3.052 Nanomechanics of Materials and Biomaterials LECTURE #18 : ELASTICITY OF SINGLE MACROMOLECULES II Modifications to the FJC and Experimental Measurements Prof. Christine Ortiz DMSE, RM 13-4022 Phone : (617) 452-3084 Email : cortiz@mit.edu WWW : http://web.mit.edu/cortiz/www

The Inextensible Freely-Jointed Chain Model Review : 3.11 The Inextensible Freely-Jointed Chain Model 1. Assumptions : (1) random walk : all bond angles are equally probable and uncorrelated to the directions of all other bonds in the chain (2) free rotation at bond junctions (3) no self-interactions or excluded volume effects two parameters : a = “statistical segment length” or local chain stiffness n =number of statistical segments Lcontour = na = fully extended length of chain 2. General Statistical Mechanical Formulas : W = number of chain conformations P(r) = probability function for a given component of length in a fixed direction in space~W S(r) = configurational entropy=kBlnP(r) A(r) = Helmholtz free energy =U(r)-TS(r) F(r) = -dA(r)/dr k(r) = dF(r)/dr 3. Gaussian Formulas : P(r) = 4b3r2/pexp(-b2r2) where b=[3/2na2]1/2 S(r) = kBln[4b3r2/pexp(-b2r2)] A(r) = [3kBT/2na2]r2 F(r) = -[3kBT/na2]r k(r) = 3kBT/na2 (1) 4. Non-Gaussian Formulas : F(r) = kBT/a L*(x) where : x=r/na=“extension ratio” (2) where : L(x)= “Langevin function”=coth(x-1/x) L*(x)= “inverse Langevin function”= 3x+(9/5)x3+(297/175)x5+(1539/875)x7 r(F) = Lcontourcoth(y-1/y) where : y=Fa/kBT (3) low stretches : Gaussian high stretches : F(r) = kBT/a (1-r/Lcontour)-1(4) F r Felastic  r1 F1 x y z

Comparison of Inextensible Non-Gaussian FJC Equations (*large force scale) (a)  F F Felastic r Felastic (1) Gaussian (4) High Stretch Approx Force (nN) (2) Langevin (3) COTH exact Distance (nm)

Comparison of Inextensible Non-Gaussian FJC Equations (*small force scale) (a)  F F Felastic r Felastic (1) Gaussian (4) High Stretch Approx Force (nN) (2) Langevin (3) COTH exact Distance (nm)

Effect of a and n in FJC (a) (b) Felastic (nN) Felastic (nN) r (nm) Effect of Statistical Segment Length Effect of Chain Length a = 0.1 nm a = 0.2 nm a = 0.3 nm a = 0.6 nm a = 1.2 nm a = 3.0 nm Felastic (nN) Felastic (nN) n=100 n=200 n=300 n=400 n=500 r (nm) r (nm) (a) Elastic force versus displacement as a function of the statistical segment length, a, for the non-Gaussian FJC model (Lcontour = 200 nm) and (b) elastic force versus displacement as a function of the number of chain segments, n , for the non-Gaussian FJC model (a = 0.6 nm)

Extensibility of Chain Segments Modification of FJC : Extensibility of Chain Segments  F F Felastic r Felastic

Comparison of Extensible and Inextensible FJC Models (a)  F F Felastic r Felastic -0.5 -0.4 -0.3 -0.2 -0.1 100 200 300 400 (a) Schematic of the stretching of an extensible freely jointed chain and (b) the elastic force versus displacement for the extensible compared to non-extensible non-Gaussian FJC (a = 0.6 nm, n = 100, ksegment = 1 N/m) extensible non- Gaussian FJC (b) Felastic (nN) non- Gaussian FJC r (nm)

Effect of a and n on Extensible FJC Models Effect of Statistical Segment Length Effect of Chain Length Felastic (nN) a = 0.1 nm a = 0.2 nm a = 0.3 nm a = 0.6 nm a = 1.2 nm a = 3.0 nm Felastic (nN) n=100 n=200 n=300 n=400 n=500 r (nm) r (nm) (a) Elastic force versus displacement for the extensible non-Gaussian FJC as a function of the statistical segment length, a (Lcontour= 200, ksegment = 2.4 N/m) and (b) the elastic force versus displacement for the extensible non-Gaussian FJC as a function of the number of chain segments, n (a = 0.6 nm, ksegment = 1 N/m)

