Problems 50 pts.
Electrostatics 10 pts.
1. Compare the linearized and full Poisson-Boltzmann equations by plotting the electrical potential (ψ) versus distance (x) for surface potentials (ψo = 50, 100, 150 mV) for an aqueous solution of 2 mM KCl. Prepare both dimensional and dimensionless graphs (i.e. scale the potential (ψ) by the surface potential (ψο) and the distance (x) by the Debeye length (κ)
Electrokinetics 20 pts.
1. A glass capillary has a radius (R=10 μm) and length (L=5 cm). The zeta (ζ) potential of the glass in 0.01 M KCl at neutral pH is (ζ= -30 mV). An electromotive force (V= 5 Volts) is applied across the capillary. How fast and in what direction does the fluid move? What is the volumetric flow rate of fluid in the capillary?
2. Consider the flow field described above. Small spherical polystryrene particles (R=50 nm) are now added to the fluid as tracers. The charged sulfate groups on the surface of the particles generate a ζ-potential on the surface of the particles (ζ= -20 mV). How fast and in what direction do these particles flow? Good markers travel at the same velocity as the fluid. Are these good markers?
Surface Forces 20 pts.
1. Using the Derjaguin approximation (Butt (6) 93), calculate the total van der Waals force between a parabolic (AFM) probe tip and a planar surface separated by a distance, D.
The probe is axisymmetric and has a radius of curvature, R, at the tip. Hint: Use A(x) for the geometry given and integrate the following:
F(D) = ∫∞ f (x) d A(x) dx Ddx
where f(x) is the force per unit area between two planar surfaces separated by distance, x.
You can find f(x) by differentiating the appropriate potential expression; i.e. van der Waals between two solid bodies.
2. So called “hydrophobic forces” are often assigned an exponential spatial dependence,
f (x) = fo exp(−κx)
Calculate the total “hydrophobic” force between a parabolic (AFM) probe and planar surface.
