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
Phone : (617)
WWW :

2. 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

3. 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)

4. 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)

5. 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)

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

7. 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)

8. 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)

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

10. 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)

11. 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)

12. Force Spectroscopy Experiment on
Single Polymer Chains

13. 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

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.
-0.1
-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
-0.1
-20 20 60
100
140
180
220
D (nm)

17. 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.

18. 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)

19. 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)

20. 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)