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Lecture8-9

# Lecture8-9 - EMA 6165 Polymer Physics Rubber Elasticity...

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Unformatted text preview: EMA 6165 - Polymer Physics Rubber Elasticity Lecture 8 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 • Swollen State • Deviations from Classical Theories – – – Affine Deformation Phantom Behavior Mooney-Rivlin Model EMA 6165 Polymer Physics – AB Brennan EMA 2 Chain Statistics Chain Statistical Mechanics Statistical • Objective: predict mechanical properties from first Objective: principles, i.e., equilibrium thermodynamics principles, q Two Approaches -Gibbs Free Energy ∆G = ∆H −T∆S q Helmholtz Free Energy ∆A = ∆E − T∆S EMA 6165 Polymer Physics – AB Brennan EMA 3 Consider: Consider dA << 1 Thermodynamics Thermodynamics dA = dE − TdS − SdT One knows that: dE = dq + dw for an Ideal Gas: Ideal for Hence: dw = −PdV + fdl dE = dq − PdV + fdl EMA 6165 Polymer Physics – AB Brennan EMA 4 Thermodynamics q If one assumes reversibility, i.e., dq = TdS, simple arithmetic gives: dA = fdl − SdT − PdV q Practically, though one is interested in: ∂ A =f ∂ l V ,T ∂A ∂ T = −S l ,V * ? EMA 6165 Polymer Physics – AB Brennan EMA 5 Thermodynamics So let’s evaluate the critical values: ∂2 A ∂f = ∂ l∂ T V ∂ T Requires constant Requires volume volume l ,V Also, one can show that: ∂ A ∂ l∂ T 2 ∂ S Requires =− constant temp ∂ l constant V ,T V EMA 6165 Polymer Physics – AB Brennan EMA 6 Thermodynamics Using Maxwell’s equation: ∂2 A ∂ f ∂T ∂ l = ∂T l ,V ∂S =− ∂l V ,T ∂2 A = ∂ l ∂T One can write: ∂ f = ∂ T l ,V ∂ S − ∂ l V ,T EMA 6165 Polymer Physics – AB Brennan EMA 7 Thermodynamics Thus: ∂ A = ∂ l T ,V ∂ S ∂ E − T ∂l ∂ l V ,T T ,V And since thermal reversibility is a boundary And condition, dq = TdS Substitution provides: ∂ E ∂ S − T f= ∂ l T ,V ∂ l V ,T Equation of State EMA 6165 Polymer Physics – AB Brennan EMA 8 Thermodynamics This Equation of State, using the identity given This Equation previously, can be rewritten: previously, ∂ E ∂ f f= +T ∂ l T ,V ∂ T l ,V Which is a second Equation of State. Which Equation EMA 6165 Polymer Physics – AB Brennan EMA 9 Thermodynamics Hence, one can measure Hence, the internal energy of the polymer through equilibrium stress measurements. measurements. ∂ S −T ∂l V,T f ∂E ∂l T, V Can you explain the Can behavior? behavior? Mathematically? EMA 6165 Polymer Physics – AB Brennan EMA 10 Thermodynamics Let’s first examine the relationship: ∂ S f = − T ∂ l V ,T Evaluate the direction given by the sign, i.e.: ( +) = − ( + ) ( −) Does it make sense? EMA 6165 Polymer Physics – AB Brennan EMA 11 Thermodynamics S = k ln Ω Now consider: ∂ f f = +T ∂ T V , l ( +) = + (+) (+) EMA 6165 Polymer Physics – AB Brennan EMA 12 Thermodynamics This confirms our earlier discussion, i.e. force This increases with increasing temperature. This was observed and reported by John Gough in 1805 using natural rubber. 