Lecture12

# Lecture12 - EMA 6165 – Polymer Physics Rubber Elasticity...

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Unformatted text preview: EMA 6165 – Polymer Physics Rubber Elasticity Lecture 12 Dr. Anthony Brennan University of Florida Department of Materials Science & Department Engineering 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 Rubber Elasticity Rubber Deviations From Ideality Deviations • First, consider the functionality: – Nc = # of active chains between network γ fV junctions: junctions: Nc = where: 2 f = functionality γ = crosslink site, i.e. junction V = volume This can be illustrated by: There are two junctions: EMA 6165 Polymer Physics – AB Brennan EMA 3 Thermodynamics Thermodynamics Hence: Hence: N c γ * f *V 2 * 4 =4 = Nv = = 2 2 *V V A total of six junctions per unit volume Each with a functionality of 4: 6* 4 Nv = =12 2 EMA 6165 Polymer Physics – AB Brennan EMA 4 Rubber Elasticity Rubber Effect of chain ends Effect q A front factor is added to the stress relationship for uniaxial deformation: f * − 2 −2 σ= N v RT ( λ − λ ) f* q Thus, for a functionality of 4, the stress is reduced by a factor of 2 compared to the standard form. q Next, lets evaluate the effect of chain ends: 2N A N v ( total ) = N v ( ideal ) − V 2N A = ( free ends per chain) where: V EMA 6165 Polymer Physics – AB Brennan EMA 5 Rubber Elasticity Chain Ends q Hence, Nv is given by the following relationship: Nv = ρ NA − MC 2ρ N A Mo M O = ( primary chain) where: Now we can develop the function: ρ N A 2 ρ N A σ o = − M C M o k t ( λ − λ−2 ) substitution: ρ R T 2ρ R T σ o = − ( λ − λ−2 ) Mo M C EMA 6165 Polymer Physics – AB Brennan EMA 6 Rubber Elasticity Graphical Analysis of Effect of Chain Ends q q ρ RT Which is also written as: σ o = MC ( λ − λ−2 ) thus, Mo is critical in determining the stress and furthermore: 2 M C >> M o q 2 MC 1 − Mo σ ≤0 Graphically, this relationship can be shown as: N V kT G or q ρRT MC − M O1 Other y-axis values used are: EMA 6165 Polymer Physics – AB Brennan EMA 7 q Thermodynamics or as illustrated in the text: σO Experimental Theoretical λ q Now, let’s finish by evaluation of a full statistical function that can account for the chain dimensions: k T −1 r f= L nl l 7 k T r 9 r 295 r 1539 r 3 + + + + l nl 5 nl 175 nl 875 nl 3 f= 5 EMA 6165 Polymer Physics – AB Brennan EMA 8 Rubber Elasticity Statistical Length Function q which can be simplified using a Langevin function: L( x ) = q 1 coth x − x The following is the result for a chain that obeys Gaussian Statistics: f= kT l r 3 nl σO r nl EMA 6165 Polymer Physics – AB Brennan EMA 9 Rubber Elasticity Phantom Networks q Another common correction used to correlate with end to end distance is: σo = q ρ RT MC 2 M C ri 2 1 − r2 Mo 0 ( λ − λ−2 ) Now, let’s evaluate the Phantom Network: σo = 1− 2 ρ R T f MC σO 2 M C ri 2 1 − M o r02 λ−1 = 1 Gaussian Chain N v kT 2 N v kT 3 ( λ − λ−2 ) Phantom Chain λ−1 EMA 6165 Polymer Physics – AB Brennan EMA fA =4 fA = 3 fP = 3 10 Summary • Developed relationships for biaxial stress Developed functions of elastomers functions • Developed relationships for Deviations for Developed non Ideal Behavior non • Demonstrated how retractive force of an Demonstrated elastomer scales with the end to end distance. distance. • Evaluated the Phantom Network behavior Evaluated of elastomers of EMA 6165 Polymer Physics – AB Brennan EMA 11 References • Introduction to Physical Polymer Science, 3rd Introduction Edition, Lesley H. Sperling, Wiley Interscience (2001) ISBN 0-471-32921-5 • Principles of Polymer Chemistry, P.J. Flory (1953) Principles Cornell University Press, Inc., New York. Cornell • 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 (1953) Principles Cornell University Press, Inc., New York. Cornell EMA 6165 Polymer Physics – AB Brennan EMA 12 ...
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