Soft Condensed Matter

1) Calculate the van der Waal interaction potential between two spheres of radius a, considering that the attractive potential between two neutral atoms is inversely proportional to the sixth power of distance, U = -C/r^2, where C is a constant and r is the distance between the atoms.

2) Charged colloidal particles are suspended in solution containing salt. Derive the Poisson-Boltzmann equation that describes the distribution of ions and electric potential around each particle. Discuss the linearised (Debye-Heckel) form of the equation and give the expression of the Debye screening length for the charged surface.

3) Two equally charged plates are separated by a distance d in a solvent containing counter ions. Express the counter ions density n(x) at the plates as a function of the counter ions density at the mid plane.

4) Within Flory's approach, the dependence of the free energy F of real polymers on their end-to-end distance R and on the number of monomers N is:

F(R,N) = 3KbTR^2/ 2Nb^2 + kbTb^3(1-2X) N^2/R^3

Here kb is the Boltzmann's constant, T the temperature, b the linear size of the monomer, and x Flory's interaction parameter. Determine how the force required to stretch a polymer scales with its end-to-end distance R.