1805 ∂ f f = +T ∂ T V , l EMA 6165 Polymer Physics – AB Brennan EMA 13 Thus, force on a good elastomer Thus, scales with: scales Thermodynamics kT T This is limited to above the Tg for the This elastomer elastomer –meets the requirement of equilibrium meets conditions conditions –assumes no crystallization EMA 6165 Polymer Physics – AB Brennan EMA 14 Thermodynamics Force ∂ f +T ∂ T V , l Tg For a good elastomer: Temp ∂S =T f = −T ∂l T ,V ∂ f ∂T EMA 6165 Polymer Physics – AB Brennan EMA V , l 15 Thermodynamics Consider how entropy Consider changes for a single chain: chain: ∂ S ∝∂ l ? S = k ln Ω Ω ≈ P(r ) = ? EMA 6165 Polymer Physics – AB Brennan EMA 16 Thermodynamics First, let’s evaluate the inflection in the curve? S ≈ k ln Ω ≈ k ln C ' e −3r 2 2 2 nl Note this is only the density function, Note since we are only interested in the # of conformations! Obviously the function scales with <r2>. <r EMA 6165 Polymer Physics – AB Brennan EMA 17 Thermodynamics So we will rewrite: f= ∂ S −T ∂ r V ,T ∂ 3r ≈ −T k ln C '−k 2 2nl ∂r 2 f Thru further analysis: f= r 3kT 2 nl HOOKE’S LAW EMA 6165 Polymer Physics – AB Brennan EMA 18 Thermodynamics Force Real Chain 3kT slope = nl 2 Note the front factor kT Elongation ∝ r Elongation However, a real system has a slight deviation. However, Why? Why? EMA 6165 Polymer Physics – AB Brennan EMA 19 Thermodynamics And since we have shown And that: that: nl ≈ r 2 2 o Then, it is clear that the following holds true: 3kT 3kT m= 2 = 2 nl ro for f vs. r Force per area EMA 6165 Polymer Physics – AB Brennan EMA 20 Thermodynamics Consider the the following: ∂ E ∂ S f= − T ∂l ∂ l T ,V V ,T ∂ E = ∂ l T ,V ∂ S f + T ∂l V ,T EMA 6165 Polymer Physics – AB Brennan EMA 21 Thermodynamics And from the other Equation of State: ∂ E ∂ f = f − T ∂ l T ,V ∂ T V ,l Apply the 3D Statistical Chain Dimension: ∂ E = ∂ l T ,V 3kT ( r ) − 2 r ∂ f T ∂ T V ,l EMA 6165 Polymer Physics – AB Brennan EMA 22 Thermodynamics ∂ E = ∂ l T ,V ∂ 3kT ( r ) T − 2 ∂T nl 3kT ( r ) nl 2 V ,l Now we need to evaluate <r2> with respect to the effect of with Temperature Temperature 3kT ( r ) ∂ E − ( 3kT ( r ) = 2 ∂ l T ,V ro ∂ T 1 2 + ∂ T ro V ,l ∂ 1 2 T ∂ T ro V ,l EMA 6165 Polymer Physics – AB Brennan EMA 23 Thermodynamics Which simplifies to: ∂ E 3kT ( r ) ∂ ro − = 2 2 ∂ l T ,V ∂ T V ,l ro 2 EMA 6165 Polymer Physics – AB Brennan EMA 2 24 Thermodynamics Now we’ve shown that: ∂ E ∝ ∂l T ,V ∂ ro ∂ T V ,l 2 Another way to express this relationship is: f E Retractive force due to internal energy = Retractive force due to entropy fS EMA 6165 Polymer Physics – AB Brennan EMA 25 Thermodynamics 3kT ( r ) 2 fE = fS 22 o r ∂ ro ∂ T V ,l 2 − () 3kT r 2 22 o r Which leads to: f E ∂ ln ro = fS ∂ ln T 2 Think of the role of chemistry in this function. EMA 6165 Polymer Physics – AB Brennan EMA 26 Summary • Thermodynamics explain the retractive Thermodynamics force dependence in elastomers on: force – Temperature – Entropy • A relationship correlating mean square relationship end to end distance was developed end • The ratio of internal energy to entropy was The shown to relate directly with chain dimensions. dimensions. EMA 6165 Polymer Physics – AB Brennan EMA 27 References • Introduction to Physical Polymer Science, 3rd Edition, Introduction Lesley H. Sperling, Wiley Interscience (2001) ISBN 0-471Lesley 32921-5 Principles of Polymer Chemistry, P.J. Flory (1953) 32921-5 Cornell University Press, Inc., New York. Cornell • The Physics of Polymers, Gert Strobl (1996) SpringerVerlag, New York. • Some figures were reproduced from Some • Principles of Polymer Chemistry, P.J. Flory (1953) Cornell Principles University Press, Inc., New York. University EMA 6165 Polymer Physics – AB Brennan EMA 28 ...
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