The Worm-Like Chain (WLC) (*Kratky-Porod Model)  g (lw)1 (lw)n r

Review : Elasticity Models for Single Polymer Chains SCHEMATIC FORMULAS Gaussian : Felastic = [3kBT /Lcontoura] r Non-Gaussian : Felastic= (kBT/a) L*(r/Lcontour) low stretches : Gaussian, L*(x)= “inverse Langevin function”= 3x+(9/5)x3+(297/175)x5+(1539/875)x7+... high stretches : Felastic=(kBT/a)(1-r/Lcontour)-1 Freely-Jointed Chain (FJC) (Kuhn and Grün, 1942 James and Guth, 1943) F  F r Felastic Felastic (a, n) Extensible Freely-Jointed Chain (Smith, et. al, 1996)  F F Non-Gaussian : Felastic = (kBT/a) L*(r/Ltotal ) where : Ltotal = Lcontour+ nFelastic /ksegment r Felastic Felastic (a, n, ksegment) Worm-Like Chain (WLC) (Kratky and Porod, 1943 Fixman and Kovac, 1973 Bustamante, et. al 1994) Exact : Numerical solution Interpolation Formula : Felastic = (kBT/p)[1/4(1-r/Lcontour)-2-1/4+r/Lcontour] low stretches : Gaussian, Felastic = [3kBT /2pLcontour] r high stretches : Felastic = (kBT/4p)(1-r/Lcontour)-2 F  F Felastic r Felastic (p, n) Extensible Worm-Like Chain (Odijk, 1995) F  F Interpolation Formula : Felastic = (kBT/p)[1/4(1-r/Ltotal)-2 -1/4 + r/Ltotal] low stretches : Gaussian high stretches : r = Lcontour [1-0.5(kBT /Felasticp)1/2 + Felastic/ksegment] r Felastic Felastic (p, n, ksegment)

Comparison of FJC and WLC  F F Felastic r Felastic (b) (a) Schematic of the extension of a worm-like chain and (b) the elastic force versus displacement for the worm-like chain model compared to inextensible non-Gaussian FJC models non-Gaussian FJC Felastic (nN) WLC r (nm)

Force Spectroscopy Experiment on Single Polymer Chains

AFM Image of Isolated, Covalently-Bound Single Polymer Chains on Gold (*solvent=toluene) HS-[CH2]12-CH3 0.5 mm dodecanethiol monolayer on gold terrace edge of gold terrace CH CH2 n HS one PS chain

Typical Force Spectroscopy Experiment on Single Polystyrene Chain AFM probe tip substrate Force (nN) Distance (nm)

Force Spectroscopy Experiment on a Single Polystyrene Chain : APPROACH RF 0.3 0.2 F (nN) 0.1 I. -0.1 -20 20 60 100 140 180 220 D (nm)

Force Spectroscopy Experiment on a Single Polystyrene Chain : APPROACH Lo 0.3 0.2 II. F (nN) 0.1 Lo2RF -0.1 -20 20 60 100 140 180 220 D (nm)

Force Spectroscopy Experiment on a Single Polystyrene Chain : APPROACH / RETRACT -0.1 0.1 0.2 0.3 -20 20 60 100 140 180 220 D (nm) F (nN) III.

Force Spectroscopy Experiment on a Single Polystyrene Chain : RETRACT Lo 0.3 0.2 F (nN) 0.1 Lo2RF IV. -0.1 -20 20 60 100 140 180 220 D (nm)

Force Spectroscopy Experiment on a Single Polystyrene Chain : RETRACT Lo Lchain 0.3 0.2 Fchain F (nN) Lchain 0.1 Lo2RF V. -0.1 -20 20 60 100 140 180 220 D (nm)

Force Spectroscopy Experiment on a Single Polystyrene Chain : RETRACT 0.3 Fbond VI. Fchain 0.2 F (nN) 0.1 Fadsorption -0.1 -20 20 60 100 140 180 220 D (nm) •Since Fadsorption<< Fbond (AU-S) = 2-3 nN* chain always desorbs from tip (*based on Morse potential using Eb(AU-S)=170 kJ/mol; Ulman, A. Chem. Rev. 1996, 96, 1553)