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EMA6165-lect-11 - Mntcria‘ls>23 Hindered Rotating...

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Unformatted text preview: Mntcria‘ls' >23 Hindered Rotating Chain Mode| 1 A ' fianfiflkfi ssumptlons n h be. .. “ O H. EMA 6165 Polymer Physics - AB Brennan 10 Materials \. ' ' - Scic%-\t‘ HIndered Rotating Chaln Model ,.. - 3—“~E1..~""\_T\-§\fi Assumptions Engl‘n- be u n - Hindered rotating model: —torsion (valence) angle dependence —first order interactions —high level interactions EMA 6165 Polymer Physics - AB Brennan 10 Z > M‘gl‘criails' X ’L ~— 2 f , —— Hindered Rotating wamd‘eh O “gmemg <r2> : n12[1+cos(180— 1)][Hc—os<¢)] 1— 005(180 — r) 1— cos<¢> <r2> : [cl-{n c056] 1+ cos<¢> 1— c036 1— cos<¢> I _ - Mslcfials \- Chaln Statlstlcs I - Maxwellian Demon — Random walk - NW5 on direction — Fjggd segment - Begin by Coin toss EMA 6165 Polymer Physics — AB Brennan 3 One Dimensional Random Walk Binomial Distribution - Binomial Distribution - Rewrite in displacements — n=nH+ nT n! l ,,' — x=(nH+ nT)l p(x.n)=x—VA. ~Thus, “UNI! — nH=1IZ(n+x/I) 2 2 — nT=1I2(n-xll) ' Substitute intoa binomial probability funcfion: EMA 6165 Polymer Physics - AB Brennan 7 Matcria s Critical Poilgjsfixk .p Englhéémg - r2 factor increases with r, reflecting higher probability of finding end, i.e., more available positions for segments 2n!2 6 o decreases with r, indicating large displacements are less probable. EMA 6165 Polymer Physics - AB Brennan 14 Materials 3:: Three Dimensional Randéfi‘iflfitk‘i Gaussian Di ngme'mg Hence : _ [(5)] pm): 2x272 47rlr4e 2”” dr 0 which upon integratio 11 yields : 2 ' 2 Which is the mean 5 . _ . quare end-to-end <r — n/ distance for a freely rotating, freely jointed chain EMA 6165 Polymer Physics - AB Brennan 16 (0&8; 0K5> C(AM ofpwfim 61PM“; (ll/mm Mg“ 0‘ M a lag.“ g) Shag} '? ...
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