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

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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 ...
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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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