Lecture6 - EMA 6165 - Polymer Physics Chain Statistics...

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 Chain Statistics Chain Dr. Anthony Brennan Dr. University of Florida Department of Materials Science & Department Engineering 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 Random Walk Random • Maxwellian Demon Maxwellian – Random walk – No restrictions on No direction direction – Fixed segment • Begin by Coin toss EMA 6165 Polymer Physics – AB Brennan EMA 3 q 10 Steps Chain Statistics Chain Random Ten Step Random – H = +1 step – T = -1 step EMA 6165 Polymer Physics – AB Brennan EMA 4 Freely Rotating Chain Model Freely Vector Analysis Vector • Binomial Distribution – where: n = no. of tosses – nH = no. of heads no. – nT = no. of tails no. – p = probability of result n! n H nT p(nH , nT ) = pH pT nH ! nT ! pH = pT = 1 / 2 EMA 6165 Polymer Physics – AB Brennan EMA 5 Freely Rotating Chain Model Freely Vector Analysis Vector • Total count (p) is too low – Thus, the n! • n! yields an overcount – HHHTTT – HTHTHT – HTTHHT • n! n H nT p(nH , nT ) = pH pT nH ! nT ! Thus, nH!nT! pH = pT = 1 / 2 EMA 6165 Polymer Physics – AB Brennan EMA 6 One Dimensional Random Walk One Binomial Distribution Binomial • Binomial Distribution • Rewrite in Rewrite displacements displacements – n = nH+ nT – x = (nH+ nT) l • Thus, – nH = 1/2(n + x/l) – nT = 1/2(n - x/l) n! p( x , n) = n+ x n+ x l ! l 2 2 1 2 ! • Substitute into a Substitute binomial probability function: function: EMA 6165 Polymer Physics – AB Brennan EMA 7 n Three Dimensional Random Walk Three Gaussian Distribution Gaussian For n>>1, i.e, n -> infinity, Sterlings For Approximation can be applied. Approximation nl + x x nl + x x − ln p ( x, n) ≈ ln1 + + ln1 − 2l nl 2l nl given n is very large, one can simplify : 2 x x 1 x ln1 + ≈ − +...which yields : nl nl 2 nl EMA 6165 Polymer Physics – AB Brennan EMA 8 Three Dimensional Random Walk Three “Normalized” Gaussian Distribution “Normalized” Gaussian − x2 2 nl 2 p(x, n) = k e where k is a normalization factor, defined as : − x2 +∞ 2 nl 2 k∫e −∞ dx = 1 EMA 6165 Polymer Physics – AB Brennan EMA 9 Three Dimensional Random Walk Three Gaussian Distribution Gaussian − x2 +∞ 2 nl 2 k∫e −∞ dx = 1 given that : − x2 +∞ 2 nl 2 ∫e dx, is a gamma function −∞ EMA 6165 Polymer Physics – AB Brennan EMA 10 Three Dimensional Random Walk Three Gaussian Distribution Gaussian which upon integration and substitution becomes : ( k = 2π nl 2 ) −1 2 EMA 6165 Polymer Physics – AB Brennan EMA 11 Three Dimensional Random Walk Three Gaussian Distribution Gaussian p( x , n) =( 2π nl 2 ) −1 2 e − x2 2 2 nl =r Now consider 3D i+2 yi i+1 i zi+1 θ xi zi φ i i+1 i-1 yi+1 EMA 6165 Polymer Physics – AB Brennan EMA 12 Three Dimensional Random Walk Three 3D Gaussian Distribution Gaussian • Cartesian Coordinates ( − 3 x2 + y2 + z 2 −3 2 2 nl 2 2 n p ( x, y, z, n) = 2π l 3 q e ) dx dy dz Polar Coordinates: Three Dimensional Random Polar Walk Walk −3 ( r 2 ) −3 2 2 nl 2 n 2 2 p ( r , n) = 2π l 3 4π r e EMA 6165 Polymer Physics – AB Brennan EMA dr 13 Critical Points • Normalized Probability Function • r2 factor increases with r, reflecting higher r2 probability of finding end, i.e., more available positions for segments available ( ) e q −3 r 2 2 nl 2 decreases with r, indicating large displacements are less probable. EMA 6165 Polymer Physics – AB Brennan EMA 14 Three Dimensional Random Walk Three Gaussian Distribution Gaussian • Consider: ∑ i f i ( m − ms ) r 2 k = ∑ f i ( ri ) , where : 2 i =1 f i = P( r , n ) dr EMA 6165 Polymer Physics – AB Brennan EMA 15 Three Dimensional Random Walk Three Gaussian Distribution Gaussian Hence : −3 n 2 2 p (r , n) = 2π l 4π ∫ r e 3 0 which upon integration yields : r 2 = nl 2 ( ) −3 r 2 ∞ 2 4 2 nl dr Which is the mean square end-to-end Which distance for a freely rotating, freely jointed chain jointed EMA 6165 Polymer Physics – AB Brennan EMA 16 Summary • A one dimensional random walk model one was derived using statistical methods was • A three dimensional random walk three model was derived using statistical methods methods • Statistical functions define limits for Statistical chain dimensions. chain EMA 6165 Polymer Physics – AB Brennan EMA 17 References • Introduction to Physical Polymer Science, 4th Introduction Edition, Lesley H. Sperling, Wiley Interscience (2006) ISBN 13-978-0-471-70606-9 (2006) • 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 18 ...
View Full Document

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.

Ask a homework question - tutors are online