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Homework 4
Material Surfaces
1
The 2007 Nobel prize in chemistry was awarded to Gerhard Ertl for his investigations of
fundamental processes at gas-solid interfaces. Google ”chemical processes on solid surfaces
Nobel”, dowload the commentary on Ertl’s work produced by the Nobel committee and
create a ≈ 250 word essay highlighting the main points in the commentary.
2
The attached Excel worksheet contains raw measured asperity elevation data as a function
of distance along the surface for a certain material surface (values in micrometers).
a) Compute the mean asperity height, adopt the computed value as the reference asperity
height and create a new column of transformed data showing the difference between the local
asperity height and the reference value.
b) Use the data to compute the values of Ra , and Rq .
c) Use the data to construct a histogram. Check that the data is approximately normally
distributed and find a closed form expression for a good fit Gaussian. Compute then the
mean µ, variance σ, Skewness Sk and Kurtosis K. Comment on your results.
d) Compute values of the autocorrelation function and discuss.
e) Compute the bearing ratio tp for various preselected asperity heights h∗ and also the
bearing area curve for the surface. Comment on your results.
Answer:
See Excel and MINITAB worksheets.
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3
The contact angle of water on platinum has been measured to be θ = 40 degrees, while that
on paraffin is θ = 110 degrees. Estimate the work of adhesion of water on platinum and
water on paraffin and comment.
Answer:
Dupre’s equation is
WSL = γLV (1 + cos θ)
where WSL is the work of adhesion of liquid to solid and γLV is the surface tension liquid-air
(γLV = 0.0728 J/m2 .
Therefore for water over platinum WSL = 0.128 J/m2 .
And for water over paraffin WSL = 0.047 J/m2 .
Thus, the adhesion of water to platinum is almost three times stronger than to paraffin.
4
A cylindrical water bridge (radius r = 0.01 m, thickness h = 0.0001 m) forms between two
flat platinum surfaces. Estimate the adhesive force due to the bridge.
Answer:
The calculated adhesive force is F = 0.35 N.
5
Carry out calculations to simulate the combined effect of a flat punch and a Hertzian pressure
distribution. Compare against the results obtained for the Hertzian pressure alone.
Answer:
See Maple worksheets and COMSOL program.
6
The surface of a sample of Al-7075 alloy has been investigated using profilometry. The
standard deviation of the asperity heights was measured as s ≈ 1.4µm and the average tip
2
radius of curvature was R = 5µm. Compute the value of the plasticity index ψ and show that
the asperities are expected to deform plastically. How small would the standard deviation
of asperity heights would have to be in order for the asperities to deform elastically?
Answer:
The plasticity index is defined by
ψ=
E σ 1/2
( )
H R
For the given alloy, E = 5 × 1010 , σY = 500 MPa and H = 3σY = 1500 × 106 Pa, so that
ψ = 17.63, therefore the asperities will undergo plastic deformation.
To get ψ = 1, the standard deviation of asperity heights must not exceed
R(
H 2
) = 5(0.15/5)2 = 0.0045
E
micrometers.
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