in water has to be compared against.
From the source
The scattering approach is extended to two dielectric half-spaces separated by an electrolyte solution, with the nonlocal electromagnetic response of the solution taken into account: because the ions give the medium a spatially dispersive response, longitudinal modes exist alongside the usual transverse ones, and the two are coupled by reflection at the surface of the local dielectric. The interaction energy follows from the matrix describing one round trip of these coupled waves.Abstract; Section II, reflection matrix for the electrolyte-dielectric interface; Section III
The contributions at nonzero Matsubara frequencies are approximately unaffected by the presence of ions, because the plasma frequency associated with the ions is always far below the thermal frequency scale — those fluctuations are too fast for the ions to follow.Introduction; Section III, the paragraph beginning ’In conclusion, we find that the modification of the nonzero Matsubara frequency contributions…’
At zero frequency the interaction splits in two: a screened contribution from longitudinal channels, written in terms of the longitudinal reflection coefficient and suppressed beyond a distance of the order of the Debye length, which reproduces the earlier linear Poisson-Boltzmann results; and an unscreened contribution from transverse-magnetic modes with the universal value three quarters of the Riemann zeta function of three times the thermal energy, about 0.9 times k-B T, which is not suppressed even in the limit of strong screening and defines the long-distance asymptotic limit.Abstract; Section III, Equations 25 and 26; Section V, Conclusion
Worked for two polystyrene surfaces across an aqueous solution at 293 kelvin, the zero-frequency part of the Hamaker coefficient is a universal function of distance divided by the Debye length, with two plateaus and a crossover near one Debye length; a 90 millimolar monovalent salt gives a Debye length of 1 nanometre and 0.9 millimolar gives 10 nanometres.Section IV, A numerical example; Figure 2
The unscreened transverse-magnetic term is large enough to measure: taken to the force between two polystyrene microspheres in the proximity force approximation, the scattering result departs from the Poisson-Boltzmann plus Lifshitz model by 35 percent at 1 nanometre, 48 percent at 10 nanometres and an order of magnitude at 100 nanometres — a range the authors say optical tweezers should reach, the same instrument having already measured double-layer forces of the order of 10 femtonewtons.Section IV, Equations 27 and 28; Figure 3 and the discussion following it
The derivation uses the bulk model of the electrolyte’s response, so it holds for separations larger than about the Debye length; a more microscopic theory built on charge fluctuations would be needed for ion-specific effects and density correlations at very short distances. That same distance range is where the double-layer interaction between charged dielectric surfaces is suppressed, which is what makes it the right window for isolating the Casimir interaction from electrostatic signals.Section I, the paragraph on the bulk approximation; Section V, Conclusion, final two paragraphs
The way in
https://doi.org/10.1140/epjd/e2019-100225-8How to cite it
Paulo A. Maia Neto, Felipe S. S. Rosa, Luis B. Pires, Anna B. Moraes, Antoine Canaguier-Durand, Romain Guérout, Astrid Lambrecht, Serge Reynaud (2019) Scattering theory of the screened Casimir interaction in electrolytes doi:10.1140/epjd/e2019-100225-8