Lecture15

# Lecture15 - EMA 6165 Polymer Physics Polymer Solubility Lecture 15 Dr Anthony Brennan University of Florida Department of Materials Science&

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Unformatted text preview: EMA 6165 - Polymer Physics Polymer Solubility Lecture 15 Dr. Anthony Brennan University of Florida Department of Materials Science & Engineering EMA 6165 Polymer Physics – AB Brennan EMA 1 Agenda • Introduction • Free Energy of Mixing • Entropic Contributions • Enthalpic Contributions • Cohesive Energy Density • Solubility Parameter • Polymer/Polymer Solubility EMA 6165 Polymer Physics – AB Brennan EMA 2 Polymer Solubility Entropy of Mixing The Classical Approach has been... The mixing of an ∆G = ∆H − T∆S “Ideal” Solution V = V1 + V2 where V = Molar Volume For ∆T = 0 , ∆P = 0 ∆Vmix = 0 ∆Hmix = 0 for gases, acceptable Raoult’s Law Obeyed, i.e. P = P o ni i i ni -mole fraction P -vapor pressure i ∆Hmix ≅ 0 for liquids EMA 6165 Polymer Physics – AB Brennan EMA 3 Polymer Solubility Entropy of Mixing V ∆ 1 = N 1k ln S V1 where N = # of molecules 1 ∆Smix = ∆S1 + ∆S2 V V = N1k ln + N1k ln V1 V2 EMA 6165 Polymer Physics – AB Brennan EMA 4 Polymer Solubility Entropy of Mixing or from kT V1 = N 1 P then kT V = ( N1 + N 2 ) P EMA 6165 Polymer Physics – AB Brennan EMA 5 Polymer Solubility Entropy of Mixing Therefore... ∆S mix n1 n2 = −N1k ln − N 2 k ln n1 + n2 n1 + n2 And by dividing by Avogadro’s Number... ∆ mix = −x1 R ln x1 − x2 R ln x2 S where x1 is the mole fraction of solvent EMA 6165 Polymer Physics – AB Brennan EMA 6 Polymer Solubility Entropy of Mixing ∆Smix = S solution − Scomponents or ∆S solution = Smix + Scomponents S1sol but since then o =1 S −R ln x1 ln x1 <0 S1sol >S o 1 EMA 6165 Polymer Physics – AB Brennan EMA 7 Polymer Solubility Entropy of Mixing ∆Gmix = N 1 ∆G1 + N 2 ∆G2 Where N1 = # of molecules and n1 = mole fraction then ∆Gmix = − kT ( N 1 ln n1 + N 2 ln n2 ) Where n1 = mole fraction of solvent n2 = mole fraction of solute EMA 6165 Polymer Physics – AB Brennan EMA 8 Polymer Solubility Entropy of Mixing Then ∆ mix = −k ( N 1 ln n1 + S N 2 ln n2 ) or ∆Smix n1 = −N 1k ln − n1 + n2 n2 N 2 k ln n1 + n2 EMA 6165 Polymer Physics – AB Brennan EMA 9 Polymer Solubility Entropy of Mixing Statistical Approach - Lattice ∆ mix = k ln Ω S N1 identical molecules No # of lattice sites identical in size Ω= (N No ! 1 ! N 2 !) EMA 6165 Polymer Physics – AB Brennan EMA 10 Polymer Solubility Entropy of Mixing Using Stirling’s approximation ln N ! =N ln N −N Hence: n ∆ mix =k [ ( N +N ) ln( n + ) − S 1 2 1 2 N 1 ln n1 − 2 ln n2 ] N which yields: ∆ mix =− k ( N 1 ln x1 + N 2 ln x2 ) S EMA 6165 Polymer Physics – AB Brennan EMA 11 Polymer Solubility Entropy of Mixing Or by summing all the sites of the lattice, Or one can define the Configurational Entropy or Combinatorial Entropy: Entropy ∆Smix = −k ∑ N i ln N i i =1 or in terms of moles: ∆Smix = − R ∑ ni ln ni i =1 EMA 6165 Polymer Physics – AB Brennan EMA 12 Polymer Solubility Entropy of Mixing • Now, one can use this to define the free Energy of Mixing for an Athermal Process: ∆ Gmix = RT ∑n ln n Restrictions: i =1 a. Solvent - Solute b. Solvent - Solvent c. Solute - Solute } i i Interactions are equal Consider: head to head head to tail d. Size of lattice sites identical EMA 6165 Polymer Physics – AB Brennan EMA 13 Two Small Molecules Bethe Lattice X O X X O X O X O O X X X O O O EMA 6165 Polymer Physics – AB Brennan EMA 14 Two Small 0ligomers (Chains) Bethe Lattice X X X X X X X X O O O O O O O O EMA 6165 Polymer Physics – AB Brennan EMA 15 One 0ligomer with Small Molecules Bethe Lattice X X X X X X X X X O O O O O O O O O EMA 6165 Polymer Physics – AB Brennan EMA O O O X O 16 Polymer Solubility Entropy of Mixing 1. Place Polymer on Lattice 2. Show Decrease in Entropy No N1 Lattice is Solvent is Polymer is xN 2 First segment has where ( i) sites N −x( i ) is first segment placed EMA 6165 Polymer Physics – AB Brennan EMA 17 Polymer Solubility Entropy of Mixing Every other has N −x(i ) N Thus a coordination # defines where the next segment ( i +1) is with respect to EMA 6165 Polymer Physics – AB Brennan EMA ( i) 18 Polymer Solubility Entropy of Mixing These conditions simulate concentrated solutions, However, we assume very dilute for ideality. Another restriction. Sum all sites to get: N − x i w1 =( N − x i ) z * N ( x −2 ) N − x i ( z −1) N EMA 6165 Polymer Physics – AB Brennan EMA 19 Polymer Solubility Entropy of Mixing and 1 Ω= N2 N2 ∏w i= 1 i which leads to the following in terms of volume fraction: ∆Smix = − R ( N1 lnυ 1 + N 2 ln υ2 ) EMA 6165 Polymer Physics – AB Brennan EMA 20 Summary Developed relationships for both entropic and enthalpic contributions for polymer mixing. Defined chi parameter. Developed relationships for cohesive energy density and polymer solubility parameter. EMA 6165 Polymer Physics – AB Brennan EMA 21 References • Introduction to Physical Polymer Science, 4th Introduction Edition, Lesley H. Sperling, Wiley Interscience (2006) ISBN 13-978-0-471-70606-9 • Principles of Polymer Chemistry, P.J. Flory (1953) Principles Cornell University Press, Inc., New York. Cornell • Polymer Chemistry, The Basic Concepts, P. C. Polymer Hiemenz (1984) Marcel Dekker, Inc., New York. Hiemenz • Principles of Polymer Chemistry, P.J. Flory (1953) Principles Cornell University Press, Inc., New York. Cornell EMA 6165 Polymer Physics – AB Brennan EMA 22 ...
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## This note was uploaded on 07/20/2011 for the course EMA 6165 taught by Professor Brennan during the Spring '08 term at University of Florida.

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