3.052 Nanomechanics of Materials and Biomaterials

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

2. Random Coil Configuration of Polymers Less Disorder More Disorder Entropy - a natural law that expresses the driving force towards disorder poly(styrene) <r2>1/2

3. Random Coil Configuration of Polymers simulation DNA (*FEBS Lett. 371:279-282) DNA (*Z. Shao, http://www.people.Virginia.EDU/~js6s/zsfig/figureindex.html)

4. The Freely-Jointed Chain (FJC) Model real polymer chain random walk statistical representation of real polymer chain   a3 a2 an-2 a1 an-1 r an

5. The Freely-Jointed Chain (FJC) Model : Entropic Elasticity consider stretching a single random coil polymer chain :  F F r Fchain Fchain (*http://align.physik.tu-berlin.de/~ronald/zib.html)

6. z F1 r1 y x 0 Review : 3.11 : Formulas For Gaussian FJC 1. General Statistical Mechanical Formulas : 2. Gaussian Formulas For Stretching a Single FJC : Linear Elasticity(1)

7. z F1 r1 y x Review : 3.11 : Formulas For NonGaussian FJC 3. Non Gaussian Formulas For Stretching a Single FJC Chain :

8. Comparison of Inextensible Non-Gaussian FJC Equations (*large force scale) (a)  F F r Felastic Felastic Force (nN) Distance (nm)

9. Comparison of Inextensible Non-Gaussian FJC Equations (*small force scale) (a)  F F r Felastic Felastic Force (nN) Distance (nm)

10. 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)

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

12. 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)

13. 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)