Lecture13

# Lecture13 - EMA 6165 – Polymer Physics Swollen Networks...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EMA 6165 – Polymer Physics Swollen Networks Rubber Elasticity Lecture 13 Dr. Anthony Brennan University of Florida Department of Materials Science & Engineering EMA 6165 Polymer Physics – AB Brennan EMA 1 Agenda • • • Introduction Thermo-Elastic Behavior Statistical Mechanical Approach – Uniaxial Deformation – Biaxial Deformation • Deviations from Classical Theories – Affine Deformation – Phantom Behavior – Mooney-Rivlin Model • Swollen State EMA 6165 Polymer Physics – AB Brennan EMA 2 Consider the following: Rubber Elasticity Swelling Behavior Increase solvent to increase volume, but what happens to stress? EMA 6165 Polymer Physics – AB Brennan EMA 3 Rubber Elasticity Swelling Behavior q First we must examine the Gibbs Free Energy of Mixing. Why? ∆Gtotal = ? δ 2 χ q How would one model this behavior? ∆Gtotal = ∆Gmixing + ∆Gelastic q What are the critical factors? – Mixing and limit of swelling q The problem is you do not have all the necessary info. So... q Examine the mixing between solvent and chain... EMA 6165 Polymer Physics – AB Brennan EMA 4 Rubber Elasticity Swelling Behavior q Evaluate the effect of mixing first: ∆Gmixing what is the sign? ∆Gelastic what is the sign? - } + ∆V = + Evaluate: ∆Gmix = ∆Hmix − T∆Smix Assume: ∂ ∆GTotal ∂ ∆G ∂ ∆G + = µ A − µ Ao = 0 = ∂n ∂n ∂ na i.e., benzene and natural rubber ∆Hmix = 0 mixing a Then: elastic a ∆Gmix = −T∆Smix ∆Smix = + k ln Ω EMA 6165 Polymer Physics – AB Brennan EMA 5 Rubber Elasticity q Swelling Behavior From thermo of solutions: ∆Smix = + k ln Ω AAAAAA BBBBBB AAAAAA BBBBBB AAAAAA . . . + AAAAAA ∆Smix = − kΣ( ni ln ϕi ) BBBBBB BBBBBB ABAAAA ABBAAA AABBAA AAABBB q This is corrected by a simple analysis we will see next: ∆Gmix = RT ( na ln ϕ a + nb ln ϕb + na χa ϕb ) ∆Hmix = kTχna ϕb EMA 6165 Polymer Physics – AB Brennan EMA 6 Rubber Elasticity Swelling Behavior q q Furthermore, ∆Hmix = kTχna ϕb ∆Gelastic = ∆Helastic − T∆Selastic Assume Gaussian chains: ∆Selastic − Nck = f ( λ) 2 ∆Gelastic = − T∆S = N c kT f ( λ ) 2 ∂∆Gelastic l x l y lz V = λx * λ y * λz = * * = ∂na l0 l0 l0 V0 (By inspection) EMA 6165 Polymer Physics – AB Brennan EMA 7 Rubber Elasticity Swelling Behavior q Since it is an isotropic deformation, one can write λx = λ y = λ z = q Thus, we have: ∆Gelastic = V V0 1/ 3 = ϕB [ N v kT 3ϕ B −2 / 3 − 3) − ( ln ϕ B −1 ) ( 2 −1/ 3 q The material stops swelling. Why? q Elastic Term! What is the factor of interest? ∆Gelastic ρRT ( 3ϕ B −2 / 3 − 3) − ( ln ϕ B −1 ) = 2 Mc [ EMA 6165 Polymer Physics – AB Brennan EMA 8 Rubber Elasticity Swelling Behavior - Chain Rule q The chain rule given as: ∂G ∂ϕ ∂ Gelastic * = ∂ϕ ∂n ∂ na elastic b q a Assume 1 cc of dry rubber, Evaluate: ∂ϕb = 1 → 1 + na Va ∂ϕb 2 − V ϕb = ∂na q b ∂ϕb V =− ∂na (1+ n V ) ∂ ϕb ∂ na a a 2 a substitution: ∂ 3ϕ −2 / 3 −3 − ln ϕb −1 b ρ RT ∂ ∆Gelastic = ∂ na 2 Mc ∂ na [( )( EMA 6165 Polymer Physics – AB Brennan EMA )] 9 Rubber Elasticity Swelling Behavior: Flory-Reiner q Which leads to: ∂ ∆Gelastic V ρ RT = ( 2ϕ 2M ∂ na 1/ 3 a b − ϕb ) c q Combine the terms for entropy and elasticity: ∆Gmix + ∆Gelastic = 0 = q ∂ ∆Gmixing ∂ ∆Gelastic + ∂ na ∂ na Flory-Reiner Relationship: [ ∆GSW = 0 = RT ln( 1 − ϕ B ) + ϕ B + χ Aϕ B + 2 VA ρRT 1/ 3 2ϕ B − ϕ B 2 Mc EMA 6165 Polymer Physics – AB Brennan EMA ( ) 10 Rubber Elasticity Swelling Behavior: Flory-Reiner q Hence, one can easily evaluate these terms q Drop the RT term – Measure ϕ b in the laboratory – Measure ρ in the laboratory – Measure Va in the laboratory q Difficulty is in the χ term. – χ can be estimated – Will evaluate in the next section on solution behavior. EMA 6165 Polymer Physics – AB Brennan EMA 11 Summary • Developed relationships for swelling Developed of elastomers based upon thermodynamics. thermodynamics. EMA 6165 Polymer Physics – AB Brennan EMA 12 References • Introduction to Physical Polymer Science, 2nd Introduction Edition, Lesley H. Sperling, Wiley Interscience (1992) ISBN 0-471-53035-2 (1992) • Principles of Polymer Chemistry, P.J. Flory Principles (1953) Cornell University Press, Inc., New York. (1953) • The Physics of Polymers, Gert Strobl (1996) The Springer-Verlag, New York. Springer-Verlag, • Some figures were reproduced from Some • Principles of Polymer Chemistry, P.J. Flory Principles (1953) Cornell University Press, Inc., New York. (1953) EMA 6165 Polymer Physics – AB Brennan EMA 13 ...
View Full Document

## 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.

Ask a homework question - tutors are online