ch9 - Advanced Topics in Equilibrium Statistical Mechanics...

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Unformatted text preview: Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 16 Recap: semi-dilute, 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 non-uniform 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 gas-liquid 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 first-order 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 z-axis is arbitrary. Let’s insert a so-called “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 z-dependent 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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ch9 - Advanced Topics in Equilibrium Statistical Mechanics...

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