3.11 Recitation #9 November 4, Rubber Elasticity
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1 3.11 Recitation #9 Novembe 4, 2003 Moe Disode Rubbe Elasticity The Entopic Sping Entopy - a natual law that expesses the diving foce towads disode Less Disode Rubbe bands ae made fom polymes, but the chains ae cosslinked to povide a netwok. The amophous phase in PE is also said to be ubbey it is above its T g but is constained by the suounding cystals and so cannot be said to be liquid-like. o the ubbe bands, it is the cosslinks which detemine the popeties. The cosslinks povide a 'memoy'. When the netwok is stetched, entopic foces come into play which favou etaction, etuning the netwok to its oiginal unstetched/equilibium state.
2 Changes to the Rubbe Netwok upon stetching Loss of entopy upon stetching, means that thee is a etactive foce fo ecovey when extenal stess emoved. This is why a ubbe band etuns to its oiginal shape. Moe Disode Less Disode Entopy - a natual law that expesses the diving foce towads disode poly(styene) < 2 > 1/2
3 Random Coil Configuations of Polymes: DNA simulation (*EBS Lett. 371: ) DNA (*Z. Shao, Let s look at eely Jointed Chain Model conside stetching a single andom coil polyme chain : chain chain Polyme Chain : Random walk in space. (Gaussian) b mean Can be thought of as a feely jointed chain. Joint length is b. An independently oiented segment. It is NOT usually a monome length, usually 4 o 5 monomes long. A simple eminde of polyme statistics.
4 Suppose the walk has N links: End to end distance R(N) om Rubbe elasticity : = instantaneous chain end-to-end sepaation distance (Daw on boad--- squiggly lines with beginning and end sepaated by ) < 2 > = na 2 oot mean squae end to end distance a = statistical segment length local chain stiffness n = # of a s Lc = contou length length of fully extended chain. Pobability of finding a fee chain end a adial distance,, away fom a fixed chain end (oigin) ~ omega = P(R) = (4b 3 2 )/sqt(pi)*exp(-b 2 2 ) whee b = sqt (3/(2na 2 )) This is Gaussian fom Macostate is defined by the length. Micostates ae the diffeent andom walks. So.. P 3R 2 2 2Nb ( R) ~ e ~ Ω( R) (# of µstates with length R) (N=n and b=a) Configuational Entopy (measue of disode) = S = kb*ln[p()] Helmholtz ee Enegy = A o H = -Tkb*ln[P()] Entopic elastic foce, linea elasticity (hookean sping) f o = -da()/d^2 Entopic chain stiffness = k = d()/d o second deivative of A. a andom walking polyme at finite T is a Hookean sping Why is this useful? Because these equations define the stetching of a single polyme chain. The following chat defines seveal types of Elasticity Models fo Single Polyme Chains you may need to descibe the diffeence between a couple of these on you poblem set.
5 MODEL SCHEMATIC ORMULAS eely-jointed Chain (JC) (Kuhn and Gün, 1942 James and Guth, 1943) elastic (a, n) elastic Gaussian : elastic =[3k B T /L contou a] Non-Gaussian : elastic = (k B T/a) L*(/L contou ) low stetches : Gaussian, L*(x)= invese Langevin function = 3x+(9/5)x 3 +(297/175)x 5 +(1539/875)x high stetches : elastic =(k B T/a)(1-/L contou ) -1 Extensible eely-jointed Chain (Smith, et. al, 1996) elastic elastic (a, n, k segment ) Non-Gaussian : elastic = (k B T/a) L*(/L total ) whee : L total = L contou + n elastic /k segment Wom-Like Chain (WLC) (Katky and Pood, 1943 ixman and Kovac, 1973 Bustamante, et. al 1994) elastic (p, n) elastic Exact : Numeical solution Intepolation omula : elastic = (k B T/p)[1/4(1-/L contou ) -2-1/4+/L contou ] low stetches : Gaussian, elastic =[3k B T /2pL contou ] high stetches : elastic = (k B T/4p)(1-/L contou ) -2 Extensible Wom-Like Chain (Odijk, 1995) elastic elastic (p, n, k segment ) Intepolation omula : elastic = (k B T/p)[1/4(1-/L total ) -2-1/4 + /L total ] low stetches : Gaussian high stetches : = L contou [1-0.5(k B T / elastic p) 1/2 + elastic /k segment ] eely Jointed Chain Equations: Gaussian : elastic = [3kBT /Lcontoua] Non-Gaussian : elastic= (kbt/a) L*(/Lcontou) low stetches : Gaussian, L*(x)= invese Langevin function = 3x+(9/5)x^3+(297/175)x^5+(1539/875)x^7+... high stetches : elastic=(kbt/a)(1-/lcontou)-1 Wom-like chain Equations: Exact : Numeical solution Intepolation omula : elastic = (kbt/p)[1/4(1-/lcontou)-2-1/4+/lcontou] low stetches : Gaussian, elastic = [3kBT /2pLcontou] high stetches : elastic = (kbt/4p)(1-/lcontou)-2
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