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

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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 ...
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## 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.

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