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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 : [email protected] WWW : http://web.mit.edu/cortiz/www.

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slide1

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 : [email protected]

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

slide2

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

slide3

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)

slide4

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)

slide5

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)

slide6

Modification of FJC :

Extensibility of Chain Segments

F

F

r

Felastic

Felastic

slide7

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)

slide8

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)

slide9

The Worm-Like Chain (WLC)

(*Kratky-Porod Model)

g

(lw)1

(lw)n

r

slide10

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)

slide11

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)

slide13

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

slide14

Typical Force Spectroscopy Experiment

on Single Polystyrene Chain

AFM

probe tip

substrate

Force (nN)

Distance (nm)

slide15

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)

slide16

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)

slide17

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)

slide18

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)

slide19

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)

slide20

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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