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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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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 : [email protected]

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


3 052 nanomechanics of materials and biomaterials

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


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

Modification of FJC :

Extensibility of Chain Segments

F

F

r

Felastic

Felastic


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

The Worm-Like Chain (WLC)

(*Kratky-Porod Model)

g

(lw)1

(lw)n

r


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

Force Spectroscopy Experiment on

Single Polymer Chains


3 052 nanomechanics of materials and biomaterials

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


3 052 nanomechanics of materials and biomaterials

Typical Force Spectroscopy Experiment

on Single Polystyrene Chain

AFM

probe tip

substrate

Force (nN)

Distance (nm)


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

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)


3 052 nanomechanics of materials and biomaterials

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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