PHY-606 / Soft Matter – Concepts and Methods
Problem Set for Practice
Colloids
1. Consider a dispersion of ideal hard rods of length L. Answer the following questions in
order to calculate the depletion interaction between two hard plates immersed in that
dispersion.
(a) Considering random orientations of the rods, calculate and
plot the density of the rods n(x) in terms of the concentration
at the bulk, nb.
(b) Calculate the osmotic pressure in-between the plates, Pin(h),
as a function of the separation, h. (consider P = nkBT)
(c) Calculate the depletion attraction force per unit area, F(h) =
(Pin (h) – Pout), where Pout denotes the pressure outside and is
given by Pout = nbkBT.
2. Considering the figure given below, derive the depletion force and
potential between a sphere and a flat surface / plate, 𝐾𝑠𝑃 (ℎ) and
𝑊𝑠𝑃 (ℎ), where h is the separation between the plate and the
colloidal sphere.
Simplify the results for 𝑅 ≫ 𝜎.
3. If the van der Waals interaction between two volume elements
𝐶
at a separation r is given by 𝑈(𝑟) = ∫ ∫ − 𝑟6 𝜌1 𝑑𝑉1 𝜌2 𝑑𝑉2,
derive the interaction between a sphere of radius R and a semiinfinite medium of same material of density 𝜌.
Page |1
PHY-606 / Soft Matter – Concepts and Methods
4. Starting from the definition of pair correlation function
(also called as radial distribution function), g(r), address
the following.
(a) Calculate the position of first three peaks in g(r) for a
hexagonal colloidal crystal in 2D. The lattice constant
and the particle diameter are a and d respectively.
(b) Further calculate the relative peak heights of those
three peaks.
(c) As per definition, one needs ensemble averaging to
calculate g(r). Discuss how you did the ensemble averaging in your calculation.
Page |2