3.052 Nanomechanics of Materials and Biomaterials

Prof. Christine OrtizDMSE, RM 13-4022

Phone : (617) 452-3084Email : cortiz@mit.edu

WWW : http://web.mit.edu/cortiz/www

LECTURE #18 : ELASTICITY OF SINGLE MACROMOLECULES II

Modifications to the FJC and Experimental Measurements

Review : 3.11The 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 : = number of chain conformations

P(r) = probability function for a given component of length in a fixed direction in space~S(r) = configurational entropy=kBlnP(r)A(r) = Helmholtz free energy =U(r)-TS(r)F(r) = -dA(r)/drk(r) = dF(r)/dr

3. Gaussian Formulas :

P(r) = 4b3r2/exp(-b2r2) where b=[3/2na2]1/2

S(r) = kBln[4b3r2/exp(-b2r2)] A(r) = [3kBT/2na2]r2

F(r) = -[3kBT/na2]rk(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)

FrFelastic Felastic

r1

F1

x

y

z0

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 10 20 30 40 50 60 70 80 90 100

Comparison of Inextensible Non-Gaussian FJC Equations (*large force scale)

For

ce (

nN

)

Distance (nm)

FrFelastic Felastic

F

(a)

(1) Gaussian

(2) Langevin

(4) High Stretch Approx

(3) COTH exact

Comparison of Inextensible Non-Gaussian FJC Equations

(*small force scale)F

orce

(n

N)

-0.05

-0.04

-0.03

-0.02

-0.01

0

0 10 20 30 40 50 60 70

Distance (nm)

FrFelastic Felastic

F

(a)

(1) Gaussian

(2) Langevin

(4) High Stretch Approx

(3) COTH exact

Effect of a and n in FJC

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 50 100 150 200-0.5

-0.4

-0.3

-0.2

-0.1

0

0 100 200 300

(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)

Fel

astic

(nN

)

r (nm)F

elas

tic (n

N)

r (nm)

n=100 n=200 n=300 n=400 n=500

(a) (b)

a = 0.1 nma = 0.2 nma = 0.3 nma = 0.6 nma = 1.2 nma = 3.0 nm

Effect of Statistical Segment Length Effect of Chain Length

Modification of FJC :Extensibility of Chain Segments

FrFelastic Felastic

F

Comparison of Extensible and Inextensible FJC Models

(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)

(a)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 100 200 300 400

Fel

astic

(nN

)

r (nm)

(b)

non-Gaussian

FJC

extensiblenon-

GaussianFJC

FrFelastic Felastic

F

Effect of a and n on Extensible FJC Models

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 50 100 150 200 250 300 350-0.5

-0.4

-0.3

-0.2

-0.1

0

0 100 200 300 400 500

(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)

Fel

astic

(nN

)

r (nm)

n=100 n=200 n=300 n=400 n=500

(a) (b)

Fel

astic

(nN

)r (nm)

a = 0.1 nma = 0.2 nma = 0.3 nma = 0.6 nma = 1.2 nma = 3.0 nm

Effect of Statistical Segment Length Effect of Chain Length

The Worm-Like Chain (WLC)(*Kratky-Porod Model)

r

(lw)1

(lw)n

Review : Elasticity Models for Single Polymer Chains

Freely-Jointed Chain (FJC)(Kuhn and Grün, 1942 James and Guth, 1943)

ExtensibleFreely-Jointed

Chain(Smith, et. al, 1996)

Worm-Like Chain (WLC)(Kratky and Porod, 1943Fixman and Kovac, 1973Bustamante, et. al 1994)

ExtensibleWorm-Like

Chain (Odijk, 1995)

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

Non-Gaussian : Felastic = (kBT/a) L*(r/Ltotal )

where : Ltotal = Lcontour+ nFelastic /ksegment

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

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]

FFrFelastic Felastic

FFrFelastic Felastic

FFr

Felastic Felastic

(a, n)

(a, n, ksegment)

(p, n)

(p, n, ksegment)

MODEL SCHEMATIC FORMULAS

Fr

Felastic Felastic

F

Comparison of FJC and WLC

-1

-0.8

-0.6

-0.4

-0.2

0

0 20 40 60 80 100

(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

FrFelastic

Felastic

F

(a)

(b)

Fel

astic

(nN

)

r (nm)

non-Gaussian FJC

WLC

Force Spectroscopy Experiment on Single Polymer Chains

AFM Image of Isolated, Covalently-BoundSingle Polymer Chains on Gold

(*solvent=toluene)

0.5 m

one PS chain

dodecanethiol monolayer

on gold terrace

edge of gold terrace

HS-[CH2]12-CH3

CHCH2n

HS

-0.1

0

0.1

0.2

0.3

-20 20 60 100 140 180 220

For

ce (

nN

)

Distance (nm)

Typical Force Spectroscopy Experiment on Single Polystyrene Chain

AFM probe tip

substrate

Force Spectroscopy Experiment on a Single Polystyrene Chain :

APPROACH

D (nm)

F (

nN

)

-0.1

0

0.1

0.2

0.3

-20 20 60 100 140 180 220

RF

I.

-0.1

0

0.1

0.2

0.3

-20 20 60 100 140 180 220

Force Spectroscopy Experiment on a Single Polystyrene Chain :

APPROACH

D (nm)

F (

nN

)

Lo2RF

II.

Lo

Force Spectroscopy Experiment on a Single Polystyrene Chain :APPROACH / RETRACT

-0.1

0

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

D (nm)

F (

nN

)

Lo2RF IV.-0.1

0

0.1

0.2

0.3

-20 20 60 100 140 180 220

Lo

Force Spectroscopy Experiment on a Single Polystyrene Chain :

RETRACT

Lo2RF

Lchain

D (nm)

F (

nN

)

V.-0.1

0

0.1

0.2

0.3

-20 20 60 100 140 180 220

LchainLo

Fchain

Force Spectroscopy Experiment on a Single Polystyrene Chain :

RETRACT

D (nm)

F (

nN

)

VI.

-0.1

0

0.1

0.2

0.3

-20 20 60 100 140 180 220

Fchain

Fadsorption

Fbond

•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)