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3.052 Nanomechanics of Materials and Biomaterials

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.

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3.052 Nanomechanics of Materials and Biomaterials

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  1. 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

  2. Review : 3.11 The Inextensible Freely-Jointed Chain Model z  F r F1 Felastic Felastic r1 y x 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 : Gaussianhigh stretches : F(r) = kBT/a (1-r/Lcontour)-1(4) 0

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

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

  5. Effect of a and n in FJC (a) (b) 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)

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

  7. 0 -0.1 -0.2 -0.3 -0.4 -0.5 0 100 200 300 400 Comparison of Extensible and Inextensible FJC Models (a)  F F r Felastic Felastic (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)

  8. Effect of a and n on Extensible FJC Models (a) (b) 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)

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

  10. Review : Elasticity Models for Single Polymer Chains MODEL 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  r Felastic 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)

  11. Comparison of FJC and WLC (a)  F F r Felastic 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)

  12. Force Spectroscopy Experiment on Single Polymer Chains

  13. HS CH2 CH n 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 one PS chain

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

  15. Force Spectroscopy Experiment on a Single Polystyrene Chain : APPROACH RF 0.3 0.2 F (nN) 0.1 I. 0 -0.1 -20 20 60 100 140 180 220 D (nm)

  16. Force Spectroscopy Experiment on a Single Polystyrene Chain : APPROACH Lo 0.3 0.2 II. F (nN) 0.1 Lo2RF 0 -0.1 -20 20 60 100 140 180 220 D (nm)

  17. Force Spectroscopy Experiment on a Single Polystyrene Chain : APPROACH / RETRACT 0.3 III. 0.2 F (nN) 0.1 0 -0.1 -20 20 60 100 140 180 220 D (nm)

  18. Force Spectroscopy Experiment on a Single Polystyrene Chain : RETRACT Lo 0.3 0.2 F (nN) 0.1 Lo2RF 0 IV. -0.1 -20 20 60 100 140 180 220 D (nm)

  19. Force Spectroscopy Experiment on a Single Polystyrene Chain : RETRACT Lo Lchain 0.3 0.2 Fchain F (nN) 0.1 Lchain 0 Lo2RF V. -0.1 -20 20 60 100 140 180 220 D (nm)

  20. Force Spectroscopy Experiment on a Single Polystyrene Chain : RETRACT 0.3 Fbond VI. 0.2 Fchain F (nN) 0.1 0 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)

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