3.052 Nanomechanics of Materials and Biomaterials Prof. Christine Ortiz DMSE, RM 13-4022 Phone :...
-
date post
21-Dec-2015 -
Category
Documents
-
view
219 -
download
2
Transcript of 3.052 Nanomechanics of Materials and Biomaterials Prof. Christine Ortiz DMSE, RM 13-4022 Phone :...
3.052 Nanomechanics of Materials and Biomaterials
Prof. Christine OrtizDMSE, RM 13-4022
Phone : (617) 452-3084Email : [email protected]
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)