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

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online