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Unformatted text preview: Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 16 Recap: semidilute, good solvent ξ ( c ) independent of N ! for c ∗ ¿ c ¿ b − 3 ξ ∼ b ( cb 3 {z} φ p ¿ 1 ) − 3 / 4 ∼ c − 3 / 4 So, b (1 A ◦ ) ¿ ξ (20 A ◦ ) ¿ R (100 A ◦ ) Π ∼ k B T ξ 3 ∼ c 9 / 4 9. Interfacial Statistical Mechanics Thus far, we have restricted our attention to homogeneous , or uniform liq uids. However, there are many nonuniform systems of interest (solids, sur faces, interfaces, etc.) and the equilibrium SM of such systems has been studied in some detail. In this course we will limit our attention to the simplest of inhomogeneities —a planar interface between coexisting gas and liquid phases: 1 The density profile turns out not to be sharp, but has a width of order the bulk correlation length ξ . We will prove that this is indeed the proper interfacial width scale shortly, but now simply focus on the problem of determining the equilibrium ρ ( z ) and the interfacial energetics . A. Thermo of Interfaces Let’s now review surface thermodynamics. Consider the gasliquid interface as above in an open system: dE = T dS − pdV + µdN + γdσ {z} work to create more interface area σ 1st Law The “surface tension” γ is thus γ = ( ∂E/∂σ ) S,V,N Now, S, V, N are extensive ; T, p, µ are intensive . E ( λS, λV, λN, λσ ) = λE ( S, V, N, σ ) so E is a firstorder homogeneous function of these variables. By Euler’s theorem for such functions: E ( { x o } ) = X i ( ∂E/∂x i ) x i so E = T S − pV + µN + γσ Notice that the origin of the zaxis is arbitrary. Let’s insert a socalled “dividing surface” at z d , such that we call everything for z < z d gas and everything with z > z d liquid . Then imagine a hypothetical ﬂuid with a step function ρ ( z ): For this ﬂuid, E L = T S L − pV 2 + µN L + L ↔ G 2 and V = V L + V G Now, define surface excess properties X S for any extensive property X : X S = X − ( X L + X G ) Clearly V S = V − ( V L + V G ) = 0. We can write N S = σ Z L − L dzρ ( z )  {z } N − σ Z z d − L dzρ d ( z )  {z } N G − σ Z L z d dzρ d ( z )  {z } N L N S −−−→ L →∞ σ Z ∞ −∞ dz [ ρ ( z ) − ρ d ( z )] Clearly, by adjusting z d , we can cause this integral to vanish . This choice −→ Gibbs dividing surface . Now, E S = E − E L − E G = T S S − p ( V S ) + µ ( N S ) + γσ Thus A S = γσ or γ = A S /σ surface Helmholtz f.e. per unit area. Another way to write this is in terms of a zdependent free energy density A V ( z ): γ = Z −∞ dz [ A V ( z ) − A V,G ] + Z ∞ dz [ A V ( z ) − A V,L ] where we have shifted the origin of z to z d , or: γ = Z ∞ −∞ dz { A V ( z ) − θ ( − z ) A V,G − θ ( z ) A V,L } where θ ( z ) = ½ 1 , z > unit , z < step fnc....
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 Spring '09
 Surface tension, dz, zd, density functional theory, density profile, equilibrium density profile

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