3.052 Nanomechanics of

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
[phone] Email cortiz_at_mit.edu WWW
http//web.mit.edu/cortiz/www
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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 spaceW S(r)
configurational entropykBlnP(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
b3/2na21/2 S(r) kBln4b3r2/?pexp(-b2r2)
A(r) 3kBT/2na2r2 F(r) -3kBT/na2r k(r)
3kBT/na2 (1) 4. Non-Gaussian Formulas
F(r) kBT/a L(x) where xr/naextension
ratio (2) where L(x) Langevin
functioncoth(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 yFa/kBT
(3) low stretches Gaussian high 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)
n100
n200
n300
n400
n500
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)
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Modification of FJC Extensibility of Chain
Segments
?
F
F
r
Felastic
Felastic
7
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)
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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)
n100
n200
n300
n400
n500
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)
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The Worm-Like Chain (WLC) (Kratky-Porod Model)

?
g
(lw)1
(lw)n
r
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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)x
5(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-
1/4r/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)
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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)
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Force Spectroscopy Experiment on Single Polymer
Chains
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AFM Image of Isolated, Covalently-Bound Single
Polymer Chains on Gold (solventtoluene)
HS-CH212-CH3
0.5 mm
dodecanethiol monolayer on gold terrace
edge of gold terrace
one PS chain
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Typical Force Spectroscopy Experiment on Single
Polystyrene Chain
AFM probe tip
substrate
Force (nN)
Distance (nm)
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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)
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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)
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Force Spectroscopy Experiment on a Single
Polystyrene Chain APPROACH / RETRACT
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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)
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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 Fadsorptionltlt 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)