Range of the double-layer force and implications

Just to complicate matters further, we remark on the range of double-layer forces. All theories take it as axiomatic that the force between surfaces or between charged molecules should decay as exp(-jd) where is the Debye length defined by 

This result affects profoimdly our intuition on double-layer forces, especially in mixed electrolytes [26]. (It has been confirmed by direct measurement of the forces acting between molecularly smooth mica surfaces in solutions of the protein cytochrome-c [48]. This protein has a charge of 12+ or 8+ depending on pH, so the effects show up very strongly. The experiments have an interesting consequence. In many applications a trace amount of a 

If the solution between two surfaces contains surfactants that form highly charged micelles, a different effect occurs. Theory, confirmed by measurement [49-51], shows that the Debye length is then to be calculated as if the micelles and their boimd coimter-ions are simply ignored. The doublelayer force is here much longer ranged than it would be on the basis of standard theory. These forces are sometimes called depletion forces. The mechanism is the same as that for the oscillatory forces discussed in section 3.5.1. 

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