# Lecture7 - EMA 6165 Polymer Physics Random Flight Analysis...

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Unformatted text preview: EMA 6165 - Polymer Physics Random - Flight Analysis Lecture 7 Dr. Anthony Brennan University of Florida Department of Materials Science & Engineering EMA 6165 Polymer Physics – AB Brennan EMA 1 Agenda • • • • • • • • Rotational States Bond Rotational Energetics Spatial Relationships Characteristic Dimensions Freely Jointed Model Freely Rotating Model Hindered Rotation Random Flight Statistical Model EMA 6165 Polymer Physics – AB Brennan EMA 2 Chain Statistics Chain Statistical Mechanics Statistical One dimensional chain ( ) −x 2 2 nl 2 − 1 2 p ( x, n)= 2πnl e 3D chain - Cartesian Coordinates 2 −3 ( −3 x 2 + y 2 + z 2 2 nl 2 n 2 2 p ( x, y, z , n) = 2π l e 3 q 3D chain - Polar Coordinates n 2 p( r , n) = 2π l 3 −3 2 4π r e 2 dx dy dz ( ) −3 r 2 2 nl 2 EMA 6165 Polymer Physics – AB Brennan EMA ) dr 3 Three Dimensional Random Walk Three 3D Random Walk 3D ∑ f (m−m ) r = ∑ f (r ) i i k s 2 i =1 i i 2 , where: f i = P( r , n) dr , hence: n 2 p(r , n) = 2π l 3 −3 2 ∞ ( ) −3 r 2 2 4 2 nl 4π ∫ r e dr 0 which upon integration yields: r = nl 2 2 3D Random Walk EMA 6165 Polymer Physics – AB Brennan EMA 4 Three Dimensional Random Walk Three Statistical Segments Statistical Similarly, one can define : ' n =n m mass of repeat units that compose ' the statistical segment n , and ∑ i f i ( m − ms ) k EMA 6165 Polymer Physics – AB Brennan EMA 5 Three Dimensional Random Walk Three Statistical Segments Statistical which can be shown to equal : r 2 ' '2 = nl From which we define the radius of gyration : s 2 = r 2 ' '2 6 = nl 6 EMA 6165 Polymer Physics – AB Brennan EMA 6 Three Dimensional Random Walk Three RIS Now consider: 1 − cos φ Apply the Boltzman Distribution of states, Apply i.e., RIS i.e., −V ( φ ) 2π RT b = cos φ = ∫0 e 2π ∫0 e −V ( φ ) EMA 6165 Polymer Physics – AB Brennan EMA cos φ dφ RT d φ 7 Three Dimensional Random Walk Three RIS • b iintroduces a temperature ntroduces dependence dependence – <r2> ~ f{n, l, θ,, φ } f{n, θ – molecular weight – chemistry – temperature – environment EMA 6165 Polymer Physics – AB Brennan EMA 8 Three Dimensional Random Walk Three RIS • b accounts for multiple valence accounts angles angles – trans, cis – +/- gauche – eclipsed EMA 6165 Polymer Physics – AB Brennan EMA 9 Three Dimensional Random Walk Three Calculated Dimensions Calculated EMA 6165 Polymer Physics – AB Brennan EMA 10 Three Dimensional Random Walk Three Calculated Dimensions • Assume: Out – Molar mass = 106 g/mol – Mo = 104 g/mol – <r2>0.5 = 73.5 nm2 (literature) (literature) Solvent In • n = {106/104} x 2 = 1.92 x 104 • l = 0.154 nm • <r2>o = 455 nm2 • <r2>FR = 910 nm2 • C = <r2>meas/ <r2>o = 5.92 5.92 • Chain Expansion: • <r2>0.5 calc/ <r2>o0.5 = 2.44 calc EMA 6165 Polymer Physics – AB Brennan EMA 11 Three Dimensional Random Walk Three Calculated Dimensions EMA 6165 Polymer Physics – AB Brennan EMA 12 Summary • The statistical 3D random walk model has The been developed been • The three dimensional random walk model The accounts for the distribution of RIS that exist in macromolecules. exist • The statistical model predicts The experimental results better than a vector based model. based EMA 6165 Polymer Physics – AB Brennan EMA 13 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 • The Physics of Polymers, Gert Strobl (1996) The Springer-Verlag, New York. Springer-Verlag, • Some figures were reproduced from Polymer Some Physics, (1996) Ulf Gedde, Chapman & Hall, New York. York. EMA 6165 Polymer Physics – AB Brennan EMA 14 ...
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