3.052 Nanomechanics of Materials and Biomaterials
13
3.052 Nanomechanics of Materials and Biomaterials Prof. Christine Ortiz DMSE, RM 13-4022 Phone : (617) 452-3084 Email : [email protected] WWW : http://web.mit.edu/cortiz/www LECTURE #17 : ELASTICITY OF SINGLE MACROMOLECULES : The Freely-Jointed Chain (FJC) Model
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3.052 Nanomechanics of Materials and Biomaterials. LECTURE #17 : ELASTICITY OF SINGLE MACROMOLECULES : The Freely-Jointed Chain (FJC) Model. Prof. Christine Ortiz DMSE, RM 13-4022 Phone : (617) 452-3084 Email : [email protected] WWW : http://web.mit.edu/cortiz/www. - PowerPoint PPT Presentation
Transcript of 3.052 Nanomechanics of Materials and Biomaterials
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 #17 : ELASTICITY OF SINGLE MACROMOLECULES :
The Freely-Jointed Chain (FJC) Model
Random Coil Configuration of Polymers
More Disorder Less Disorder
Entropy - a natural law that expresses the driving force towards disorder
<r2>1/2
poly(styrene)
DNA simulation
(*FEBS Lett. 371:279-282)
(*Z. Shao, http://www.people.Virginia.EDU/~js6s/zsfig/figureindex.html)
Random Coil Configuration of Polymers
DNA
The Freely-Jointed Chain (FJC) Model
a1
ran
a2
a3
an-1
an-2
real polymer chain
random walk statistical
representationof real polymer chain
The Freely-Jointed Chain (FJC) Model :
Entropic Elasticity
FrFchain
Fchain
F
consider stretching a single random coil polymer chain :
(*http://align.physik.tu-berlin.de/~ronald/zib.html)
Review : 3.11 : Formulas For Gaussian FJC1. General Statistical Mechanical Formulas :
2. Gaussian Formulas For Stretching a Single FJC :
r1
F1
x
y
z
B
= number of chain conformations
P(r) = probability of finding a free chain end a radial distance, r, away from fixed chain end (origin) ~ Ω
S(r) = configurational entropy = k lnP(r)
A(r) = Helmholtz free ener
B
2
2
gy = U(r) - TS(r) = - Tk lnP(r)
-dA(r)f(r) = entropic elastic force =
dr
dF(r) -d A(r)k(r) = (global) entropic chain stiffness = =
dr dr
0
3 22 2
2
3 22 2
B
2 2B B2
c
B B2
c
B B2
c
4b r 3P(r) exp[ b r ] where b =
2na
4b rS(r) = k ln exp[ b r ]
3k T 3k TA(r) = r = r
2L a2na
3k T 3k Tf(r) = - r = r
L ana
3k T 3k Tk(r) = = cons tan t
L ana
Linear Elasticity(1)
Review : 3.11 : Formulas For NonGaussian FJC
3. Non Gaussian Formulas For Stretching a Single FJC Chain :
r1
F1
x
y
z
contour
3 5
1 a ( ) coth where: x=
r r( ) *( ) where : x= 3
na L
9 297 1539L*(x) = "inverse Langevin function" = 3x+
5 175 875
B
B
fr f x (2)
x k T
k Tf r L x ( )
a
x x
Exact Formula :
Langevin Expansion :
7
1
...
( ) 1B
contour
x
k T rf r
a L
High Stretch Approximation : ( 4)
-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)
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)
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
