Ch7 - Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 7 Critical Phenomena The subject of critical phenomena i.e the study

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Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 7. Critical Phenomena The subject of critical phenomena , i.e., the study of the equilibrium and nonequi- librium properties of systems near a critical point (e.g., a gas-liquid critical point), has a long history, but many notable recent advances (culminating in a Nobel prize to Robert Wilson in 1982). A. Mean-Field Theory: The VDW fluid The simplest theories to construct of critical phenomena are so-called mean-feld theories —approaches that average the environment around a given molecule, thus decoupling that molecule from its neighbors and making the statistical mechanics calculation simple, like that of an ideal gas . Indeed the Van der Waals EOS for a fluid can be “derived” using this sort of reasoning: " p + a µ N V 2 # [ V Nb ]= Nk B T and is one of the simplest MF theories of the gas-liquid critical point. Let’s begin by rewriting the equation in terms of ρ = N V rather than V : ( p + 2 )(1 )= ρk B T. Qualitatively, we have: plane: pT plane: 1
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plane: Let’s now locate the critical point in the VDW model. Clearly ∂p ∂ρ ! T =0 , 2 p 2 ! T , VDW equation give three equations that can give p c , T c , ρ c in terms of a, b . Simpler is to write out the VDW equation as abρ 3 + 2 ( bp + k B T ) ρ + p or ρ 3 1 b ρ 2 + µ p a + k B T ab ρ p ab . Above T c , this has 1 real, 2 imaginary roots; these merge to 3 equal real roots at T c .Thu s ,a t( T c ,p c ): ρ 3 1 b ρ 2 + µ p c a + k B T c ab ρ p c ab =( p p c ) 3 = p 3 3 ρ 2 ρ c +3 ρρ 2 c ρ 3 c ρ c = 1 3 b c = a 27 b 2 ,k B T c = 8 a 27 b ,Z c = β c p c ρ c = 3 8 Next, we introduce reduced variables: ρ r = ρ/ρ c r = p/p c ,T r = T/T c which allows the VDW equation to be rewritten as: ( p r ρ 2 r )(3 ρ r )=8 T r ρ r . The absence of any material-dependent parameters is the basis for the familiar Law of Corresponding States ”, which you have undoubtedly studied in your thermo classes. Next, let’s zoom in on the critical region by further changing variables and expanding the VDW equation: t T T c T c = T r 1 “temp diFerence” ψ ρ ρ c ρ c = ρ r 1 “order parameter” (density diFerence) π p p c p c = p r 1 “pressure diFerence” 2
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[ π +1+3( ψ +1) 2 ][3 ( ψ + 1)] = 8( t + 1)( ψ This reduces to: π =4 t ³ 1+ ψ 1 1 2 ψ ´ + 3 2 ψ 3 1 1 1 2 ψ π 4 t ( 1+ 3 2 ψ + ... ) + 3 2 ψ 3 + for | t |¿ 1, | ψ 1. Next, we do a thermodynamic integration to get the Helmholtz f.e.: write A VA V ,A V = A V ( ρ, T ) Helmholtz f.e. per unit volume (intensive) p = ∂A ∂V ´ N,T = A V V V ´ N,T = A V V ∂ρ ´ N V ´ T = A V + ρ V ´ T p c ( π +1)= A V +(1+ ψ ) V ∂ψ ´ t or in terms of a dimensionless f.e. density F A V /p c : F ψ ) ∂F ! t =1+ π ( t, ψ ) Integrate this ODE subject to F ( t, 0) F 0 ( t ) .
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Ch7 - Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson 7 Critical Phenomena The subject of critical phenomena i.e the study

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