PHYC6250: Topics in Experimental Physics
AFM module Assignment
Kreplak@dal.ca
Part 1: Graphite imaging (5pts)
The scanning tunneling microscope (STM) physical principle is the tunneling of
electrons through a small gap between two conductors. The tunneling current
decreases exponentially with the gap width. In a STM the tunneling current is used
to control the distance between an atomically sharp metal tip and a conductive
surface. By scanning the tip across the surface one can generate a topographic image
at the atomic scale. In this lab you will use a STM to image the surface of graphite.
Structure of graphite:
Use the STM image of graphite provided to answer the following questions using
Gwyddion (http://gwyddion.net/).
a) Using the cristallographic structure of graphite shown on the previous page find
the width of the carbon rings. (Note: the rings are perfect hexagons). (1 pt)
b) Use the fourier transform tool (FFT) in Gwyddion to demonstrate that you are
imaging an hexagonal lattice (Note: the fourier transform of a perfect 2D square
lattice shows four spots forming the corners of a square). (2 pts)
c) Using the cross-sectional tool in Gwyddion, find the size of the black spots on the
image (Note: for a periodic signal, the more periods you count the more precise
the measurement is). Compare your result to the width of the ring that you
obtained in a). Where are the carbon atoms on your image? (2 pts)
Part 2: A cantilever-based force sensor. (5pts)
We are currently building a macroscopic force sensor using a XX mm long Quartz
plate. The strip is clamped at one end and we measure the deflection using an HeNe
laser (633 nm wavelength) spot centered YY mm away from free end. We will use
the plate to measure forces applied at the free end, perpendicular to its length like in
an AFM system. However, instead of using the standard light-lever geometry of
commercial AFM systems, we built a Michelson interferometer to measure the plate
deflection. The interference pattern is a set of bright fringes that move past a fixed
photodetector has the cantilever is being deflected. The light path length difference
between two consecutive fringes is half a laser wavelength.
a) Using results of beam mechanics for a clamped cantilever with a free end, find the
ratio between the deflection measured by the laser interferometer and the
deflection at the free end of the plate? (2pts)
b) To calibrate the spring constant of the plate we use the added mass method
where we measure the frequency of free oscillations of the plate as a function of a
mass added to its free end. Note that the cantilever is vertical and that we have an
inverted T-shape at the free end of the plate to accommodate small pieces of lead of
known mass. See below a table of measured values, find a linear relationship
between a function of frequency and the added mass and use it to fit the data and
estimate the spring constant with error bar. (2 pts)
c) If I apply 1 mN of force at the end of the plate that I calibrated in b), how many
bright fringes will pass by the photodetector? (1pt)
If you are unfamiliar with beam mechanics, a good refresher can be found here:
https://mechanicalc.com/reference/beam-analysis
Part 3: Nanomechanical testing. (4pts)
I want to perform the micromechanical characterization of a smooth 1 mm thick
piece of reinforced rubber with a Young’s modulus of 5 MPa and a Poisson ratio of
0.5. I am using a cantilever with a spring constant of 0.87N/m, frequency 44 kHz, Q
factor is 33, with a pyramidal tip with a radius of curvature of 100 nm.
a) Find a theoretical expression for the force F felt by the cantilever as the tip
penetrates into the material by an amount x smaller than the tip radius. Plot F(x) for
a maximum penetration depth of 50 nm. (1pt)
b) Compare your theoretical result with the VEDA simulation of an approach curve
using the Force distance curve application in the absence of any adhesion force.
Load example 1 and adjust parameters as required including a final z separation of
minus 50 nm. For this imposed z separation how much did the tip indent the
reinforced rubber? (1 pt)
c) Let’s assume that the adhesion force between the tip and the sample is 15 nN, is
the maximum indentation increasing or decreasing compared to the no adhesion
case? (Use VEDA again). Knowing the cantilever spring constant is 0.87N/m can you
predict exactly what the difference of indentation between the two measurements
should be? (2pt)
Part 4: Nanomechanical mapping of a collagen fibril (6 pts)
Background:
Collagen fibrils are linear aggregates of 300 nm long tropocollagen molecules that
have a distinctive triple-helical structure. Tens to hundreds thousand of
tropocollagen molecules are covalently bound to each other by cross-links and
packed radially in a semi-crystalline fashion to form 50 to 500 nm wide fibrils.
There is an intermediate stage of organization between the molecules and the fibril
that is the subfibril with a diameter around 20 nm. Both the individual molecules
and the subfibrils run almost parallel to the fibril axis and are staggered with
respect to each other giving rise to 67 nm wide bands also named D-periods. The
current molecular model of the D-period involves two regions of almost equal
width, one where all the molecules overlap and one where 20% of the molecules are
missing due to an axial gap. This model was proposed on the basis of electron
microscopy images of negatively stained Collagen fibrils and further refined with
small angle X-ray scattering data of rat-tail tendon (see figure below).
We have acquired force curves at a tip velocity of 1.2 mm/s on a grid of 256 by 256
pixels. The grid covers an area of 2 by 2 micrometers squared. The tip had a spring
constant of 0.96 N/m, a sensitivity of 12 nm/V and can be approximated as a cone of
half-angle 18 degrees. The sample is a collagen fibril extracted from a bovine tendon
and absorbed to a silicon rubber, the Poisson’s ratio of both materials can be
approximated to 0.5. The grid was acquired in an aqueous solution (Phosphate
Buffer Saline).
Use the two files provided and the Nanoscope software to answer the following
questions:
a) How much does the tip penetrate the silicon rubber and the fibril in average?
How tall is the fibril? (3pt) (Use the diagram below that shows how the undisturbed
height of the fibril can be calculated using the measure height and the deformations
of the substrate and fibril).
b) What is the maximum force applied to the sample? What is the elastic modulus
for the fibril and the elastomer? (2pts)
c) Use the images to estimate the length of the D-band repeat. We claim that the
elastic modulus measured along the length of the fibril scales with the local
molecular density, can you substantiate this claim? (1pt)