Tolman length

The Tolman length δ {displaystyle delta } (also known as Tolman's delta) measures the extent by which the surface tension of a small liquid drop deviates from its planar value. It is conveniently defined in terms of an expansion in 1 / R {displaystyle 1/R} , with R = R e {displaystyle R=R_{e}} the equimolar radius of the liquid drop, of the pressure difference across the droplet's surface: Δ p = 2 σ R ( 1 − δ R + … ) {displaystyle Delta p={frac {2sigma }{R}}left(1-{frac {delta }{R}}+ldots ight)} (1) σ ( R ) = σ ( 1 − 2 δ R + … ) {displaystyle sigma (R)=sigma left(1-{frac {2delta }{R}}+ldots ight)} (2) δ σ = 2 k R 0 {displaystyle delta sigma ={frac {2k}{R_{0}}}} Δ p = 2 σ s R s {displaystyle Delta p={frac {2sigma _{s}}{R_{s}}}} δ = Γ s Δ ρ 0 {displaystyle delta ={frac {Gamma _{s}}{Delta ho _{0}}}} δ = lim R s → ∞ ( R e − R s ) = z e − z s {displaystyle delta =lim _{R_{s} ightarrow infty }(R_{e}-R_{s})=z_{e}-z_{s}} The Tolman length δ {displaystyle delta } (also known as Tolman's delta) measures the extent by which the surface tension of a small liquid drop deviates from its planar value. It is conveniently defined in terms of an expansion in 1 / R {displaystyle 1/R} , with R = R e {displaystyle R=R_{e}} the equimolar radius of the liquid drop, of the pressure difference across the droplet's surface: In this expression, Δ p = p l − p v {displaystyle Delta p=p_{l}-p_{v}} is the pressure difference between the (bulk) pressure of the liquid inside and the pressure of the vapour outside, and σ {displaystyle sigma } is the surface tension of the planar interface, i.e. the interface with zero curvature R = ∞ {displaystyle R=infty } . The Tolman length δ {displaystyle delta } is thus defined as the leading order correction in an expansion in 1 / R {displaystyle 1/R} . Another way to define the tolman length is to consider the radius dependence of the surface tension, σ ( R ) {displaystyle sigma (R)} . To leading order in 1 / R {displaystyle 1/R} one has: Here σ ( R ) {displaystyle sigma (R)} denotes the surface tension (or (excess) surface free energy) of a liquid drop with radius R, whereas σ {displaystyle sigma } denotes its value in the planar limit. In both definitions (1) and (2) the Tolman length is defined as a coefficient in an expansion in 1 / R {displaystyle 1/R} and therefore does not depend on R. Furthermore, the Tolman length can be related to the radius of spontaneous curvature when one compares the free energy method of Helfrich with the method of Tolman: Any result for the Tolman length therefore gives information about the radius of spontaneous curvature, R 0 {displaystyle R_{0}} . If the Tolman length is known to be positive (with k > 0) the interface tends to curve towards the liquid phase, whereas a negative Tolman length implies a negative R 0 {displaystyle R_{0}} and a preferred curvature towards the vapour phase. Apart from being related to the radius of spontaneous curvature, the Tolman length can also be linked to the surface of tension'. The surface of tension, positioned at R = R s {displaystyle R=R_{s}} , is defined as the surface for which the Laplace equation holds exactly for all droplet radii: where σ s = σ ( R = R s ) {displaystyle sigma _{s}=sigma (R=R_{s})} is the surface tension at the surface of tension. Using the Gibbs adsorption equation, Tolman himself showed that the Tolman length can be expressed in terms of the adsorbed amount at the surface of tension at coexistence