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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here.
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This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby
here. Click here for the most recent version of the errata for the print version.
[Return to Table of Contents]
Jump To:
[Kinematics]
[Couette/Poiseuille Flow]
[Fluid Circuits]
[Mixing]
[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
[Separations]
[Particle Electrophoresis]
[DNA]
[Nanofluidics]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
 Write a MATLAB routine to solve the OrnsteinZernike equation with hypernetted chain closure to
find the radial distribution function for a homogeneous LennardJones fluid.
Proceed as follows:
 Use an iterative technique that, in turn, uses the hypernetted chain closure (Equation H.20) to
solve for f_{tc} and the OrnsteinZernike equation (Equation H.19) to solve for f_{dc}.
 Start by setting f_{tc}(r) = f_{dc}(r) = 0 on a domain that ranges from r = 0 to r = 512σ.
 In each step, define a new f_{tc} by using the hypernetted chain relation:
 (H.32) 
Note that e_{1}(r) in this case is the LennardJones potential.
 In each step, define a new f_{dc} by Fourier transforming f_{tc} and f_{dc}, applying the Fouriertransformed
OrnsteinZernike equation to get a new , and inverse Fouriertransforming to get a new f_{dc}. We
do this because the Fouriertransformed OrnsteinZernike equation is much easier to deal with (the
spatial integral becomes a product when Fouriertransformed):
 (H.33) 
Here k is the frequency variable and is the Fourier transform of f. This can be rearranged to
give
 (H.34) 
So we Fouriertransform f_{tc} and f_{dc} to get and , apply Equation H.34, and then transform
back.
 The two previous steps are repeated until the solutions for f_{tc} and f_{dc} are no longer changing.
Some attention to numerical stability is needed, especially if ρ^{*} is high and T ^{*} is low.
Plot your results for nine cases as follows: three values of ρ^{*} (0.1, 0.4, 0.8) and three values of T ^{*} (0.5, 1.0,
1.5).
 Given the results from Exercise H.1, calculate the potential of mean force that atoms see in this case.
Plot the e_{mf} for the nine cases from Exercise H.1.

Calculate the magnitude of the dipole moment (in Debye) for a water molecule given the geometry of
the SPC model.
 Consider hard spheres of radius a.
 What is the closest approach of the centers of two spheres?
 What is the doublet potential for the interaction between the two spheres?
 Calculate the excluded volume. How does the excluded volume compare with the volume of one
of the spheres?
 Show that the nonintegrability of the Coulomb pair potential guarantees that physical systems must be overall
electroneutral.
 Using the equation for monopole interaction potentials, explain why sodium chloride might be expected to be
a crystalline solid when dry but dissolves when exposed to water.
[Return to Table of Contents]
Jump To:
[Kinematics]
[Couette/Poiseuille Flow]
[Fluid Circuits]
[Mixing]
[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
[Separations]
[Particle Electrophoresis]
[DNA]
[Nanofluidics]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections.
This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.
This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby
here. Click here for the most recent version of the errata for the print version.
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