3.04 [Special Problems in Materials Science and Engineering] S00 :
Nanomechanics of Materials and Biomaterials
SAMPLE EXAM #2 : 04/10/00
Force
1. Q. An atomic force spectroscopy experiment is conducted at room temperature on an
individual polymer chain which is attached at the tip and surface by covalent bonds as
shown below. The bond at the tip is the weakest and its interatomic interaction energy
can be approximated by the Lennard-Jones potential with A=10-77 Jm6 and ro=0.15 nm.
approach
0
1 nN
retract
0
Distance
(a) Calculate the distance at which the polymer chain detaches, Ddetach, from the tip if the
chain is modeled as an inextensible worm-like chain with p=5 nm and n=300.
(b) If the cantilever spring constant, k, is 0.01 N/m, how much of the Gaussian regime
data is lost?
(c) Indicate on the above figure how you would obtain the elastic energy of deformation.
How would you increase this energy?
2. Q. The adhesion force was calculated for the nanomechanical experiment shown below
using four different theories and the following results were obtained :
F
Fadhesion (nN)
0
10
13
R=50 nm
K=1 N/m
(a) List the appropriate theory for each simulation.
(b) What is ao (describe in words and/or with a schematic)? Calculate ao (nm) for each
theory.
(c) An experiment is conducted on two loosely crosslinked elastomers and seen to deviate
from all of these theories at high strains, what are some possible explanations for this?
3.04 [Special Problems in Materials Science and Engineering] S00 :
Nanomechanics of Materials and Biomaterials
SAMPLE EXAM #2 SOLUTIONS : 04/10/00
1. A. (a) If we model the polymer as an inextensible, worm-like chain, at high stretches (as
seen in the Figure) the appropriate equation describing the force versus distance curve is :
F=(kBT/4p)(1-(r/Lcontour))-2 (1)
Rearranging eq. (1) and solving for r=D we obtain :
D=Lcontour(1-[kBT/4Fp]1/2) (2)
We can rewrite eq.(2) for the point of chain detachment :
Ddetach=Lcontour(1-[kBT/4Fdetachp]1/2) (3)
Since Lcontour=np=1500 nm, all of the parameters in eq. (3) are known except the force at
detachment, Fdetach. Since all of the bonds in the polymer chain are in series, Fdetach is equal
to the bond rupture force of the weakest bond (i.e. at the tip) :
Fdetach=Frupture
Frupture can be calculated using the equations given on page 9 of Israelachivili and Lecture
Notes #5 :
Frupture=[126A2/169B]/(26B/7A)1/6 (4)
Since A is given, we need to calculate B :
ro=[2B/A]1/6 Þ B=A[ro]6/2 (5)
Substituting A=10-77 Jm6 and ro=0.15 nm into eq. (5) :
B=10-77 Jm6• [0.15•1E-9 (m)] 6/2=5.695•1E-137 Jm12
Then, substituting A=10-77 Jm6 and B=5.695•1E-137 Jm12 into eq. (4) one obtains :
Frupture=7.87 nN=Fdetach
Substituting all of the known values into equation (3), F=Frupture=Fdetach=7.87 nN, p=5nm,
Lcontour=1500 nm, kB= 1.38106•1E-23 J/K, and T=293 K :
Ddetach=499.744 nm (99.9% of Lcontour)
(b) The slope of the cantilever instability line is equal to the cantilever spring constant, k :
k=∆F/∆D (6)
Rearranging eq. (6), one can obtain the distance range of lost data due to surface
adhesion :
∆D=∆F/k
Setting the surface adhesion force, ∆F=Fadhesion=1 nN (which is read off the given
graph) and k=0.01 N/m :
∆D=1 nN/0.01 N/m=100 nm
The Gaussian regime takes place up to approximately :
(1/3)Lcontour=(1/3)1500nm=500 nm
Hence, the first 1/5 of the Gaussian regime is lost due to surface adhesion.
(c) The elastic (Helmholtz) energy of deformation is equal to the area of the force
versus distance curve up until detachment :
Force
A(r)=-òF(r)dr
0
1 nN
0
Distance
One could increase this energy by decreasing the statistical segment length, a, or by
inducing some supramolecular structure in the chain which would cause the curve to
shift to higher forces.
2. A. (a)
Fadhesion(nN)
0
10
13
Theory
Hertz
JKR
DMT
Formula, Fadhesion
Fadhesion=0
Fadhesion=1.5πRW12
Fadhesion=2πRW12
(b) ao is the radius of the contact area at zero force for the compression of two
bodies (e.g. spheres).
Fadhesion(nN)
0
10
13
Theory
Hertz
JKR
DMT
Formula, ao
ao=0
ao=[(3FadhesionR)/K] 1/3
ao=[(FadhesionR)/K] 1/3
ao(m) at F=0
0
1.15*•1E-5
8.76*•1E-6
(c) All of these theories are based on linear elasticity, as we have seen in our treatment
of the FJC and ELC models, polymers such as elastomers exhibit nonlinear elasticity at
high strains.