Lecture16 - EMA 6165 - Polymer Physics Polymer Solubility...

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Unformatted text preview: EMA 6165 - Polymer Physics Polymer Solubility Lecture 16 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 • We finished the last lecture with the following: ∆ Gmix = RT ∑n ln n Restrictions: i =1 i i a. Solvent - Solute b. Solvent - Solvent c. Solute - Solute } Interactions are equal Consider: head to head head to tail d. Size of lattice sites identical EMA 6165 Polymer Physics – AB Brennan EMA 3 Polymer Solubility Entropy of Mixing And we began with the lattice structure: Lattice is Solvent is Polymer is xN2 No N1 sites First segment has where ( i) N − x( i ) 4 is first segment placed EMA 6165 Polymer Physics – AB Brennan EMA Polymer Solubility Entropy of Mixing Such that: Every other has N − x( i ) N is with respect to Thus a coordination # defines where the next segment ( i +1) ( i) 5 EMA 6165 Polymer Physics – AB Brennan EMA Polymer Solubility Entropy of Mixing These conditions simulate concentrated solutions, However, we assume very dilute for ideality. Another restriction. Sum all sites to get: w1 N − xi ( z − 1) N − xi = ( N − xi ) z * N N ( x −2 ) EMA 6165 Polymer Physics – AB Brennan EMA 6 Polymer Solubility Entropy of Mixing And 1 Ω= N2 ∏w i= 1 N2 i which leads to the following in terms of volume fraction: ∆Smix = − R( N1 lnυ1 + N 2 ln υ2 ) EMA 6165 Polymer Physics – AB Brennan EMA 7 Polymer Solubility Entropy of Mixing Where: x1 N1 υ1 = x1 N1 + x2 N 2 x2 N 2 υ2 = x1 N1 + x2 N 2 xi Ni = = # of segments or repeat units # of molecules Subscript 1 designates Solvent Subscript 2 designates polymer EMA 6165 Polymer Physics – AB Brennan EMA 8 Polymer Solubility Entropy of Mixing ∆ Smix = − k ( N 1 ln υ1 + N 2 ln υ 2 ) The Total change in Entropy for Mixing in terms of volume fraction EMA 6165 Polymer Physics – AB Brennan EMA 9 Polymer Solubility Entropy of Mixing Entropy per unit volume ∆ mix S ∑N i ln υi =− kV ∑υi where: ∑N i ln υ i 10 Which is Configurational or Combinatorial Entropy EMA 6165 Polymer Physics – AB Brennan EMA Polymer Solubility Enthalpy of Mixing Since Then ∆ =0 H for an ideal solution ∆Gmix = kT ∑ N i ln ni i =1 Normalization with respect to Avogadro’s No. yields ∆ Gmix = RT ∑ N i lnni EMA 6165 Polymer Physics – AB Brennan EMA i =1 11 Polymer Solubility Enthalpy of Mixing Combinatorial Entropy ∆S = −k ∑N i ln υi ∆ =− k ( N 1 ln υ 1 + N 2 ln υ2 ) S and based upon the assumption of Ideality, Which for a two-phase system ∆ =0 H EMA 6165 Polymer Physics – AB Brennan EMA 12 Polymer Solubility Enthalpy of Mixing Therefore: ∆Gmix = kT ( N 1 ln υ + 1 Where N1 = # N2 = # N 2 ln υ ) 2 of molecules of segments υ= volume fraction EMA 6165 Polymer Physics – AB Brennan EMA 13 Polymer Solubility Enthalpy of Mixing ∆ S R X1 UCST n=500 n=100 n=50 Xi= mole fraction of segments X2 Why? ∆ ≠0 H Consider: London van der Waals Dipole H-bonding Acid-Base, etc. 14 EMA 6165 Polymer Physics – AB Brennan EMA Polymer Solubility Enthalpy of Mixing Use pairwise interaction energy and arguments similar to the Combinatorial Entropy ∆ =2 w1,2 −w1,1 −w2 ,2 w Segment i is surrounded by zυ and 1 zυ 2 An interaction energy can be calculated for each segment and solvent molecule, w 15 EMA 6165 Polymer Physics – AB Brennan EMA Polymer Solubility Enthalpy of Mixing Since we are interested in 1,2 pairwise interactions, a new term is defined 1∆ w χ= z 2 kT Chi Parameter ∆ mix = N 1υ χk T H 2 Flory-Huggins Pairwise interaction 16 EMA 6165 Polymer Physics – AB Brennan EMA Polymer Solubility Enthalpy of Mixing ∆Gmix = kT [ N υυ 1 χ− 2 ( N1 lnυ1 + N 2 lnυ2 ) ] EMA 6165 Polymer Physics – AB Brennan EMA Positive } Positive Zero Negative 17 } Polymer Solubility Enthalpy of Mixing In terms of polymer (solute) volume fractions: ∆G 1 1 2 = RT [ ln(1 −υ2 ) + 1 − x υ2 + χ1 (υ2 ) 1 2 RT [ ln (υ1 ) + 1 − (1 −υ1 ) + χ1 (1 − υ1 ) x In terms of solvent volume fractions: ∆G 1 = To examine this important quantity in terms of pairwise interaction. EMA 6165 Polymer Physics – AB Brennan EMA 18 Polymer Solubility Enthalpy of Mixing ∆ = Nυ υ χ H kT 1 2 = N 1υ χ kT 2 ∆ w = 2 w1,2 − w1,1 − w2 ,2 ∆V E 1 = zNwii 2 Cohesive energy density, Vaporization 19 EMA 6165 Polymer Physics – AB Brennan EMA Polymer Solubility Enthalpy of Mixing It can be shown that... ∆ ∆ ∝ ( EV ,1 ) − w 2 1 2 E ( ∆ V ,2 ) 1 2 Since we are interested in the difference in sizes, put this on a volume basis: EMA 6165 Polymer Physics – AB Brennan EMA 20 Polymer Solubility Enthalpy of Mixing Substitute in for ∆Hmix 1 2 ∆Hmix ∆E ∆E 2 1 − υ1υ2 = Vm V2 V1 1 2 ∆ 1 E V1 Hildebrand and Scott Cohesive Energy Density 21 EMA 6165 Polymer Physics – AB Brennan EMA Polymer Solubility Enthalpy of Mixing Which provides a definition for the Solubility Parameter: δ= ∆E1 V1 1 2 cal 3 cm 1 2 which for an Athermal Process: δ1 = δ2 EMA 6165 Polymer Physics – AB Brennan EMA 22 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 23 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 24 ...
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